Mixed-norm inequalities and
operator space Lp embedding theory
Marius Junge
Department of Mathematics
University of Illinois at Urbana-Champaign
273 Altgeld Hall, 1409 W. Green Street. Urbana, IL 61801. USA
[email protected]
and
Javier Parcet
Departamento de Matemáticas
Instituto de Matemáticas y Fı́sica Fundamental
Consejo Superior de Investigaciones Cientı́ficas
C/ Serrano 121. 28006, Madrid. Spain
[email protected]
Abstract
Let f1 , f2 , . . . , fn be a family of independent copies of a given random variable
f in a probability space (Ω, F, µ). Then, the following equivalence of norms holds
whenever 1 ≤ q ≤ p < ∞
½ ³Z
n
³Z hX
i p ´ p1
´ r1 ¾
1
q q
r
r
(Σpq )
|fk |
dµ
∼ max n
|f | dµ
.
Ω
k=1
r∈{p,q}
Ω
We prove a noncommutative analogue of this inequality for sums of free random
variables over a given von Neumann subalgebra. This formulation leads to new
classes of noncommutative function spaces which appear in quantum probability as
square functions, conditioned square functions and maximal functions. Our main
tools are Rosenthal type inequalities for free random variables, noncommutative
martingale theory and factorization of operator-valued analytic functions. This
allows us to generalize (Σpq ) as a result for noncommutative Lp in the category
of operator spaces. Moreover, the use of free random variables produces the right
formulation of (Σ∞q ), which has not a commutative counterpart. In the last part of
the paper, we use our mixed-norm inequalities to construct a completely isomorphic
embedding of Lq –equipped with its natural operator space structure– into some
sufficiently large Lp space for 1 ≤ p < q ≤ 2. The construction of such embedding
has been open for quite some time. We also show that hyperfiniteness and the
QWEP are preserved in our construction.
Junge is partially supported by the NSF DMS-0301116.
Parcet is partially supported by ‘Programa Ramón y Cajal, 2005’
and also by Grants MTM2007-60952 and CCG06-UAM/ESP-0286, Spain.
2000 Mathematics Subject Classification: 46L07, 46L09, 46L51, 46L52, 46L53, 46L54.
Contents
Introduction
1
Chapter 1. Noncommutative integration
1.1. Noncommutative Lp spaces
1.2. Pisier’s vector-valued Lp spaces
1.3. The spaces Lrp (M, E) and Lcp (M, E)
13
13
17
20
Chapter 2. Amalgamated Lp spaces
2.1. Haagerup’s construction
2.2. Triangle inequality on ∂∞ K
2.3. A metric structure on the solid K
27
29
31
38
Chapter 3. An interpolation theorem
3.1. Finite von Neumann algebras
3.2. Conditional expectations on ∂∞ K
3.3. General von Neumann algebras I
3.4. General von Neumann algebras II
3.5. Proof of the main interpolation theorem
43
44
48
55
61
66
Chapter 4. Conditional Lp spaces
4.1. Duality
4.2. Conditional L∞ spaces
4.3. Interpolation results and applications
71
72
73
74
Chapter 5. Intersections of Lp spaces
5.1. Free Rosenthal inequalities
5.2. Estimates for BMO type norms
5.3. Interpolation of 2-term intersections
5.4. Interpolation of 4-term intersections
79
79
83
99
103
n
Chapter 6. Factorization of Jp,q
(M, E)
6.1. Amalgamated tensors
6.2. Conditional expectations and ultraproducts
n
6.3. Factorization of the space J∞,1
(M, E)
107
108
112
115
Chapter 7. Mixed-norm inequalities
n
7.1. Embedding of Jp,q
(M, E) into Lp (A; `nq )
7.2. Asymmetric Lp spaces and noncommutative (Σpq )
119
119
126
Chapter 8. Operator space Lp embeddings
8.1. Embedding Schatten classes
8.2. Embedding into the hyperfinite factor
8.3. Embedding for general von Neumann algebras
129
129
132
144
Bibliography
155
Introduction
Probabilistic methods play an important role in harmonic analysis and Banach
space theory. Let us just mention the relevance of sums of independent random
variables, p-stable processes or martingale inequalities in both fields. The analysis
of subspaces of the classical Lp spaces is specially benefited from such probabilistic
notions. Viceversa, Burkholder’s martingale inequality for the conditional square
function has been discovered in view of Rosenthal’s inequality for the norm in
Lp of sums of independent random variables. This is only one example of the
fruitful interplay between harmonic analysis, probability theory and Banach space
geometry carried out mostly in the 70’s by Burkholder, Gundy, Kwapień, Maurey,
Pisier, Rosenthal and many others.
More recently it became clear that a similar endeavor for noncommutative Lp
spaces requires an additional insight from quantum probability and operator space
theory [15, 17, 50]. A noncommutative theory of martingale inequalities finds its
beginnings in the work of Lust-Piquard [36] and Lust-Piquard/Pisier [37] on the
noncommutative Khintchine inequality. The seminal paper of Pisier and Xu on the
noncommutative analogue of Burkholder-Gundy inequality [51] started a new trend
in quantum probability. Nowadays, most classical martingale inequalities have a
satisfactory noncommutative analogue, see [16, 28, 41, 53]. In the proof of these
results the classical stopping time arguments are no longer available, essentially
because point sets disappear after quantization. These arguments are replaced by
functional analytic or combinatorial arguments. In the functional analytic approach
we often encounter new spaces. Indeed, maximal functions in the noncommutative
context can only be understood and defined through analogy with vector-valued Lp
spaces. A careful analysis of these spaces is crucial in establishing basic results such
as Doob’s inequality [16] for noncommutative martingales and the noncommutative
maximal theorem behind Birkhoff’s ergodic theorem [29]. The proof of maximal
theorems and noncommutative versions of Rosenthal’s inequality often uses square
function and conditioned square function estimates, see [26] and the references
therein. These are examples of more general classes of noncommutative function
spaces to be defined below. However, all of them illustrate our main motto in
this paper. Namely, certain problems can be solved by finding and analyzing the
appropriate class of Banach spaces. We shall develop in this paper a new theory
of generalized noncommutative Lp spaces with three problems in mind for a given
von Neumann algebra A.
Problem 1. Calculate the Lp (A; `q ) norm for sums of free random variables.
Problem 2. If 1 ≤ p < q ≤ 2, find a cb-embedding of Lq (A) into some Lp space.
Problem 3. Any reflexive subspace of A∗ embeds into some Lp for certain p > 1.
1
2
INTRODUCTION
Our main contribution in this paper is the calculation of mixed norms for sums
of free random variables and its application to construct a complete embedding
of Lq into Lp . Unfortunately, the generalization of Rosenthal’s theorem [57] to
noncommutative Lp spaces –whose simplest version is the content of Problem 3
above– is beyond the scope of this paper and we analyze it in [23]. We should
nevertheless note that its solution is also deeply related to the main results in this
paper. Namely, the interplay of interpolation and intersection is at the heart of both
results. On the other hand, operator space Lp embedding theory is motivated by
the classical notion of q-stable variables and norm estimates for sums of independent
random variables. Let us briefly explain this. The construction of the cb-embedding
for (p, q) = (1, 2) was obtained in [17]. The simplest model of 2-stable variables
is provided by normalized gaussians (gk ). In this particular case and after taking
operator coefficients (ak ) in some noncommutative L1 space, the noncommutative
Khintchine inequality [37] tells us that
°X
°
°¡ X
°¡ X
¢1 °
¢1 °
°
°
°
°
°
°
ak gk ° ∼
inf °
rk rk∗ 2 ° + °
(1)
E°
c∗k ck 2 ° .
k
1
ak =rk +ck
k
k
1
1
Let us point out that operator spaces are a very appropriate framework for analyzing
noncommutative Lp spaces and linear maps between them. Indeed, inequality (1)
describes the operator space structure of the subspace spanned by the gk ’s in L1 as
the sum R + C of row and column subspaces of B(`2 ). We refer to [11] and [47] for
background information on operator spaces. In the language of noncommutative
probability many operator space inequalities translate into module valued versions
of scalar inequalities, this will be further explained below. The only drawback of
inequality (1) is that it does not coincide with Pisier’s definition of the operator
space `2
(2)
°X
°
°
°
ak ⊗ δk °
°
k
L1 (M;`2 )
=
inf
ak =αγk β
kαkL4 (M)
³X
k
kγk k2L2 (M)
´ 12
kβkL4 (M) .
However, it was proved by Pisier that the right side in (2) is obtained by complex
interpolation between the row and the column square functions appearing on the
right of (1). One of the main results in this article is a far reaching generalization
of this observation. In fact, the solution of Problem 2 in full generality is closely
related to this analysis.
Following our guideline we will now introduce and discuss the new class of
spaces relevant for these problems and martingale theory. These generalize Pisier’s
theory of Lp (Lq ) spaces over hyperfinite von Neumann algebras. We begin with a
brief review of some noncommutative function spaces which have lately appeared in
the literature, mainly in noncommutative martingale theory. We refer to Chapter
1 below for a more detailed exposition.
1. Noncommutative function spaces.
Inspired by Pisier’s theory [46], several noncommutative function spaces have
been recently introduced in quantum probability. The first motivation comes from
some of Pisier’s fundamental equalities, which we briefly review. Let N1 and N2
be two hyperfinite von Neumann algebras. Then, given 1 ≤ p, q ≤ ∞ and defining
1/r = |1/p − 1/q|, we have
INTRODUCTION
3
¡
¢
i) If p ≤ q, the norm of x ∈ Lp N1 ; Lq (N2 ) is given by
o
n
¯
inf kαkL2r (N1 ) kykLq (N1 ⊗N
¯ 2 ) kβkL2r (N1 ) ¯ x = αyβ .
¡
¢
ii) If p ≥ q, the norm of x ∈ Lp N1 ; Lq (N2 ) is given by
n
o
¯
sup kαxβkLq (N1 ⊗N
¯ 2 ) ¯ α, β ∈ BL2r (N1 ) .
On the other hand, the row and column subspaces of Lp are defined as follows
n
nX
o
¯
¡
¢
n
¯
Lp (M; Rp ) =
xk ⊗ e1k ¯ xk ∈ Lp (M) ⊂ Lp M⊗B(`
2) ,
k=1
Lp (M; Cpn ) =
n
nX
o
¯
¡
¢
¯
xk ⊗ ek1 ¯ xk ∈ Lp (M) ⊂ Lp M⊗B(`
2) ,
k=1
where (eij ) denotes the unit vector basis of B(`2 ). These spaces are crucial in
the noncommutative Khintchine/Rosenthal type inequalities [26, 37, 40] and in
noncommutative martingale inequalities [28, 51, 53], where the row and column
spaces are traditionally denoted by Lp (M; `r2 ) and Lp (M; `c2 ). Now, considering a
von Neumann subalgebra N of M with a normal faithful conditional expectation
E : M → N , we may define Lp norms of the conditional square functions
n
n
³X
´ 21
³X
´ 12
∗
E(xk xk )
and
E(x∗k xk ) .
k=1
k=1
The expressions E(xk x∗k ) and E(x∗k xk ) have to be defined properly for 1 ≤ p ≤ 2,
see [16] or Chapter 1 below. Note that the resulting spaces coincide with the row
and column spaces defined above when N is M itself. When n = 1 we recover the
spaces Lrp (M, E) and Lcp (M, E), which have been instrumental in proving Doob’s
inequality [16], see also [21, 29] for more applications.
2. Amalgamated Lp spaces
The definition of amalgamated Lp spaces is algebraic. We recall that by
Hölder’s inequality Lu (M)Lq (M)Lv (M) is contractively included in Lp (M) when
1/p = 1/u + 1/q + 1/v. Let us now assume that N is a von Neumann subalgebra
of M with a normal faithful conditional expectation E : M → N . Then we have
natural isometric inclusions Ls (N ) ⊂ Ls (M) for 0 < s ≤ ∞ and we may consider
the amalgamated Lp space
Lu (N )Lq (M)Lv (N )
as the subset of elements x in Lp (M) which factorize as x = αyβ with α ∈ Lu (N ),
y ∈ Lq (M) and β ∈ Lv (N ). The natural “norm” is then given by the following
expression
n
o
¯
kxku·q·v = inf kαkLu (N ) kykLq (M) kβkLv (N ) ¯ x = αyβ .
However, the triangle inequality for the homogeneous expression k ku·q·v is by no
means trivial. Moreover, it is not clear a priori that this subset of Lp (M) is indeed
a linear space. Before explaining these difficulties in some detail, let us consider
some examples. We fix an integer n ≥ 1 and the subalgebra N embedded in the
4
INTRODUCTION
diagonal of the direct sum M = N ⊕∞ N ⊕∞ · · · ⊕∞ N with n terms. The natural
conditional expectation is
n
¡
¢
1X
En x1 , . . . , xn =
xk .
n
k=1
Then it is easy to see that for 1 ≤ p ≤ 2 and 1/p = 1/2 + 1/w we have
√
n Lw (N )L2 (M)L∞ (N ),
Lp (N ; Rpn ) =
√
n
Lp (N ; Cp ) =
n L∞ (N )L2 (M)Lw (N ),
isometrically. Here we use the notation γX to denote the space X equipped with
the norm γk · kX . At the time of this writing and with independence of this paper,
a result of Pisier [44] on interpolation of these spaces for p = ∞ was generalized
by Xu [69] for arbitrary p’s
n
n
°X
°
³X
´1
°
°
2 2
xk ⊗ δk °
=
inf
kαk
ky
k
kβkvθ .
°
u
k
2
θ
xk =αyk β
[Lp (N ;Rpn ),Lp (N ;Cpn )]θ
k=1
k=1
Here (1/uθ , 1/vθ ) = ((1 − θ)/w, θ/w) and√for θ = 1/2 we find Pisier’s definition of
Lp (N ; `2 ). That is, we obtain the space n Luθ (N )L2 (M)Lvθ (N ). Our definition
is flexible enough to accommodate the conditional square function. Indeed, given a
von Neumann subalgebra N0 of N with a normal faithful conditional expectation
E0 : N → N0 , we find
n
n
°X
°
° X
¢1 °
1 °¡
°
°
°
x k ⊗ δk °
= n− 2 °
E0 (xk x∗k ) 2 °
,
°
k=1
n
°X
°
°
k=1
Lw (N0 )L2 (M)L∞ (N0 )
°
°
x k ⊗ δk °
L∞ (N0 )L2 (M)Lw (N0 )
°
1 °¡
= n− 2 °
k=1
n
X
Lp (N0 )
E0 (x∗k xk )
¢ 12 °
°
°
k=1
Lp (N0 )
.
Xu’s interpolation does not apply in this more general setting, which appears in the
context of the noncommutative Rosenthal inequality. This illustrates how certain
amalgamated Lp spaces occur naturally in quantum probability. Now we want to
understand for which range of indices (u, q, v) we have the triangle inequality. In
fact, our proof intertwines with the proof of our main interpolation result which
can be stated as follows. Let us consider the solid K in R3 defined by
n
o
¯
K = (1/u, 1/v, 1/q) ¯ 2 ≤ u, v ≤ ∞, 1 ≤ q ≤ ∞, 1/u + 1/q + 1/v ≤ 1 .
Theorem A. The amalgamated space Lu (N )Lq (M)Lv (N ) is a Banach space
for any (1/u, 1/v, 1/q) ∈ K. Moreover, if (1/uj , 1/vj , 1/qj ) ∈ K for j = 0, 1, the
space Luθ (N )Lqθ (M)Lvθ (N ) is isometrically isomorphic to
h
i
Lu0 (N )Lq0 (M)Lv0 (N ), Lu1 (N )Lq1 (M)Lv1 (N ) .
θ
The triangle inequality follows from Theorem A. On the other hand we will need
the triangle inequality in order to apply interpolation and factorization techniques
in proving Theorem A. This intriguing interplay makes our proof quite involved.
Our first step is showing that the triangle inequality holds in the boundary region
n
o
¯
©
ª
∂∞ K = (1/u, 1/v, 1/q) ∈ K ¯ min 1/u, 1/q, 1/v = 0 .
INTRODUCTION
5
Our argument uses the operator-valued analogue of Szegö’s factorization theorem,
a technique which will be used repeatedly throughout this paper. The triangle
inequality for other indices follows by convexity since K is the convex hull of ∂∞ K,
so that any other point in K is associated to an interpolation space between two
spaces living in ∂∞ K. Another technical difficulty is the fact that the intersection
of two amalgamated Lp spaces is in general quite difficult to describe. Thus, any
attempt to use a density argument meets this obstacle. The second step is to prove
Theorem A for finite von Neumann algebras, where the intersections are easier to
handle. Moreover, most of the factorization arguments (as Szegö’s theorem) a priori
only apply in the finite setting. In the third step we consider general von Neumann
algebras using Haagerup’s crossed product construction [12] to approximate σ-finite
von Neumann algebras by direct limits of finite von Neumann algebras. Finally, we
need a different argument for the case min(q0 , q1 ) = ∞, which is out of the scope
of Haagerup’s construction. The main technique here is a Grothendieck-Pietsch
version of the Hahn-Banach theorem.
Let us observe that in the hyperfinite case Pisier was able to establish many of
his results using the Haagerup tensor product. Though similar in nature, we can
not directly use tensor product formulas for our interpolation results due to the
complicated structure of general von Neumann subalgebras. Theorem A will also
be useful in understanding certain interpolation spaces in martingale theory. Let
us mention some open problems, for partial results see Chapter 5 below.
Problem 4. Let M be a von Neumann algebra and denote by Hpr (M) and Hpc (M)
the row and column Hardy spaces of noncommutative martingales over M. Let us
consider an interpolation parameter 0 < θ < 1.
(a) Calculate the interpolation norms [Hpr (M), Hpc (M)]θ .
(b) If x ∈ [H1r (M), H1c (M)]θ , the maximal function is in L1 .
3. Conditional Lp spaces
Once we know which amalgamated Lp spaces are Banach spaces it is natural
to investigate their dual spaces. We assume as above that N is a von Neumann
subalgebra of M and E : M → N is a normal faithful conditional expectation. Let
1/s = 1/u + 1/p + 1/v ≤ 1.
The conditional Lp space
Lp(u,v) (M, E)
is defined as the completion of Lp (M) with respect to the norm
n
o
¯
kxkLp(u,v) (M,E) = sup kaxbkLs (M) ¯ kakLu (N ) , kbkLv (N ) ≤ 1 .
In our next result we show that amalgamated and conditional Lp are related by
anti-linear duality. This will allow us to translate the interpolation identities in
Theorem A in terms of conditional Lp spaces. In this context the correct set of
parameters is given by
n
o
¯
K0 = (1/u, 1/v, 1/q) ∈ K ¯ 2 < u, v ≤ ∞, 1 < q < ∞, 1/u + 1/q + 1/v < 1 .
6
INTRODUCTION
Theorem B. Let 1 < p < ∞ given by 1/q 0 = 1/u + 1/p + 1/v, where the
indices (u, q, v) belong to the solid K0 and q 0 is conjugate to q. Then, the following
isometric isomorphisms hold via the anti-linear duality bracket hx, yi = tr(x∗ y)
¡
¢∗
Lu (N )Lq (M)Lv (N ) = Lp(u,v) (M, E),
¡ p
¢∗
L(u,v) (M, E) = Lu (N )Lq (M)Lv (N ).
In particular, we obtain isometric isomorphisms
h
i
Lp(u0 0 ,v0 ) (M, E), Lp(u1 1 ,v1 ) (M, E) = Lp(uθ θ ,vθ ) (M, E).
θ
As we shall see, Theorem B generalizes the interpolation results obtained by
Pisier [44] and Xu [69] mentioned above. Pisier and Xu’s results provide an explicit
expression for the operator space structure of [Cpn , Rpn ]θ with 0 < θ < 1. Theorem B
also provides explicit formulas for [`np , Cpn ]θ and [`np , Rpn ]θ . In fact, a large variety of
interesting formulas of this kind arise from Theorem B. A detailed analysis of these
applications is out of the scope of this paper. On the other hand, the analogue
of Theorem B for p = ∞ (which we will investigate separately) has been already
applied to study the noncommutative John-Nirenberg theorem [21].
Exactly as it happens with amalgamated Lp spaces, several noncommutative
function spaces arise as particular forms of conditional Lp spaces. Let us review
the basic examples in both cases.
(a) The spaces Lp (M) satisfy
Lp (M) = L∞ (N )Lp (M)L∞ (N ) and
Lp (M) = Lp(∞,∞) (M, E).
(b) The spaces Lp (N1 ; Lq (N2 )):
• Let p ≤ q and 1/r = 1/p − 1/q. Then
¯ 2 )L2r (N1 ).
Lp (N1 ; Lq (N2 )) = L2r (N1 )Lq (N1 ⊗N
• Let p ≥ q and 1/r = 1/q − 1/p. Then
¯ 2 , E),
Lp (N1 ; Lq (N2 )) = Lp(2r,2r) (N1 ⊗N
¯ 2 → N1 is given by E = 1N1 ⊗ ϕN2 .
where E : N1 ⊗N
(c) The spaces Lrp (M, E) and Lcp (M, E):
• Let 1 ≤ p ≤ 2 and 1/p = 1/2 + 1/s. Then
Lrp (M, E) = Ls (N )L2 (M)L∞ (N ),
Lcp (M, E) = L∞ (N )L2 (M)Ls (N ).
• Let 2 ≤ p ≤ ∞ and 1/p + 1/s = 1/2. Then
Lrp (M, E) = Lp(s,∞) (M, E)
Lcp (M, E) = Lp(∞,s) (M, E).
P
In particular, taking En (x1 , ..., xn ) = n1 k xk we find
√ r n
Lp (M; Rpn ) =
n Lp (`∞ (M), En ),
√ c n
n
Lp (M; Cp ) =
n Lp (`∞ (M), En ).
INTRODUCTION
7
(d) As we shall see through the text, asymmetric Lp spaces (a non-standard
operator space structure on Lp which will be crucial in this paper) also
have representations in terms of amalgamated or conditional Lp spaces.
4. Intersection spaces
Intersection of Lp spaces appear naturally in the theory of noncommutative
Hardy spaces. These spaces are also natural byproducts of Rosenthal’s inequality
for sums of independent random variables. Let us first illustrate this point in the
commutative setting and then provide the link to the spaces defined above. Let
us consider a finite collection f1 , f2 , . . . , fn of independent random variables on a
probability space (Ω, F, µ). The Khintchine inequality implies for 0 < p < ∞
n
n
°X
°
³Z hX
i p2 ´ p1
°
°
|fk |2 dµ
∼cp E °
εk fk ° .
Ω
k=1
p
k=1
Therefore, Rosenthal’s inequality [56] gives for 2 ≤ p < ∞
n
n
n
³Z hX
i p2 ´ p1
´ p1 ³ X
´ 12
³X
|fk |2 dµ
kfk k22 .
(Σp2 )
∼cp
kfk kpp
+
Ω
k=1
k=1
k=1
Here (εk ) is an independent sequence of Bernoulli random variables equidistributed
on ±1. We can easily generalize this result for calculating `q sums of independent
random variables. Indeed, consider 1 ≤ q ≤ p < ∞ and define g1 , g2 , . . . , gn by the
relation gk = |fk |q/2 for 1 ≤ k ≤ n. Then we have the following identity for the
index s = 2p/q
n
n
2
³Z hX
i p ´ p1
³Z hX
i 2s ´ qs
q q
|fk |
dµ
=
|gk |2 dµ
.
Ω
Ω
k=1
k=1
Therefore (Σp2 ) implies
( n Z
)
n
³Z hX
i p ´ p1
³X
´ r1
q q
r
(Σpq )
|fk |
dµ
∼cp max
|fk | dµ
.
Ω
r∈{p,q}
k=1
k=1
Ω
In particular, Rosenthal’s inequality provides a natural realization of
1
1
n
Jp,q
(Ω) = n p Lp (Ω) ∩ n q Lq (Ω)
into Lp (Ω; `nq ). More precisely, if f1 , f2 , . . . , fn are taken to be independent copies
of a given random variable f , the right hand side of (Σpq ) is the norm of f in the
n
intersection space Jp,q
(Ω) and inequality (Σpq ) provides an isomorphic embedding
n
of Jp,q (Ω) into the space Lp (Ω; `nq ).
Quite surprisingly, replacing independent variables by matrices of independent
variables in (Σpq ) requires to intersect four spaces using the so-called asymmetric
n
Lp spaces. In other words, the natural operator space structure of Jp,q
comes from
a 4-term intersection space. We have already encountered such a phenomenon in
[22] for the case q = 1. To justify this point, instead of giving precise definitions
we note that Hölder inequality gives Lp = L2p L2p , meaning that the p-norm of f
is the infimum of kgk2p khk2p over all factorizations f = gh. If Lrp and Lcp denote
the row and column quantizations of Lp (see Chapter 1 for the definition), the
operator space analogue of the isometry above is given by the complete isometry
8
INTRODUCTION
Lp = Lr2p Lc2p , see Chapter 7 for more details. In particular, according to the
n
algebraic definition of Lp (`q ), the space Jp,q
has to be redefined as the product
´³
´
³ 1
1
1
1
n
Jp,q
= n 2p Lr2p ∩ n 2q Lr2q n 2p Lc2p ∩ n 2q Lc2q .
We shall see in this paper that
1
1
1
1
1
1
n
Jp,q
= n p Lr2p Lc2p ∩ n 2q + 2p Lr2q Lc2p ∩ n 2p + 2q Lr2p Lc2q ∩ n q Lr2q Lc2q .
On the Banach space level we have the isometries
Lr2p Lc2q = Ls = Lr2q Lc2p
with 1/s = 1/2p + 1/2q.
Moreover, again by Hölder inequality it is clear that
n 1
o
1
1
n s kf ks ≤ max n p kf kp , n q kf kq .
Therefore, the two cross terms in the middle disappear in the Banach space level.
However, replacing scalars by operators in the context of independence/freness over
a given von Neumann subalgebra, the Banach space estimates from above are no
longer valid and all four terms may have a significant contribution.
n
It is the operator space structure of Jp,q
what originally led us to introduce
amalgamated and conditional Lp spaces. To be more precise, we consider a von
Neumann algebra M equipped with a normal faithful state ϕ and a von Neumann
subalgebra N with associated normal faithful conditional expectation E. Then, if
we fix 1 ≤ q ≤ p ≤ ∞, we define
\
1
1
1
n
Jp,q
(M, E) =
n u + p + v Lp(u,v) (M, E)
with
1/r = 1/q − 1/p.
u,v∈{2r,∞}
This definition is motivated by the fundamental isometry
¡ n
¢
¡
¢
n
(3)
Spm Jp,q
(M) = Jp,q
Mm ⊗ M, 1Mm ⊗ ϕ ,
n
which will be proved in Chapter 7. Some preliminary results on Jp,q
(M) (and
vector-valued generalizations) are already contained in the recent paper [22]. We
extend many results from [22] to the realm of free random variables including the
n
limit case p = ∞. Our main result for the spaces Jp,q
(M, E) shows that we have
an interpolation scale with respect to the index 1 ≤ q ≤ p.
Theorem C. If 1 ≤ p ≤ ∞, then
£ n
¤
n
n
Jp,1 (M, E), Jp,p
(M, E) θ ' Jp,q
(M, E)
with 1/q = 1 − θ + θ/p and with relevant constants independent of n.
There seems to be no general argument to make intersections commute with
complex interpolation. For commutative Lp spaces or rearrangement invariant
spaces one can often find concrete formulas of the resulting interpolation norms,
see [35]. However, in the noncommutative context these arguments are no longer
valid and we need genuinely new tools.
5. Mixed-norm inequalities
The central role played by Rosenthal inequality partially justifies why the index
p must be finite in the commutative form of (Σpq ). This also happens in [22], where
we used the noncommutative Rosenthal inequality from [28]. However, mainly
INTRODUCTION
9
motivated by Problems 2 and 3, one of our main goals in this paper is to obtain a
right formulation of (Σ∞q ). As in several other inequalities involving independent
random variables, such as the noncommutative Khintchine inequalities, the limit
case as p → ∞ holds when replacing classical independence by Voiculescu’s concept
of freeness [67]. Therefore, it is not surprising that we shall use in our proof the free
analogue of Rosenthal inequality [26]. Following (Σpq ) we have a natural candidate
n
for a complemented embedding of Jp,q
in Lp (`nq ) using free probability.
We define Ak to be the direct sum M⊕M for 1 ≤ k ≤ n. Then we consider the
reduced amalgamated free product A = ∗N¡ Ak where the ¢conditional expectation
EN : A → N has the form EN (x1 , x2 ) = 12 E(x1 ) + E(x2 ) when restricted to the
algebra Ak . Let πk : Ak → A denote the natural embedding of Ak into A. Moreover,
given x ∈ M we shall write xk as an abbreviation of πk (x, −x). Note that xk is a
mean-zero element for 1 ≤ k ≤ n. Our main embedding result reads as follows.
Theorem D. Let 1 ≤ q ≤ p ≤ ∞. The map
u:x∈
n
(M, E)
Jp,q
7→
n
X
xk ⊗ δk ∈ Lp (A; `nq )
k=1
is an isomorphism with complemented image. The constants are independent of n.
The map u is of course reminiscent of the fundamental mappings employed in
[15, 17, 22] constructing certain embeddings of Lp spaces. Theorem C follows as
a consequence of Theorem D using the fact that
¤
£
Lp (A, `nq ) = Lp (A, `n1 ), Lp (A, `np ) θ .
We have tried in vain to prove Theorem D directly using uniquely tools from free
probability. The methods around Voiculescu’s inequality seem to work perfectly fine
in the limit case p = ∞, for which there is no commutative version. However, basic
tools from free harmonic analysis are still missing for a direct proof of Theorem
D. Instead, apart from the free analogue [26] of Rosenthal inequality, we also use
factorization techniques and interpolation results for noncommutative Hardy and
BMO spaces, see Chapter 5 for further details.
The interested reader might be surprised that we have not formulated our
results in the category of operator spaces. However, as so often in martingale
theory these results are automatic, provided the spaces carry the correct operator
space structure, given in this case by (3) and
³ ¡
³
´
¢´
Spm Lp ∗k Ak ; `nq
= Lp Mm ⊗ (∗k Ak ); `nq = Lp (A; `nq ),
where the von Neumann algebra A is now given by
¡
¢ ¡
¢
b k with A
b k = Mm ⊗ Ak = Mm ⊗ M ⊕ Mm ⊗ M .
A = ∗Mm A
6. Operator space Lp embeddings
In the last part of the paper, we shall use the mixed-norm inequalities obtained
in Theorem D to construct a completely isomorphic embedding of Lq into Lp for
1 ≤ p < q ≤ 2. Our construction solves positively a problem formulated by Pisier
in 1996, whose precise statement is the following.
10
INTRODUCTION
Theorem E. Let 1 ≤ p < q ≤ 2 and let M be a von Neumann algebra.
Then, there exists a sufficiently large von Neumann algebra A and a completely
isomorphic embedding of Lq (M) into Lp (A), where both spaces are equipped with
their respective natural operator space structures. Moreover, we have
i) If M is QWEP, we can choose A to be QWEP.
ii) If M is hyperfinite, we can choose A to be hyperfinite.
This is the second part of a series of three papers in which we investigate the
validity of some well-known results, from the embedding theory of Lp spaces, in the
category of operator spaces. We refer to the Introduction of the first part [24] for
the motivations which led to our result and also to place it in the right context. As
explained there, the presence of noncommutative Lp spaces is not only necessary
but natural. The major result in [24] is a complete embedding of discrete Lq
spaces into the predual of a von Neumann algebra. This is the simplest case of our
complete embedding and it allowed us to give an essentially self-contained approach
for a wider audience. In the third paper [25], we shall prove that for M infinite
dimensional the von Neumann algebra A must be of type III. This generalizes a
previous result of Pisier [48] and requires different techniques from [23].
Our argument can be sketched as follows. We first construct a cb-embedding
of the Schatten class Sq into Lp (A) with A a QWEP von Neumann algebra. This
will give rise to an Lp generalization of [24, Theorem D]. The drawback of this
first approach is that this construction does not preserve hyperfiniteness. The
argument to fix this is quite involved and requires recent techniques from [18] and
[20]. We first apply a transference argument, via a noncommutative Rosenthal type
inequality in L1 for identically distributed random variables, to replace freeness in
our construction by some sort of noncommutative independence. This allows to
avoid free product von Neumann algebras and use tensor products instead. Then
we combine the algebraic central limit theorem with the notion of noncommutative
Poisson random measure to eliminate the use of ultraproducts in the process. After
these modifications in our original argument, it is easily seen that hyperfiniteness is
preserved. This more involved construction of the cb-embedding is the right one to
analyze the finite dimensional case. In other words, we estimate the dimension of
A in terms of the dimension of M, see Remark 8.15 below for details. The proof for
general von Neumann algebras requires to consider a ‘generalized’ Haagerup tensor
product since we are not in the discrete case anymore. Our approach here follows
Pisier’s method in [49].
After a quick look at the main results in [3, 15], the problem of constructing
an isometric cb-embedding of Lq into Lp arises in a natural way. This remains an
open problem.
Structure of the paper. This paper has three natural parts. Part I contains
Chapters 1–4, where we prove Theorems A and B. In Part II, which is given by
Chapters 5–7, we discuss our mixed-norm inequalities. This includes Theorems
C and D. Finally, Part III is the content of Chapter 8 where we construct our
operator space Lp embeddings stated in Theorem E. Those readers interested on
getting rapidly to the core of the paper might skip Part I –including Chapter 1
if the reader is familiar with recent advances on noncommutative integration– in
a first reading. Indeed, the arguments there are quite technical and essentially
INTRODUCTION
11
independent from the rest of the paper. Thus, the definitions and statements that
we have included in this Introduction should be enough to follow Parts II and III
without too much trouble. Of course, although very rarely, from time to time the
reader will have to go back and read the statement of some result from Part I in
order to keep track of the argument.
Background and notation. Although we shall review some basic concepts along
the paper, we assume some familiarity with some branches of operator algebra such
as von Neumann algebras, noncommutative integration, operator space theory and
free probability. Moreover, we shall also use specific techniques like Calderón’s
complex interpolation method [2], Haagerup’s approximation theorem [12], the
Grothendieck-Pietsch separation argument [43], Raynaud’s results on ultraproducts
of noncommutative Lp spaces [55], a noncommutative form of a Poisson process
[20]... Some of these techniques will be briefly introduced within the text. In any
case, our exposition intends to be as self-contained as possible. To conclude, the
non-expert reader must think of [24] as a prerequisite to read Chapter 8, since it
serves as a good motivation for most of our present approach.
We shall use standard notation from the literature such as e.g. [47, 62]. Apart
from this and the terminology introduced along the paper, our notation will be as
coherent with [24] as we can.
Acknowledgements. This work was mostly carried out in a one-year visit of
the second-named author to the University of Illinois at Urbana-Champaign. The
second-named author would like to thank the Math Department of the University
of Illinois for its support and hospitality.
CHAPTER 1
Noncommutative integration
In this chapter we review some basic notions on noncommutative integration
that will be frequently used through out this paper. We begin by recalling Haagerup
and Kosaki’s constructions of noncommutative Lp spaces. Then we briefly introduce
Pisier’s theory of vector-valued noncommutative Lp spaces, giving some emphasis
to those aspects which are relevant in this work. Finally, we analyze some basic
properties of certain Lp spaces associated to a conditional expectation, which were
recently introduced in the literature and are basic for our further purposes. We
shall assume some familiarity with von Neumann algebras. Basic concepts such as
trace, state, commutant, affiliated operator, crossed product... can be found in [31]
or [62] and will be freely used along the text.
1.1. Noncommutative Lp spaces
Noncommutative Lp spaces over non-semifinite von Neumann algebras will be
used quite frequently in this paper. In the literature there exist two compatible
constructions of Lp in such a general setting: Haagerup Lp spaces and Kosaki’s
interpolation spaces. These constructions and the associated notion of conditional
expectation will be considered in this section.
1.1.1. Haagerup Lp spaces. A full-detailed exposition of this theory is given
in Haagerup [13] and Terp [63] papers. We just present the main notions according
to our purposes with an exposition similar to [28]. A preliminary restriction is that,
in view of our aims, we can work in what follows with normal faithful (n.f. in short)
states instead of normal semifinite faithful (n.s.f. in short) weights.
Let M be a von Neumann algebra equipped with a distinguished n.f. state
ϕ. The GNS construction applied to ϕ yields a faithful representation ρ of M into
B(H) so that ρ(M) is a von Neumann algebra acting on H with a separating and
generating unit vector u satisfying ϕ(x) = hu, ρ(x)ui for all x ∈ M. Let us agree
to identify M with ρ(M) in the following. Then, the modular operator ∆ is the
(generally unbounded) operator obtained from the polar decomposition S = J∆1/2
of the anti-linear map S : Mu → Mu given by S(xu) = x∗ u, see Section 9.2 in [31]
or [60]. We denote by σt : M → M the one-parameter modular automorphism
group associated to the separating and generating unit vector u. That is, for any
t ∈ R we have an automorphism of M given by
σt (x) = ∆it x∆−it .
Then we consider the crossed product R = M oσ R, which is defined as the
von Neumann algebra acting on L2 (R; H) and generated by the representations
π : M → B(L2 (R; H)) and λ : R → B(L2 (R; H)) with
¡
¢
¡
¢
π(x)ξ (t) = σ−t (x)ξ(t)
and
λ(s)ξ (t) = ξ(t − s)
13
14
1. NONCOMMUTATIVE INTEGRATION
for t ∈ R and ξ ∈ L2 (R; H). Note that the representation π is faithful so that we
can identify M with π(M). The dual action of R on¡ R is defined
as follows. Let
¢
W : R → B(L2 (R; H)) be the unitary representation W(t)ξ (s) = e−its ξ(s). Then
we define the one-parameter automorphism group σ̂t : R → R by
σ̂t (x) = W(t)xW(t)∗ .
It turns out that M is the space of fixed points of the dual action
n
o
¯
M = x ∈ R ¯ σ̂t (x) = x for all t ∈ R .
Following [42], the crossed product R is a semifinite von Neumann algebra and
admits a unique n.s.f. trace τ satisfying τ ◦ σ̂t = e−t τ for all t ∈ R. Let L0 (R, τ )
denote the topological ∗-algebra of τ -measurable operators affiliated with R and
let 0 < p ≤ ∞. The Haagerup noncommutative Lp space over M is defined as
n
o
¯
Lp (M, ϕ) = x ∈ L0 (R, τ ) ¯ σ̂t (x) = e−t/p x for all t ∈ R .
It is clear from the definition that L∞ (M, ϕ) coincides with M. Moreover, as it
is to be expected, L1 (M, ϕ) can be canonically identified with the predual M∗ of
the von Neumann algebra M. This requires a short explanation. Given a normal
state ω ∈ M+
∗ , the dual state ω̃ : R+ → [0, ∞] is defined by
³Z
´
ω̃(x) = ω
σ̂s (x) ds .
R
Note that, by the translation invariance of the Lebesgue measure, the operator
valued integral above is invariant under the dual action. In particular, it can be
regarded as an element of M. As a n.s.f. weight on R and according to [42], the
dual weight ω̃ has a Radon-Nikodym derivative hω with respect to τ so that
ω̃(x) = τ (hω x)
for any x ∈ R+ . The operator hω so defined belongs to L1 (M, ϕ)+ . Indeed,
³Z
´
³Z
´
τ (hω σ̂t (x)) = ω
σ̂s (σ̂t (x)) ds = ω
σ̂s (x) ds = τ (hω x).
R
R
In particular,
τ (σ̂t (hω )σ̂t (x)) = e−t τ (hω x) = e−t τ (hω σ̂t (x)) for all
x ∈ R,
which implies that σ̂t (hω ) = e−t hω . Therefore, there exists a bijection between
M+
∗ and L1 (M, ϕ)+ which extends to a bijection between the predual M∗ and
L1 (M, ϕ) by polar decomposition
ω = u|ω| ∈ M∗ 7→ uh|ω| = hω ∈ L1 (M, ϕ).
In fact, after imposing on L1 (M, ϕ) the norm
khω k1 = |ω|(1) = kωkM∗ ,
we obtain an isometry between M∗ and L1 (M, ϕ). There is however a nicer way
to describe this norm. As we have seen, for any x ∈ L1 (M, ϕ) there exists a unique
ωx ∈ M∗ such that hωx = x. This gives rise to the functional tr : L1 (M, ϕ) → C
called trace and defined by
tr(x) = ωx (1).
1.1. NONCOMMUTATIVE Lp SPACES
15
¡ ¢
The functional tr is continuous since |tr(x)| ≤ tr |x| = kxk1 and satisfies the
tracial property
tr(xy) = tr(yx).
Our distinguished state ϕ can be recovered from tr as follows. First we note as
above that its dual weight ϕ̃ admits a Radon-Nikodym derivative Dϕ with respect
to τ , so that ϕ̃(x) = τ (Dϕ x) for x ∈ R+ . Then, it turns out that
ϕ(x) = tr(Dϕ x) for
x ∈ M.
According to this, we will refer in what follows to Dϕ as the density of ϕ. Moreover,
we shall write D instead of Dϕ whenever the dependence on ϕ is clear from the
context. Given 0 < p < ∞ and x ∈ Lp (M, ϕ), we define
¡
¢1/p
kxkp = tr|x|p
and kxk∞ = kxkM .
k kp is a norm (resp. p-norm) on Lp (M, ϕ) for 1 ≤ p ≤ ∞ (resp. 0 < p < 1).
Lemma 1.1. The Haagerup Lp spaces satisfy the following properties:
i) If 0 < p, q, r ≤ ∞ with 1/r = 1/p + 1/q, we have
kxykr ≤ kxkp kykq
for all
ii) If 1 ≤ p < ∞, Lp (M, ϕ)
Lp0 (M, ϕ) via
∗
x ∈ Lp (M, ϕ), y ∈ Lq (M, ϕ).
is anti-linear isometrically isomorphic to
x ∈ Lp0 (M, ϕ) 7→ tr(x∗ ·) ∈ Lp (M, ϕ)∗ .
An element x ∈ M is called analytic if the function t ∈ R 7→ σt (x) ∈ M
extends to an analytic function z ∈ C → σz (x) ∈ M. By [42] we know that the
subspace Ma of analytic elements in M is a weak∗ dense ∗-subalgebra of M. The
proof of the following result can be found in [28]. It will be useful in the sequel.
Lemma 1.2. If 0 < p < ∞, we have
i) Ma D1/p is dense in Lp (M, ϕ).
ii) D(1−η)/p Ma Dη/p = Ma D1/p for all 0 ≤ η ≤ 1.
1.1.2. Kosaki’s interpolation. The given definition of Haagerup Lp space
has the disadvantage that the intersection of Lp (M, ϕ) and Lq (M, ϕ) is trivial for
p 6= q. In particular, these spaces do not form an interpolation scale. All these
difficulties disappear with Kosaki’s construction. As above, we only consider von
Neumann algebras equipped with n.f. states. The general construction for any von
Neumann algebra can be found in [32] and [64]. Let us consider a von Neumann
algebra M equipped with a n.f. state ϕ. First we define
L1 (M) = Mop
∗ .
Note that the natural operator space structure for L1 (M) requires to consider Mop
∗
instead of M∗ , we refer the reader to [47] for a detailed explanation. Then, given
any real number t, we consider the map
¡
¢
jt : x ∈ M 7→ σt (x)ϕ ∈ L1 (M)
with
σt (x)ϕ (y) = ϕ(σt (x)y).
According to [32] there exists a unique extension jz : M → L1 (M) such that,
for any 0 ≤ η ≤ 1, the map j−iη is injective. In particular, (j−iη (M), L1 (M)) is
compatible for complex interpolation and we define the Kosaki noncommutative Lp
spaces as follows
£
¤
Lp (M, ϕ, η) = j−iη (M), L1 (M) 1
p
16
1. NONCOMMUTATIVE INTEGRATION
by specifying
−1
kxk0 = kj−iη
(x)kM
and
kxk1 = kxkL1 (M) .
The following result was proved in [13] except for the last isometry [32].
Theorem 1.3. Let 1 ≤ p ≤ ∞ and let M be any von Neumann algebra:
i) If ϕ1 and ϕ2 are two n.f. states on M, we have an isometric isomorphism
Lp (M, ϕ1 ) = Lp (M, ϕ2 ).
ii) If ϕ is a n.f. state and 0 ≤ η ≤ 1, we have an isometric isomorphism
Lp (M, ϕ) = Lp (M, ϕ, η).
More concretely, given x ∈ M
°
°
kj−iη (x)kLp (M,ϕ,η) = °Dη/p xD(1−η)/p °Lp (M,ϕ) .
According to Theorem 1.3, Haagerup and Kosaki noncommutative Lp spaces
can be identified. We shall write Lp (M) to denote in what follows any of the spaces
defined above. In particular, after the corresponding identifications, we may use
the complex interpolation method for Haagerup Lp spaces. We should also note
that Kosaki’s definition of Lp presents some other disadvantages. The main lacks in
this paper will be the absence of positive cones and the fact that the case 0 < p < 1
is excluded from the definition. In particular, we will use Haagerup Lp spaces and
we will apply Theorem 1.3 when needed.
1.1.3. Conditional expectations. Let us consider a von Neumann algebra
M equipped with a n.f. state ϕ and a von Neumann subalgebra N of M. A
conditional expectation E : M → N is a positive contractive projection. E is
called faithful if E(x∗ x) 6= 0 for any x ∈ M and normal when it has a predual
E∗ : M∗ → N∗ . According to Takesaki [61], if N is invariant under the action of
the modular automorphism group (i.e. σt (N ) ⊂ N for all t ∈ R), there exists a
unique faithful normal conditional expectation E : M → N satisfying ϕ ◦ E = ϕ.
Moreover, by Connes [6] it commutes with the modular automorphism group
E ◦ σt = σt ◦ E.
The required invariance of N under the action of σt implies that the modular
automorphism group associated to N coincides with the restriction of σ to N . It
follows that N oσ R is a von Neumann subalgebra of M oσ R. In particular, the
space Lp (N ) can be identified isometrically with a subspace of Lp (M), see [28]
for details. In this paper we shall permanently assume the existence of a normal
faithful conditional expectation E : M → N .
It is well-known that in the tracial case, the conditional expectation E extends
to a contractive projection from Lp (M) onto Lp (N ) for any 1 ≤ p ≤ ∞, which is
still positive and has the modular property E(axb) = aE(x)b for all a, b ∈ N and
x ∈ Lp (M). These properties remain valid in this context. We summarize in the
following lemma the main properties of E that will be used in the sequel and refer
the reader to [28] for a proof of these facts.
Lemma 1.4. Let M and N be as above:
i) E : Lp (M) → Lp (N ) is a positive contractive projection.
ii) If 2 ≤ p ≤ ∞ and x ∈ Lp (M), we have E(x)∗ E(x) ≤ E(x∗ x).
1.2. PISIER’S VECTOR-VALUED Lp SPACES
iii) If a ∈ Lp (N ), x ∈ Lq (M), b ∈ Lr (N ) and
In particular, if 1 ≤ p ≤ ∞ and x ∈ M
E(D1/p x) = D1/p E(x)
and
1
p
17
+ 1q + 1r ≤ 1, E(axb) = aE(x)b.
E(xD1/p ) = E(x)D1/p .
1.2. Pisier’s vector-valued Lp spaces
Noncommutative vector-valued Lp spaces Lp (M; X) appeared quite recently
with Pisier’s work [46]. The reason for the novelty of such a natural concept lies
in the fact that X must be equipped with an operator space structure rather than
a Banach space one. Moreover, many natural properties such as duality, complex
interpolation, etc... must be formulated in the category of operator spaces. Let us
begin by recalling the concept of operator space.
1.2.1. Operator spaces. Operator space theory plays a central role in this
paper. It can be regarded as a noncommutative generalization of Banach space
theory and it has proved to be an essential tool in operator algebra as well as in
noncommutative harmonic analysis. An operator space X is defined as a closed
subspace of the space B(H) of bounded operators on some Hilbert space H. Let
Mn (X) denote the vector space of n × n matrices with entries in X. If we have
an isometric embedding j : X → B(H), let us consider the sequence of norms on
Mn (X) for n ≥ 1 given by
°¡ ¢°
°¡
¢°
° xij °
= ° j(xij ) °B(Hn ) .
Mn (X)
Ruan’s axioms [11, 47, 58] describe axiomatically those sequences of matrix norms
which can occur from an isometric embedding into B(H). Any such sequence of
norms provides X with a so-called operator space structure. Every Banach space
can be equipped with several operator space structures. In particular, the most
important information carried by an operator space is not the space itself but the
way in which it embeds isometrically into B(H). For this reason, the main difference
between Banach and operator space theory lies on the morphisms rather than on the
spaces. The morphisms in the category of operator spaces are completely bounded
linear maps u : X → Y. That is, linear maps satisfying that the quantity
°
°
kukcb(X,Y) = sup °idMn ⊗ u°B(M (X),M (Y))
n
n
n≥1
is finite. In this paper we shall assume some familiarity with basic notions such
as duality, Haagerup tensor products, the OH spaces... that can be found in the
recent books [11, 47].
1.2.2. The hyperfinite case. Before any other consideration, let us recall
the natural operator space structure (o.s.s. in short) on Lp (M). Proceeding as in
Chapter 3 of [46], we regard M as a subspace of B(H) with H being the Hilbert
space arising from the GNS construction. Similarly, by embedding the predual
von Neumann algebra M∗ on its bidual M∗ , we obtain an o.s.s. for M∗ . The
o.s.s. on L1 (M) is then given by that of Mop
∗ , see page 139 in [47] for a detailed
justification of this definition. Then, the complex interpolation method for operator
spaces developed in [45] provides a natural o.s.s. on Lp (M)
£
£
¤
¤
Lp (M) = L∞ (M), L1 (M) 1/p = M, Mop
∗ 1/p .
18
1. NONCOMMUTATIVE INTEGRATION
Let us now describe the natural operator space structure for vector-valued Lp
spaces. Let M be an hyperfinite von Neumann algebra and let X be any operator
space. Then, recalling the definitions of the projective and the minimal tensor
product in the category of operator spaces, we define
b X and
L1 (M; X) = L1 (M)⊗
L∞ (M; X) = L∞ (M) ⊗min X.
Then, the space Lp (M; X) is defined by complex interpolation
£
¤
Lp (M; X) = L∞ (M; X), L1 (M; X) 1/p .
As it was explained in [46], the hyperfiniteness of M leads to obtain some expected
properties of these spaces. We shall discuss the non-hyperfinite case in the next
paragraph. We are not reviewing here the basic results from Pisier’s theory, for
which we refer the reader to Chapters 1,2 and 3 in [46].
Let us fix some notation. R and C denote the row and column operator spaces
(c.f. Chapter 1 of [47]) constructed over `2 . Identifying B(`2 ) (via the canonical
basis of `2 ) with a space of matrices with infinitely many rows and columns, R and
C are the first row and first column subspaces of B(`2 ). Similarly, if 1 ≤ p ≤ ∞ and
Sp denotes the Schatten p-class over `2 , the spaces Rp and Cp denote the row and
column subspaces of Sp . The finite-dimensional versions over `n2 will be denoted
by Rpn and Cpn respectively. Note that, as in the infinite-dimensional case, we have
n
n
Rn = R∞
and Cn = C∞
. Given an operator space X, the vector-valued Schatten
classes with values in X will be denoted by Sp (X) and Spn (X) respectively. Among
the several characterizations of these spaces given in [46], we have
Sp (X) = Cp ⊗h X ⊗h Rp
and
Spn (X) = Cpn ⊗h X ⊗h Rpn .
1.2.3. The non-hyperfinite case. One of the main restrictions in Pisier’s
theory [46] comes from the fact that the construction of Lp (M; X) requires M to
be hyperfinite. This excludes for instance free products of von Neumann algebras, a
very relevant tool in this paper. There exists however a very recent construction in
[19] of Lp (M; X) which is valid for any QWEP von Neumann algebra. Nevertheless,
since we only deal with very specific operator spaces, we briefly discuss them.
The spaces Lp (M; Rpn ) and Lp (M; Cpn ). As we explained in the Introduction,
these spaces are of capital importance. Let us recall that the spaces Rpn (X) and
Cpn (X) are defined as subspaces of Spn (X) for any operator space X. Therefore,
motivated by Fubini’s theorem for noncommutative Lp spaces [46], we obtain the
following definition valid for arbitrary von Neumann algebras
Lp (M; Rpn ) = Rpn (Lp (M))
and Lp (M; Cpn ) = Cpn (Lp (M)).
These spaces appear quite frequently in the theory of noncommutative martingale
inequalities. However, in that context they are respectively denoted by Lp (M; `r2 )
and Lp (M; `c2 ), see e.g. [51] or [53] for more details.
The spaces Lp (M; `nq ). The spaces Lp (M; `n1 ) and Lp (M; `n∞ ) were defined
in [16] for general von Neumann algebras to study Doob’s maximal inequality for
noncommutative martingales. Indeed, since the notion of maximal function does
not make any sense in the noncommutative setting, the norm in Lp (Ω) of Doob’s
maximal function
f ∗ (w) = sup |fn (w)|
n≥1
1.2. PISIER’S VECTOR-VALUED Lp SPACES
19
was reinterpreted as the norm in Lp (Ω; `∞ ) of the sequence (f1 , f2 , . . .). This was
the original motivation to study the spaces Lp (M; `1 ) and Lp (M; `∞ ). A detailed
exposition of these spaces can also be found in [29]. The spaces Lp (M; `nq ) will be
of capital importance in the second part of this paper. These spaces were recently
defined in [30] over non-hyperfinite von Neumann algebras as follows
£
¤
Lp (M; `nq ) = Lp (M; `n∞ ), Lp (M; `n1 ) 1 .
q
This formula trivially generalizes by the reiteration theorem [2]. A relevant property
of these spaces proved in [30] is Fubini’s isometry Lp (M; `np ) = `np (Lp (M)). It
is also proved in [30] that Pisier’s identities hold. In other words, defining the
auxiliary index 1/r = |1/p − 1/q| we find
i) If p ≤ q, we have
n
o
¯
kxkpq = inf kαkL2r (M) kykLq (M;`nq ) kβkL2r (M) ¯ x = αyβ .
ii) If p ≥ q, we have
n
o
¯
kxkpq = sup kαyβkLq (M;`nq ) ¯ α, β ∈ BL2r (M) .
Remark 1.5. Let M be an hyperfinite von Neumann algebra and
£
¤
Ap (M; n) = Lp (M; `n∞ ), Lp (M; `n1 ) 1 ,
¤2
£
Bp (M; n) = Lp (M; Cpn ), Lp (M; Rpn ) 1 .
2
According to [46], we have
Ap (M; n) = Lp (M; OHn ) = Bp (M; n).
It is therefore natural to ask whether these identities remain valid with our definition
of Lp (M; OHn ) for non-hyperfinite M. This is indeed the case since, following the
same terminology already defined in the Introduction, we have
i) If 1 ≤ p ≤ 2 and 1/p = 1/2 + 1/q, we have
¡
¢
Ap (M; n) = L2q (M)`n2 L2 (M) L2q (M) = Bp (M; n).
ii) If 2 ≤ p ≤ ∞ and 1/2 = 1/p + 1/q, we have
√
Ap (M; n) = n Lp(2q,2q) (`n∞ (M), En ) = Bp (M; n),
where En is given by En :
Pn
k=1
xk ⊗ δk ∈ `n∞ (M) 7→
1
n
Pn
k=1
xk ∈ M.
The isometric identities for the Ap ’s follow from Pisier’s identities above while the
identities for the Bp ’s follow from [69] or Theorem 3.2 below. Thus, we agree to
define the o.s.s. on Lp (M; OHn ) as any of the interpolation spaces above.
Remark 1.6. As we shall see through this paper, all the spaces mentioned so
far arise as particular cases of our definition below of amalgamated and conditional
Lp spaces.
20
1. NONCOMMUTATIVE INTEGRATION
1.3. The spaces Lrp (M, E) and Lcp (M, E)
The spaces Lrp (M, E) and Lcp (M, E) were introduced in [16], where they turned
out to be quite useful in the context of noncommutative probability. Both spaces
will play a central role in this paper since they are the most significant examples
of the so-called amalgamated and conditional Lp spaces, to be defined below. Let
M be a von Neumann algebra equipped with a n.f. state ϕ and let N be a von
Neumann subalgebra of M. Let E : M → N denote the conditional expectation of
M onto N . Then, given 1 ≤ p ≤ ∞ and (α, a) ∈ N × M, we define
°
°
°
°
°αD p1 E(aa∗ )D p1 α∗ °1/2
°αD p1 a° r
=
,
L (M,E)
L
(N )
p/2
p
(1.1)
° 1 °
°aD p α° c
L (M,E)
p
°
°1/2
1
1
= °α∗ D p E(a∗ a)D p α°L
p/2 (N )
.
Lrp (M, E) and Lcp (M, E) will stand for the completions with respect to these norms.
Remark 1.7. Note that for 1 ≤ p < 2 we have 0 < p/2 < 1 so that we are
forced to use Haagerup Lp spaces in the definition (1.1). On the other hand, let
us note that the assumption x ∈ N D1/p M (resp. x ∈ MD1/p N ) is not needed to
define the norm of x in Lrp (M, E) (resp. Lcp (M, E)) for 2 ≤ p ≤ ∞. Indeed, given
x ∈ Lp (M) we can define
°
°1/2
°
°1/2
kxkLr (M,E) = °E(xx∗ )°
and kxkLc (M,E) = °E(x∗ x)°
.
p
Lp/2 (N )
Lp/2 (N )
p
In that case E(xx∗ ) and E(x∗ x) are well-defined and
n°
o
°
°
°
max °E(xx∗ )1/2 °L (N ) , °E(x∗ x)1/2 °L (N ) ≤ kxkLp (M) .
p
p
1/p
1/p
Hence, by the density of N D M and MD N in Lp (M), we obtain the same
closure as in (1.1). However, in the case 1 ≤ p < 2 the conditional expectation E is
no longer continuous on Lp/2 (M) so that we need this alternative definition.
The duality of Lrp (M, E) and Lcp (M, E) was studied in [16]. For the moment
we just need to know that, given 1 < p < ∞, the following isometries hold via the
anti-linear duality bracket hx, yi = tr(x∗ y)
(1.2)
Lrp (M, E)∗
Lcp (M, E)∗
=
=
Lrp0 (M, E),
Lcp0 (M, E).
Lemma 1.8. If 2 ≤ p ≤ ∞ and 1/2 = 1/p + 1/s, we have
n
o
¯
kxkLrp (M,E) = sup kαxkL2 (M) ¯ kαkLs (N ) ≤ 1 ,
n
o
¯
kxkLcp (M,E) = sup kxβkL2 (M) ¯ kβkLs (N ) ≤ 1 .
Proof. Given x ∈ Lrp (M, E), the operator E(xx∗ ) is positive. Hence
n ¡
o
¢¯
kxk2Lrp (M,E) = sup tr aE(xx∗ ) ¯ a ≥ 0, kakL(p/2)0 (N ) ≤ 1
n ¡
o
¢¯
= sup tr αE(xx∗ )α∗ ¯ kα∗ αkL(p/2)0 (N ) ≤ 1
n ¡
o
¢¯
= sup tr αxx∗ α∗ ¯ kα∗ αkL(p/2)0 (N ) ≤ 1
1.3. THE SPACES Lrp (M, E) AND Lcp (M, E)
21
n
o
¯
= sup kαxk2L2 (M) ¯ kαkL2(p/2)0 (N ) ≤ 1 .
Finally we recall that s = 2(p/2)0 . The proof for Lcp (M, E) is entirely similar.
¤
As usual, we shall write Lr2 (M) = B(L2 (M), C) and Lc2 (M) = B(C, L2 (M))
to denote the row and column Hilbert spaces over L2 (M). Both spaces embed
isometrically in B(L2 (M)). Hence, they admit a natural o.s.s. Given any index
2 ≤ p ≤ ∞, we generalize these spaces as follows. According to the definition of
Haagerup Lp spaces, we may consider the contractive inclusions
1
jr : x ∈ M →
7
jc : x ∈ M →
7
D 2 x ∈ L2 (M),
1
xD 2 ∈ L2 (M).
Then we define for 2 ≤ p ≤ ∞
£
¤
Lrp (M) = jr (M), Lr2 (M) 2
£
¤p
(1.3)
Lcp (M) = jc (M), Lc2 (M) 2
p
°
°
with kxk0 = °D−1/2 x°M ,
°
°
with kxk0 = °xD−1/2 ° .
M
Note that Lrp (M) = Lp (M) = Lcp (M) as Banach spaces.
Let N be a von Neumann subalgebra of M. Then, given 2 ≤ u, q, v ≤ ∞,
we consider the closed ideal I1 in the Haagerup tensor product Lru (N ) ⊗h Lcq (M)
generated by the differences xγ ⊗ y − x ⊗ γy, with x ∈ Lru (N ), y ∈ Lcq (M) and
γ ∈ N . Similarly, we consider the closed ideal I2 in Lrq (M) ⊗h Lcv (N ) generated
by the differences xγ ⊗ y − x ⊗ γy, with x ∈ Lrq (M), y ∈ Lcv (N ) and γ ∈ N . Then,
we define the amalgamated Haagerup tensor products as the quotients
±
Lru (N ) ⊗N ,h Lcq (M) = Lru (N ) ⊗h Lcq (M) I1 ,
±
Lrq (M) ⊗N ,h Lcv (N ) = Lrq (M) ⊗h Lcv (N ) I2 .
Let a1 , a2 , . . . , am ∈ Lu (N ) and b1 , b2 , . . . , bm ∈ Lq (M). Using that the Haagerup
tensor product commutes with complex interpolation, it is not difficult to check
that the following identities hold
m
m
°X
°
°¡ X
¢ °
°
°
°
∗ 1/2 °
ak ⊗ e1k °
=
a
a
,
°
°
°
k k
r
k=1
m
°X
°
°
k=1
Lu (N )⊗h Rm
°
°
ek1 ⊗ bk °
Cm ⊗h Lcq (M)
=
°¡
°
°
k=1
m
X
Lu (N )
b∗k bk
¢1/2 °
°
°
Lq (M)
k=1
.
P
P
Let us write Sr (a) = ( k ak a∗k )1/2 and Sc (b) = ( k b∗k bk )1/2 . Then, following the
definition of the Haagerup tensor product, the norm of x in Lru (N ) ⊗N ,h Lcq (M)
can be written as
m
n°
o
X
°
°
°
¯
°Sc (b)°
¯x'
kxku·q = inf °Sr (a)°
ak ⊗ bk ,
Lu (N )
Lq (M)
k=1
where ' means that the difference belongs to I1 . Similarly, given e1 , e2 , . . . , em in
Lrq (M) and f1 , f2 , . . . , fm in Lcv (N ), we can write the norm of an element x in the
space Lrq (M) ⊗N ,h Lcv (N ) as follows
m
n°
o
X
°
°
°
¯
kxkq·v = inf °Sr (e)°L (M) °Sc (f )°L (N ) ¯ x '
ek ⊗ f k .
q
v
k=1
22
1. NONCOMMUTATIVE INTEGRATION
Lemma 1.9. The following identities hold
kxku·q = inf kαkLu (N ) kykLq (M) ,
x=αy
kxkq·v = inf kykLq (M) kβkLv (N ) .
x=yβ
Proof. The upper estimates
kxku·q ≤ inf kαkLu (N ) kykLq (M) ,
x=αy
kxkq·v ≤ inf kykLq (M) kβkLv (N ) ,
x=yβ
are trivial. We only prove the converse for the first identity since the second identity
can be derived in the same way. Let us assume that kxku·q < 1. Then we can find
a1 , a2 , . . . , am ∈ Lu (N ) and b1 , b2 , . . . , bm ∈ Lq (M) such that
x'
m
X
ak ⊗ bk
and
°
°
°
°
°Sr (a)°
°Sc (b)°
< 1.
Lu (N )
Lq (M)
k=1
If D denotes the density associated to ϕ, we define
m
³X
´1/2
Ser (a) =
ak a∗k + δD2/u
with
δ > 0.
k=1
Since supp D = 1, Ser (a) is invertible and we can define αk by ak = Ser (a)αk so that
m
°¡ X
°
°
¢ °
°
∗ 1/2 °
°Sr (α)°
=
α
α
≤ 1.
°
°
i i
L∞ (N )
L∞ (N )
i=1
The amalgamation over N allows us to write
m
X
ak ⊗ bk ' Ser (a) ⊗
k=1
m
¡X
αk bk
k=1
for any possible decomposition of x. Then, we have
m
°X
°
°
°
°
°
°
°
αk bk °
≤ °Sr (α)°L (N ) °Sc (b)°L
°
k=1
q (M)
∞
Lq (M)
¢
°
°
≤ °Sc (b)°Lq (M) .
Moreover, applying the triangle inequality we obtain
m
°X
°
°
°
°
°2
°
2/u °
∗
°Ser (a)°2
=
a
a
+
δD
≤ °Sr (a)°L
°
°
k
k
Lu (N )
Lu/2 (N )
k=1
In summary, we have
inf kαkLu (N ) kykLq (M) ≤
x=αy
This holds for any decomposition x '
u (N )
+δ
q°
°
°
°
°Sr (a)°2 + δ °Sc (b)°
.
u
Lq (M)
P
k
ak ⊗ bk . We conclude letting δ → 0.
¤
Lemma 1.10. Let 1 ≤ p ≤ ∞ and let M be a von Neumann algebra. Assume
that α, β are elements of Lp (M), β is positive and αα∗ ≤ Cβ 2 . Then, there exists
w ∈ M such that the identity α = βw holds.
1.3. THE SPACES Lrp (M, E) AND Lcp (M, E)
23
Proof. We first construct w in the crossed product von Neumann algebra
M oσ R. Indeed, multiplying at both sides of αα∗ ≤ Cβ 2 by 1(λ,∞) (β)β −1 for each
positive λ, we conclude that β −1 αα∗ β −1 ≤ C and hence β −1 α is in M oσ R. Now,
using the dual automorphism group, we have
σ̂t (β −1 α) = σ̂t (β −1 )σ̂t (α) = σ̂t (β)−1 σ̂t (α) = et/p β −1 e−t/p α = β −1 α,
which means that β −1 α ∈ M. This completes the proof.
¤
Proposition 1.11. Let 2 ≤ p < ∞ and 2 < s ≤ ∞ related by 1/2 = 1/p + 1/s.
Then, we have the following isometries
Lrp0 (M, E) = Lrp (M, E)∗ = Lrs (N ) ⊗N ,h Lc2 (M),
Lcp0 (M, E) = Lcp (M, E)∗ = Lr2 (M) ⊗N ,h Lcs (N ).
Proof. As we have already said, the isometries
Lrp0 (M, E) = Lrp (M, E)∗ ,
Lcp0 (M, E) = Lcp (M, E)∗ ,
were proved in [16]. Now let us consider an element x ∈ Lrs (N ) ⊗N ,h Lc2 (M) and
let x = αy be a decomposition with α ∈ Ls (N ) and y ∈ L2 (M). Then, (1.2) and
Lemma 1.8 provide the following inequality
n ¡
o
¢¯
kxkLr 0 (M,E) = sup tr α∗ zy ∗ ¯
sup
kγzkL2 (M) ≤ 1 ≤ kαkLs (N ) kykL2 (M) .
p
kγkLs (N ) ≤1
According to Lemma 1.9, this proves that
id : x ∈ Lrs (N ) ⊗N ,h Lc2 (M) 7→ x ∈ Lrp0 (M, E)
0
is a contraction. Reciprocally, let us consider an element of the form x = ξD1/p a
with ξ ∈ N and a ∈ M. Then, given any δ > 0, we consider the decomposition
x = αδ yδ with αδ and yδ given by
³
´γ/2
1
1
αδ = ξD p0 E(aa∗ + δ1)D p0 ξ ∗
and
yδ = αδ−1 x
with
p0
sγ
=
.
2
2
Let us note that yδ is a well-defined element of L2 (M). Indeed, if we set
³
´1/2
1
1
,
Aδ = ξD p0 E(aa∗ + δ1)D p0 ξ ∗
1−γ =
it is clear that
xx∗ ≤ δ −1 kak2M A2δ .
By Lemma 1.10, we know that x = Aδ w for some w ∈ M. In particular, this gives
yδ = A1−γ
w and since 1 − γ = p0 /2 we obtain an element in L2 (M). Let us now
δ
estimate the norms of αδ and yδ . We have
°
°γ/2
°
°
1
1
1
1
°ξD p0 E(aa∗ + δ1)D p0 ξ ∗ °γ/2
kαδ kLs (N ) = °ξD p0 E(aa∗ + δ1)D p0 ξ ∗ °L
=
(N )
L 0 (N )
sγ/2
and
kyδ kL2 (M)
=
=
³
£
¤−γ ´1/2
1
1
tr xx∗ ξD p0 E(aa∗ + δ1)D p0 ξ ∗
³
£
¤−γ ´1/2
1
1
1
1
tr ξD p0 E(aa∗ )D p0 ξ ∗ ξD p0 E(aa∗ + δ1)D p0 ξ ∗
p /2
24
1. NONCOMMUTATIVE INTEGRATION
≤
In particular,
³
£
¤−γ ´1/2
1
1
1
1
tr ξD p0 E(aa∗ + δ1)D p0 ξ ∗ ξD p0 E(aa∗ + δ1)D p0 ξ ∗
.
°
°1/2−γ/2
1
1
kyδ kL2 (M) ≤ °ξD p0 E(aa∗ + δ1)D p0 ξ ∗ °L 0 (N ) .
p /2
We finally conclude
inf kαkLs (N ) kykL2 (M) ≤ lim kαδ kLs (N ) kyδ kL2 (M) ≤ kxkLrp0 (M,E) .
x=αy
δ→0
Note that the case p = 2 degenerates since we obtain γ = 0 and s = ∞. However,
this is a trivial case since it suffices to consider the decomposition given by α = 1
and y = x. In summary, we have seen that the norms of Lrs (N ) ⊗N ,h Lc2 (M) and
of Lrp0 (M, E) coincide on
0
Ap0 = N D1/p M.
Moreover, Ap0 = N D1/s D1/2 M is clearly dense in Lrs (N ) ⊗N ,h Lc2 (M). Therefore,
since the density of Ap0 in Lrp0 (M, E) follows by definition, we obtain the desired
isometry. The arguments for the column case are exactly the same.
¤
Remark 1.12. Proposition 1.11 provides an o.s.s. for Lrp (M, E) and Lcp (M, E)
when 1 < p ≤ 2. Moreover, by anti-linear duality we also obtain a natural o.s.s.
for Lrp (M, E) and Lcp (M, E) when 2 ≤ p < ∞. However, we shall be interested only
on the Banach space structure of these spaces rather than the operator space one.
Therefore, we shall use the simpler notation
Lu (N )Lq (M) and
to denote the underlying Banach space of
Lq (M)Lv (N )
Lru (N )⊗N ,h Lcq (M)
or Lrq (M)⊗N ,h Lcv (N ).
We conclude by giving one more characterization of the norm of Lrp (M, E) and
for 2 ≤ p ≤ ∞. To that aim, we shall denote by LmN (X1 , X2 ) and
RmN (X1 , X2 ) the spaces of left and right N -module maps between X1 and X2 .
Lcp (M, E)
Proposition 1.13. Let N be a von Neumann subalgebra of M. Then, given
any three indices 2 ≤ u, q, v < ∞, we have the following isometric isomorphisms
¡
¢∗
0
0
¡Lu (N )Lq (M)¢∗ = RmN (Lu (N ), Lq (M)) = LmN (Lq (M), Lu (N )),
Lq (M)Lv (N )
= LmN (Lv (N ), Lq0 (M)) = RmN (Lq (M), Lv0 (N )).
Proof. The arguments we shall be using hold for both isometries. Hence, we
only prove the first one. Let us consider a linear functional Φ : Lu (N )Lq (M) → C
and let Ψ denote the associated bilinear map
Ψ : Lu (N ) × Lq (M) → C,
defined by Ψ(α, x) = Φ(α ⊗ x). Clearly, we have
n¯
o
¯¯
kΨk = sup ¯Φ(α ⊗ x)¯ ¯ kαkLu (N ) , kxkLq (M) ≤ 1 = kΦk.
On the other hand, we use the isometry
B(Lu (N ) × Lq (M), C) = B(Lu (N ), Lq0 (M))
¡
¢
defined by Ψ(α, x) = tr T(α)x . By the amalgamation over N , any such linear
map T is a right N -module map (i.e. T(αβ) = T(α)β for all β ∈ N ). Indeed, given
β ∈ N , we have
¡
¢
¡
¢
tr T(αβ)x = Φ(αβ ⊗ x) = Φ(α ⊗ βx) = tr T(α)βx
1.3. THE SPACES Lrp (M, E) AND Lcp (M, E)
25
for all x ∈ Lq (M), which implies our claim. Reciprocally, if T : Lu (N ) → Lq0 (M) is
a right N -module map, we know by the same argument that T arises from a bilinear
map Ψ satisfying Ψ(αβ, x) = Ψ(α, βx), which corresponds to a linear functional Φ
on Lu (N )Lq (M) with the same norm. This proves the first identity. For the second
we proceed in a similar way by using the adjoint map T∗ instead of T.
¤
Remark 1.14. Note that the proof given above still works when we only require
one of the two indices to be finite. In particular, we cover all combinations appearing
for Lrp (M, E) and Lcp (M, E) since in that case the M-index is always 2.
CHAPTER 2
Amalgamated Lp spaces
Let M be a von Neumann algebra equipped with a n.f. state ϕ and let us
consider a von Neumann subalgebra N of M. In what follows we shall work with
indices (u, q, v) satisfying the following property
(2.1)
1≤q≤∞
and 2 ≤ u, v ≤ ∞
and
1
1 1 1
+ + = ≤ 1.
u q
v
p
Let us define Nu Lq (M)Nv to be the space Lq (M) equipped with
o
n° 1 °
°
¯
°
¯ α, β ∈ N , y ∈ Lq (M) .
°D u α°
°βD v1 °
|||x|||u·q·v = inf
kyk
L
(M)
q
Lv (N )
Lu (N )
x=αyβ
In the sequel it will be quite useful to have a geometric representation of the
indices (u, q, v) satisfying property (2.1). To that aim, we consider the variables
(1/u, 1/v, 1/q) in the Euclidean space R3 . Then, we note that the points satisfying
(2.1) are given by the intersection of the simplex 0 ≤ 1/u + 1/v + 1/q ≤ 1 with
the prism 0 ≤ min(1/u, 1/v) ≤ max(1/u, 1/v) ≤ 1/2. This gives rise to a solid K
sketched below.
1/q
A
B
C
0
E
D
G
F
1/v
K
0 = (0, 0, 0)
D = ( 21 , 0, 12 )
A = (0, 0, 1)
E = (0, 12 , 0)
B = (0, 0, 12 )
F = ( 21 , 12 , 0)
C = (0, 12 , 12 )
G = ( 12 , 0, 0)
1/u
Figure I: The solid K.
Now, let (u, q, v) be any three indices satisfying property (2.1). Then we define
the amalgamated Lp space Lu (N )Lq (M)Lv (N ) as the set of elements x ∈ Lp (M),
with 1/p = 1/u + 1/q + 1/v, such that there exists a factorization x = ayb with
a ∈ Lu (N ), y ∈ Lq (M) and b ∈ Lv (N ). For any such element x, we define
n
o
¯
kxku·q·v = inf kakLu (N ) kykLq (M) kbkLv (N ) ¯ x = ayb .
Chapters 2, 3 and 4 are devoted to prove that Lu (N )Lq (M)Lv (N ) is a Banach space
for any (u, q, v) satisfying (2.1) and to study the complex interpolation and duality
of such spaces. In the process, it will be quite relevant to know that the spaces
Nu Lq (M)Nv embed isometrically in Lu (N )Lq (M)Lv (N ) as dense subspaces. This
will be essentially the aim of this chapter.
27
28
2. AMALGAMATED Lp SPACES
Remark 2.1. In order to justify the restriction 2 ≤ u, v ≤ ∞ on (2.1), let us
show how the triangle inequality might fail when this restriction is not satisfied.
Let us take M = `∞ (N ), then we claim that
Lw (N )L∞ (M)L∞ (N ) and
L∞ (N )L∞ (M)Lw (N )
are not normed for 1 ≤ w < 2. Indeed, given a sequence x = (xn )n≥1 in L∞ (M)
and elements a, b ∈ Lw (N ), we have by definition the following inequalities
kaxkw·∞·∞ ≤ kakLw (N ) kxkL∞ (M)
and
kxbk∞·∞·w ≤ kxkL∞ (M) kbkLw (N ) .
In particular, if these spaces were normed they should contain contractively the
projective tensor product Lw (N ) ⊗π L∞ (M). However, by the noncommutative
Menchoff-Rademacher inequality in [7], we may find (zn ) ∈ Lw (N ) ⊗π L∞ (M) of
the form
n
X
xn
zn =
εk
with xk ∈ Lw (N )
1 + log k
k=1
and such that (zn ) ∈
/ Lw (N )L∞ (M)L∞ (N ). Moreover, taking adjoints we also may
find a sequence (zn ) such that (zn ) ∈ Lw (N ) ⊗π L∞ (M) \ L∞ (N )L∞ (M)Lw (N ).
Example 2.2. As noted in the Introduction, several noncommutative function
spaces arise as particular cases of our notion of amalgamated Lp space. Let us
mention four particularly relevant examples:
(a) The noncommutative Lp spaces arise as
Lq (M) = L∞ (N )Lq (M)L∞ (N ).
Note that these spaces are represented by the segment 0A in Figure I.
(b) If p ≤ q and 1/r = 1/p − 1/q, the spaces Lp (N1 ; Lq (N2 )) arise as
¯ 2 )L2r (N1 ).
L2r (N1 )Lq (N1 ⊗N
When the index p is fixed, these spaces are represented in Figure I as
segments parallel to the upper face ACDF whose projection into the plane
xy goes in the direction of the diagonal.
(c) By Proposition 1.11, Lrp (M, E) and Lcp (M, E) (1 ≤ p ≤ 2) arise as
Lrp (M, E) =
Ls (N )L2 (M)L∞ (N )
with
1/p = 1/2 + 1/s,
Lcp (M, E)
L∞ (N )L2 (M)Ls (N )
with
1/p = 1/2 + 1/s.
=
These spaces are represented by the segments BD and BC in Figure I. As
we shall see in Chapter 4, the spaces Lp (M; Rpn ) and Lp (M; Cpn ) arise as
particular cases of the latter ones.
(d) If M = N , the so-called asymmetric noncommutative Lp spaces arise as
L(u,v) (M) = Lu (M)L∞ (M)Lv (M).
These spaces are represented by the square 0EFG in Figure I. The reader
is refereed to [22] and Chapter 7 below for a detailed exposition of the
main properties of these spaces and their vector-valued analogues.
2.1. HAAGERUP’S CONSTRUCTION
29
2.1. Haagerup’s construction
We now briefly sketch a well-known unpublished result due to Haagerup [12]
which will be essential for our further purposes. Let us consider a σ-finite von
Neumann algebra M equipped with a distinguished n.f. state ϕ and let us define
the discrete multiplicative group
[
G=
2−n Z.
n∈N
Then we can construct the crossed product R = M oσ G in the same way we
constructed the crossed product Moσ R. That is, if H is the Hilbert space provided
by the GNS construction applied to ϕ, R is generated by π : M → B(L2 (G; H))
and λ : G → B(L2 (G; H)) where
¡
¢
¡
¢
π(x)ξ (g) = σ−g (x)ξ(g)
and
λ(h)ξ (g) = ξ(g − h).
By the faithfulness of π we are
P allowed to identify M with π(M). Then, a generic
element in R has the form g xg λ(g) with xg ∈ M and we have the conditional
expectation
X
EM :
xg λ(g) ∈ R 7→ x0 ∈ M.
g∈G
This gives rise to the state
³X
´
X
ϕ
b:
xg λ(g) ∈ R 7→ ϕ ◦ EM
xg λ(g) = ϕ(x0 ) ∈ C.
g∈G
g∈G
According to [12], R is the closure of a union of finite von Neumann algebras
[
Mk
k≥1
where (Mk )k≥1 is directed by inclusion M1 ⊂ M2 ⊂ . . . and each Mk satisfies
(2.2)
c1 (k)1Mk ≤ Dϕk ≤ c2 (k)1Mk
for some constants 0 < c1 (k) ≤ c2 (k) < ∞. Here ϕk denotes the restriction of ϕ
b to
Mk and Dϕk stands for the corresponding density. Moreover, it also follows from
[12] that we can find for any integer k ≥ 1 conditional expectations
E k : R → Mk
such that the following limit holds for every x̂ ∈ R, any 1 ≤ p < ∞ and all 0 ≤ η ≤ 1
°
¢ η/p °
° (1−η)/p ¡
°
(2.3)
x̂ − Ek (x̂) Dϕb °
−→ 0
as
k → ∞.
°Dϕb
Lp (R)
Remark 2.3. The σ-finiteness assumption might be dropped if we replaced
sequences of finite von Neumann algebras by nets. However, it suffices for our aims
to consider the σ-finite case.
In the following result we provide the first application of this construction. As
usual, we write S for the strip of complex numbers z ∈ C with 0 < Re(z) < 1 where
we consider the decomposition ∂S = ∂0 ∪ ∂1 of its boundary into the sets
n
o
n
o
¯
¯
∂0 = z ∈ C ¯ Re(z) = 0
and ∂1 = z ∈ C ¯ Re(z) = 1 .
30
2. AMALGAMATED Lp SPACES
Let X be a Banach space and let x be an element of X. Then, given 0 < θ < 1, we
write A(X, θ, x) for the set of bounded analytic functions f : S → X (i.e. bounded
and continuous on S ∪ ∂S and analytic on S) which satisfy f (θ) = x.
Lemma 2.4. Let x be an element in a von Neumann algebra M. Then, given
0 ≤ η ≤ 1 and 1/pθ = (1 − θ)/p for some 0 < θ < 1 and some 1 ≤ p < ∞, we have
½
n
o¾
° 1−η
° 1−η
°
°
η °
η°
°D p f (z)D p °
°f (z)°
°D pθ xD pθ °
=
inf
max
sup
,
sup
,
L∞ (M)
Lp (M)
Lp (M)
θ
z∈∂0
z∈∂1
where the infimum runs over all analytic functions f : S → M in the set A(M, θ, x).
Proof. The upper estimate follows by Kosaki’s interpolation. To prove the
lower estimate, we assume by homogeneity that the norm on the left is 1. Then we
begin with the case where M is a finite von Neumann algebra equipped with a n.f.
state ϕ0 such that the density Dϕ0 satisfies (2.2). That is,
c1 1M ≤ Dϕ0 ≤ c2 1M
for some positive numbers 0 < c1 ≤ c2 < ∞. Note that our assumption implies
Dϕ0 ∈ M. In particular, the function z → Dλz
ϕ0 is analytic for any scalar λ ∈ C.
Then it is easy to find an optimal function. Indeed, since
θ
θ
x(η, θ) = D(1−η)/p
x Dη/p
∈M
ϕ0
ϕ0
we find by polar decomposition a partial isometry ω such that x(η, θ) = ω|x(η, θ)|.
Our optimal function is defined as
−
(1−η)(1−z)
p
f (z) = Dϕ0
ω |x(η, θ)|
pθ (1−z)
p
−
η(1−z)
p
Dϕ 0
.
Our assumption on Dϕ0 implies the boundedness and analyticity of f . On the other
hand, it is easy to check that f (θ) = x so that f ∈ A(M, θ, x). Then, recalling that
both |x(η, θ)|iλ and Diλ
ϕ0 are unitaries for any λ ∈ R, we have
°
° 1−η
° (1−η)z
η
ηz °
pθ
pθ z
°
°
°
°
sup °Dϕ0p f (z)Dϕp0 °
= sup °Dϕ0 p ω|x(η, θ)| p |x(η, θ)|− p Dϕp0 °
Lp (M)
z∈∂0
Lp (M)
z∈∂0
≤
°
°
pθ
°
°
°|x(η, θ)| p °
Lp (M)
° 1−η
η ° 1
° p
p ° 1−θ
= °Dϕ0θ xDϕθ0 °
Lpθ (M)
Similarly, we have on ∂1
° i(1−η)Imz
iηImz °
°
°
ipθ Imz
°
°
ω|x(η, θ)|− p Dϕ0p °
sup °f (z)°L (M) = sup °Dϕ0 p
∞
z∈∂1
L∞ (M)
z∈∂1
= 1.
≤ 1.
This completes the proof for finite von Neumann algebras satisfying (2.2). In
the case of a general von Neumann algebra M we use the Haagerup construction
sketched above. Let us introduce the shorter notation
° 1−η
η °
N1 (M, ϕ, x) = °D pθ x D pθ °L (M) ,
pθ
½
n
o¾
° 1−η
°
°
η°
°
°
°
°
p
p
f (z)D L (M) , sup f (z) L (M)
N2 (M, ϕ, x) = inf max sup D
.
p
∞
z∈∂0
z∈∂1
We are interested in proving N1 (M, ϕ, x) ≥ N2 (M, ϕ, x). Combining property (2.2)
with the first part of this proof, we deduce that any x ∈ M satisfies the following
identities
(2.4)
N1 (Mk , ϕk , Ek (x)) = N2 (Mk , ϕk , Ek (x))
for all
k ≥ 1.
2.2. TRIANGLE INEQUALITY ON ∂∞ K
31
By the triangle inequality
N2 (M, ϕ, x) ≤ lim sup N2 (M, ϕ, EM (Ek (x))) + N2 (M, ϕ, x − EM (Ek (x))).
k→∞
Applying the contractivity of EM , (2.3) and (2.4)
lim sup N2 (M, ϕ, EM (Ek (x))) ≤ lim sup N2 (Mk , ϕk , Ek (x))
k→∞
k→∞
= lim sup N1 (Mk , ϕk , Ek (x))
k→∞
= N1 (M, ϕ, x)
It remains to see that N2 (M, ϕ, x − EM (Ek (x))) is arbitrary small. First we note
that
N2 (M, ϕ, x − EM (Ek (x))) = N2 (M, ϕ, EM (x − Ek (x))) ≤ N2 (R, ϕ,
b x − Ek (x)).
Then we consider the bounded analytic functions
¡
¢
z
fk : z ∈ S 7→ m1−z
0k m1k x − Ek (x) ∈ R
with the constants m0k , m1k defined by
°
¢ η/p °
° (1−η)/p ¡
°−1/2
m0k = °Dϕb
,
x − Ek (x) Dϕb °
Lp (R)
m1k
=
°
¢ η/p °
° (1−η)/p ¡
°(1−θ)/2θ
x − Ek (x) Dϕb °
.
°Dϕb
Lp (R)
¡
¢
θ
Note that fk ∈ A R, θ, x − Ek (x) since m1−θ
0k m1k = 1. On the other hand,
°
°
°
¢ η/p °
° (1−η)/p
° (1−η)/p ¡
°
η/p °
sup °Dϕb
fk (z)Dϕb °
= m0k °Dϕb
x − Ek (x) Dϕb °
.
Lp (R)
z∈∂0
Similarly, we have on ∂1
°
°
sup °fk (z)°
z∈∂1
L∞ (R)
Lp (R)
°
°
= m1k °x − Ek (x)°L
∞ (R)
≤ 2m1k kxkM .
Therefore, the proof is completed since from (2.3) both terms tend to 0 with k.
¤
2.2. Triangle inequality on ∂∞ K
Let ∂∞ K be the subset of the boundary of K given by the intersection of K with
the coordinate planes (i.e. the union of the plane regions 0ACE, 0ADG and 0EFG).
This set will appear repeatedly in what follows. Note that the indices (u, q, v) which
are represented in Figure I by the points of ∂∞ K are those satisfying (2.1) and
n
o
(2.5)
min 1/u, 1/q, 1/v = 0.
In the following result, we apply a complex interpolation trick based on the
operator-valued version of Szegö’s classical factorization theorem. This result is
due to Devinatz [8]. A precise statement of Devinatz’s theorem adapted to our
aims can be found in Pisier’s paper [44], where he also applies it in the context of
complex interpolation. We note in passing that a more general result (combining
previous results due to Devinatz, Helson & Lowdenslager, Sarason and Wiener &
Masani) can be found in the survey [52], see Corollary 8.2. This interpolation
technique will be used frequently in the sequel.
Lemma 2.5. If (1/u, 1/v, 1/q) ∈ ∂∞ K, Nu Lq (M)Nv is a normed space.
32
2. AMALGAMATED Lp SPACES
Proof. Assuming (2.1), Hölder inequality gives for 1/p = 1/u + 1/q + 1/v
° 1
°
°D u xD v1 °
≤ |||x|||u·q·v
for all
x ∈ Nu Lq (M)Nv .
Lp (M)
Therefore, |||x|||u·q·v = 0 implies x = 0. Since the homogeneity over R+ is clear,
it remains to show that the triangle inequality holds in this case. Our assumption
(1/u, 1/v, 1/q) ∈ ∂∞ K reduces the possibilities to those in which at least one of the
indices is infinite. When 1/q = 0, Pisier’s factorization argument (c.f. Lemma 3.5
in [46]) suffices to obtain the triangle inequality. Indeed, it can be checked that
this factorization provides the estimate
¡
¢
|||x1 + x2 |||u·q·v ≤ 21/q |||x1 |||u·q·v + |||x2 |||u·q·v .
Hence, it remains to consider the cases 1/u = 0 and 1/v = 0. Both can be treated
with the same arguments so that we only show the triangle inequality for 1/u = 0.
In that case, the left term N∞ is irrelevant in N∞ Lq (M)Nv so that we shall ignore
it in what follows. We need to consider two different cases.
Case I. We first assume that 1 ≤ q ≤ 2. Let x1 , x2 , . . . , xm be a finite sequence
of vectors in Lq (M)Nv satisfying |||xk |||q·v < 1 and
P let us also consider a finite
sequence of positive numbers λ1 , λ2 , . . . , λm with
k λk = 1. Then, it clearly
suffices to see that
m
¯¯¯ X
¯¯¯
¯¯¯
¯¯¯
λk xk ¯¯¯ ≤ 1.
¯¯¯
k=1
q·v
By hypothesis, we may assume that xk = yk βk , with
n
o
°
1°
max kyk kLq (M) , °βk D v °L (N ) < 1.
v
On the other hand, since 1 ≤ q ≤ 2 we can consider 2 ≤ q1 ≤ ∞ defined by
1
1
1
1 1
+ = = + .
q1
2
p
q
v
It is not difficult to check that
Lq (M) =
Lv (N ) =
£
¤
Lp (M), Lq1 (M) 2/v ,
£
¤
L∞ (N ) , L2 (N ) 2/v .
Then, by the complex interpolation method we can find bounded analytic functions
f1k : S → Lp (M) + Lq1 (M)
satisfying f1k (2/v) = yk for all k = 1, 2, . . . , m and
°
°
sup °f1k (z)°L (M) < 1,
p
z∈∂0 °
°
(2.6)
sup °f1k (z)°
< 1.
z∈∂1
Lq1 (M)
Similarly, Lemma 2.4 provides us with bounded analytic functions
f2k : S → N
satisfying f2k (2/v) = βk for all k = 1, 2, . . . , m and
°
°
sup °f2k (z)°L (N ) < 1,
∞
z∈∂0
°
1°
< 1.
sup °f2k (z)D 2 °
z∈∂1
L2 (N )
2.2. TRIANGLE INEQUALITY ON ∂∞ K
33
Given δ > 0, we define the following function on ∂S
½
1
if z ∈ ∂0 ,
P
W(z) =
δ1 + k λk f2k (z)∗ f2k (z) if z ∈ ∂1 .
According to Devinatz’s factorization theorem [44], there exists a bounded analytic
function w : S → N with bounded analytic inverse and satisfying the following
identity on ∂S
w(z)∗ w(z) = W(z).
Let us consider the bounded analytic function
m
³X
´
g(z) =
λk f1k (z)f2k (z) w−1 (z).
k=1
∗
Then, since w(z) w(z) ≡ 1 on ∂0 , we have
m
X
°
°
°
°
°
°
sup °g(z)°Lp (M) ≤ sup
λk °f1k (z)°Lp (M) °f2k (z)w−1 (z)°L∞ (N ) < 1.
z∈∂0
z∈∂0
k=1
On the other hand, we can write
g(z) =
m p
X
λk f1k (z)γk (z) with
p
λk f2k (z) = γk (z)w(z).
k=1
Note that, according to the definition of w, we have for any z ∈ ∂1
m
X
(2.7)
γk (z)∗ γk (z) ≤ 1.
k=1
Therefore, the following estimate follows from (2.6) and (2.7)
m
°³ X
´1/2 °
°
°
°
°
sup °g(z)°Lq (M) ≤ sup °
λk f1k (z)f1k (z)∗
°
z∈∂1
1
z∈∂1
×
°
sup °
z∈∂1
≤
k=1
m
°³ X
sup
´1/2 °
°
γk (z)∗ γk (z)
°
L∞ (N )
k=1
m
³X
z∈∂1
Lq1 (M)
°
°2
λk °f1k (z)°Lq
´1/2
1 (M)
k=1
< 1.
Note that the last estimate uses the triangle inequality on Lq1 /2 (M) and that we
are allowed to do so since q1 ≥ 2. Combining the estimates obtained so far and
applying Kosaki’s interpolation we obtain
°
°
°g(2/v)°
< 1.
Lq (M)
On the other hand, we have
°
°
sup °w(z)°L (N ) =
z∈∂0
∞
°
1 °2
sup °w(z)D 2 °L2 (N )
=
°
°1/2
sup °w(z)∗ w(z)°L (N ) = 1,
∞
z∈∂0
° 1
1°
sup °D 2 w(z)∗ w(z)D 2 °L1 (N )
z∈∂1
z∈∂1
≤
sup
z∈∂1
m
X
k=1
°
1 °2
λk °f2k (z)D 2 °L2 (N ) + δ < 1 + δ.
34
2. AMALGAMATED Lp SPACES
Again, Kosaki’s interpolation provides the estimate
°
°
°w(2/v)D v1 °
≤ 1 + δ.
Lv (N )
In summary, recalling that
m
m
X
X
λk f1k (2/v)f2k (2/v) = g(2/v)w(2/v),
λk xk =
k=1
k=1
we obtain from our previous estimates that
m
¯¯¯ X
¯¯¯
¯¯¯
¯¯¯
λk xk ¯¯¯ < 1 + δ.
¯¯¯
q·v
k=1
Thus, the triangle inequality follows by letting δ → 0 in the expression above.
Case II. It remains to consider the case 2 ≤ q ≤ ∞. This case is simpler. Indeed,
given any family of vectors x1 , x2 , . . . , xm and scalars λ1 , λ2 , . . . , λm as above (with
xk = yk βk and yk , βk satisfying the same inequalities), we define for δ > 0
m
³X
´1/2
Sβ =
λk βk∗ βk + δ1
so that
βk = bk Sβ ,
k=1
for some b1 , b2 , . . . , bm ∈ N . Then we have a factorization
m
m
³X
´
X
λk xk =
λk yk bk Sβ .
k=1
Now, since 2 ≤ q ≤ ∞, we
m
°X
°
°
°
λk yk bk °
≤
°
k=1
Lq (M)
≤
k=1
have triangle inequality
m
°³ X
´1/2 °
°
°
λk yk yk∗
°
°
in Lq/2 (M) and
m
°³ X
´1/2 °
°
°
λk b∗k bk
°
°
Lq (M)
k=1
m
³X
λk kyk k2Lq (M)
´1/2 °
° −1
°Sβ
k=1
k=1
m
³X
L∞ (N )
°1/2
´
°
λk βk∗ βk Sβ−1 °
L∞ (N )
k=1
,
which is clearly bounded by 1. On the other hand,
m
° 1³X
´ 1°
°
°
°
° v
∗
°Sβ D v1 °2
=
λ
β
β
+
δ1
Dv °
°D
k
k
k
Lv (N )
Lv/2 (N )
k=1
≤
m
X
°
1 °2
λk °βk D v °Lv (N ) + δ < 1 + δ.
k=1
Therefore, the triangle inequality follows one more time by letting δ → 0.
¤
The following will be a key point in the proof of Theorem 3.2, the main result
in Chapter 3. Note that we state it under the assumption that Nu Lq (M)Nv is a
normed space and that for the moment we only know it (from Lemma 2.5) whenever
(1/u, 1/v, 1/q) ∈ ∂∞ K. However, since eventually we shall need to apply this result
for any point in K, we state it in full generality.
Proposition 2.6. If Nu Lq (M)Nv is a normed space, we have
i) The following map is an isometry
1
1
ju,v : x ∈ Nu Lq (M)Nv 7→ D u xD v ∈ Lu (N )Lq (M)Lv (N ).
ii) Lu (N )Lq (M)Lv (N ) is a Banach space which completes Nu Lq (M)Nv .
2.2. TRIANGLE INEQUALITY ON ∂∞ K
35
Proof. It is clear that
kju,v (x)ku·q·v ≤ |||x|||u·q·v
for all
x ∈ Nu Lq (M)Nv .
Let us see that the reverse inequality holds. Assume it does not hold, then we can
find x0 ∈ Nu Lq (M)Nv such that |||x0 |||u·q·v > 1 and kju,v (x0 )ku·q·v < 1. By the
Hahn-Banach theorem (here we use the assumption that the space Nu Lq (M)Nv is
normed) there exists a norm one functional
ϕ : Nu Lq (M)Nv → C
satisfying
(a) ϕ(x0 ) = |||x0 |||u·q·v > 1.
¯ ¡
°
¢¯ ° 1 °
1°
(b) ¯ϕ αyβ ¯ ≤ °D u α°Lu (N ) kykLq (M) °βD v °Lv (N ) .
Note that any y ∈ Lq (M) provides a densely defined bilinear map
¡ 1
¡
¢
1¢
Φy : D u α, βD v ∈ Lu (N ) × Lv (N ) 7→ ϕ αyβ ∈ C,
which satisfies kΦy k ≤ kykq and
¡ 1
¡ 1
1¢
1¢
Φn1 yn2 D u α, βD v = Φy D u αn1 , n2 βD v
for all
n1 , n2 ∈ N .
On the other hand, since kju,v (x0 )ku·q·v < 1 we must have ju,v (x0 ) = a0 y0 b0 with
n
o
max ka0 kLu (N ) , ky0 kLq (M) , kb0 kLv (N ) < 1.
If we consider the invertible elements
¡
¢1/2
a = a0 a∗0 + δD2/u
and
¡
¢1/2
b = b∗0 b0 + δD2/v
,
we can write ju,v (x0 ) = aa−1 a0 y0 b0 b−1 b = ayb. Moreover, for δ > 0 small enough
n
o
max kakLu (N ) , kykLq (M) , kbkLv (N ) < 1.
Since a2 ≥ δD2/u and b2 ≥ δD2/v , there exist bounded elements α, β ∈ N with
D1/u = aα
D1/v = βb.
and
Let us denote by e the left support of α and by f the right support of β. We
note that the right support of α and the left support of β is 1. Then, we use
polar decomposition to find strictly positive densities d1 ∈ eN e and d2 ∈ f N f and
partial isometries w1 and w2 such that
α = d1 w 1
Note that
w1∗ w1
= 1,
w1 w1∗
= e,
and
w2∗ w2
β = w2 d2 .
= f and w2 w2∗ = 1. Then we observe that
ayb = ju,v (x0 ) = D1/u x0 D1/v = aαx0 βb ⇒ y = αx0 β.
This yields
eyf = αx0 β = d1 w1 x0 w2 d2 .
In particular, taking spectral projections en = 1[ n1 ,n] (d1 ) and fn = 1[ n1 ,n] (d2 )
−1
∗
∗
∗
w1∗ en d−1
1 eyf d2 fn w2 = w1 en w1 x0 w2 fn w2 .
This implies that
¯
¯
¯ϕ(w1∗ en w1 x0 w2 fn w2∗ )¯
=
=
¯
¡
¢¯
¯Φw∗ en w1 x0 w2 fn w∗ D u1 , D v1 ¯
1
2
¯
¡ 1
1 ¢¯
¯Φ ∗ −1
u
v ¯
w en d eyf d−1 fn w∗ D , D
1
1
2
2
36
2. AMALGAMATED Lp SPACES
¯
¡
¢¯
¯Φ ∗ −1
¯
∗ ad1 w1 , w2 d2 b
w1 en d1 eyf d−1
2 fn w2
¯
¡
¢¯
= ¯Φeyf aen , fn b ¯
≤ kaen ku keyf kq kfn bkv
≤ kaku kykq kbkv < 1.
=
On the other hand, (w1∗ en w1 ) (resp. (w2 fn w2∗ )) converges to w1∗ w1 = 1 (resp.
w2 w2∗ = 1) strongly. By Lemma 2.3 in [16], this implies that (D1/u w1∗ en w1 ) (resp.
(w2 fn w2∗ D1/v )) converges to D1/u (resp. D1/v ) in the norm of Lu (N ) (resp. Lv (N )).
This combined with the continuity of Φx0 gives
¯
¯
¯
¡
¢¯
¯ϕ(x0 )¯ = lim ¯Φx D u1 w1∗ en w1 , w2 fn w2∗ D v1 ¯
0
n→∞
¯ ¡
¢¯
= lim ¯ϕ w1∗ en w1 x0 w2 fn w2∗ ¯ ≤ 1.
n→∞
This contradicts condition (a). Therefore, the map ju,v defines an isometry and
the proof of i) is completed. Next, we see that k ku·q·v is a norm ‘outside’ the
space Nu Lq (M)Nv . The homogeneity over R+ is clear and the positive definiteness
follows as in Lemma 2.5. Thus, it suffices to show that the triangle inequality
holds. As we did above, we begin by taking a family x1 , x2 , . . . , xm of elements in
Lu (N )Lq (M)Lv (N )Psatisfying kxk ku·q·v < 1 and a collection of positive numbers
λ1 , λ2 , . . . , λm with k λk = 1. By hypothesis, we may assume that xk = ak yk bk
with
n
o
max kak kLu (N ) , kyk kLq (M) , kbk kLv (N ) < 1.
1
1
Given any ξ > 1 and by the density of D u N (resp. N D v ) in Lu (N ) (resp. Lv (N )),
it is not difficult to check that both ak and bk can be written in the following way
ak
=
∞
X
1
D u aik
with
° 1
°
°D u aik °
L
≤ ξ −i ,
with
°
°
°bjk D v1 °
L
≤ ξ −j .
u (N )
i=0
bk
=
∞
X
1
bjk D v
j=0
v (N )
This gives rise to
m
X
∞
X
λk x k =
i,j=0
k=1
1
Du
m
³X
´ 1
λk aik yk bjk D v .
k=1
Assuming the triangle inequality on Nu Lq (M)Nv , we can write
1
Du
m
³X
´ 1
λk aik yk bjk D v = Aij Yij Bij
k=1
where
kAij kLu (N ) ≤ ξ −
i+j
4
,
kYij kLq (M) ≤ ξ −
i+j
2
,
kBij kLv (N ) ≤ ξ −
i+j
4
.
Then, we define for δ > 0
Sr (A) =
∞
³X
i,j=0
2
Aij A∗ij + δD u
´1/2
and
Sc (B) =
∞
³X
i,j=0
2
B∗ij Bij + δD v
´1/2
,
2.2. TRIANGLE INEQUALITY ON ∂∞ K
37
so that there exist αij , βij ∈ N given by Sr (A)αij = Aij and βij Sc (B) = Bij . Hence,
m
X
λk xk = Sr (A)
∞
³X
´
αij Yij βij Sc (B).
i,j=0
k=1
Moreover, we have
∞
³
´1/2 √
X
°
°
2
°Sr (A)°
≤
δ
+
≤ δ+
kA
k
ij
L
(N
)
u
Lu (N )
i,j=0
1
√ .
1 − 1/ ξ
The same estimate applies to Sc (B). The middle term satisfies
∞
°X
°
°
°
αij Yij βij °
°
q
i,j=0
≤
∞
∞
∞
°³ X
´ q1 °³ X
´ 12 °
´1 ° ³ X
°
°
°
∗
∗ 2°
βij °
kYij kqq
βij
αij αij
°
°
°
³
≤
∞
i,j=0
∞
X
kYij kqq
´ q1
³
≤
i,j=0
i,j=0
∞
i,j=0
´2/q
1
q/2
1 − 1/ξ
In summary,
m
°X
°
°
°
λk xk °
°
u·q·v
k=1
°
°
≤ °Sr (A)°L
∞
°X
°
°
°
α
Y
β
°
°
ij
ij
ij
u (N )
Lq (M)
i,j=0
°
°
°Sc (B)°
L
v (N )
−→ 1
as δ → 0 and ξ → ∞. This proves the triangle inequality. To prove completeness
we use again a geometric series argument. Let x1 , x2 , . . . be a countable family of
elements in Lu (N )Lq (M)Lv (N ) with kxk ku·q·v < 4−k for any integer k ≥ 1. That
is, we have xk = 4−k ak yk bk with
n
o
max kak kLu (N ) , kyk kLq (M) , kbk kLv (N ) < 1.
Then,Pby a well known characterization of completeness, it suffices to see that the
sum k xk belongs to Lu (N )Lq (M)Lv (N ). We use again the same factorization
trick
∞
∞
³X
´
X
xk = Sr (a)
2−k αk yk βk Sc (b)
k=1
k=1
where
Sr (a)
=
∞
³X
2
´1/2
2
´1/2
2−k ak a∗k + δD u
k=1
Sc (b)
=
∞
³X
2−k b∗k bk + δD v
and
Sr (a)αk
= 2−k/2 ak ,
and
βk Sc (b)
=
2−k/2 bk .
k=1
In particular, we just need to show that Sr (a) ∈ Lu (N ), Sc (b) ∈ Lv (N ) and the
middle term belongs to Lq (M). However, this follows again by applying the same
estimates as above, details are left to the reader. To conclude, we just need to
show that Nu Lq (M)Nv is dense in Lu (N )Lq (M)Lv (N ). However, this follows
easily from the density of D1/u N (resp. N D1/v ) in Lu (N ) (resp. Lv (N )) and the
triangle inequality proved above.
¤
38
2. AMALGAMATED Lp SPACES
2.3. A metric structure on the solid K
At this point, we are not able to prove that Nu Lq (M)Nv is a normed space for
any point (1/u, 1/v, 1/q) in the solid K. However, we need at least to know that
we have a metric space structure. In fact, we shall prove that ||| |||u·q·v is always a
γ-norm for some 0 < γ ≤ 1.
Lemma 2.7. If (1/u, 1/v, 1/q) ∈ K, there exists 0 < γ ≤ 1 such that
|||x1 + x2 |||γu·q·v ≤ |||x1 |||γu·q·v + |||x2 |||γu·q·v
for all
x1 , x2 ∈ Nu Lq (M)Nv .
Proof. According to Lemma 2.5, we can assume in what follows that
(1/u, 1/v, 1/q) ∈ K \ ∂∞ K.
As we did to prove the triangle inequality, let x1 , x2 , . . . , xm ∈ Nu Lq (M)Nv be a
sequence
of vectors satisfying |||xk |||u·q·v < 1 and let λ1 , λ2 , . . . , λm ∈ R+ with sum
P
λ
=
1. Then it suffices to show that
k k
m
¯¯¯ X
¯¯¯γ
¯¯¯
¯¯¯
λk xk ¯¯¯
¯¯¯
k=1
u·q·v
≤
m
X
λγk
for some
0 < γ ≤ 1.
k=1
By hypothesis, we may assume that xk = αk yk βk , with
n° 1 °
°
1°
max °D u αk °L (N ) , kyk kLq (M) , °βk D v °L
u
o
< 1.
v (N )
By Figure I we can always find 1 ≤ u1 , q0 , v1 ≤ ∞ and 0 < θ < 1 such that
(a) We have 1/q
¡ 0 = 1/u + ¢1/q ¡+ 1/v = 1/u1 +¢1/v¡1 .
¢
(b) We have 1/u, 1/v, 1/q = 0, 0, (1 − θ)/q0 + θ/u1 , θ/v1 , 0 .
Indeed, we first consider the plane P parallel to ACDF containing (1/u, 1/v, 1/q).
The point (0, 0, 1/q0 ) is the intersection of P with the segment 0A. Then, we
consider the line L passing through (0, 0, 1/q0 ) and (1/u, 1/v, 1/q). Then, the point
(1/u1 , 1/v1 , 0) is the intersection of L with the coordinate plane z = 0. Note that
the point (1/u1 , 1/v1 , 0) is not necessarily in K. However, according to (a) it always
satisfies 1/u1 + 1/v1 ≤ 1. Note also that, since we have excluded the points in
∂∞ K at the beginning of this proof, we can always assume that 0 < θ < 1. Then it
follows from (b) that
£
¤
Lu (N ) = L∞ (N ) , Lu1 (N ) θ ,
£
¤
Lq (M) = Lq0 (M), L∞ (M) θ ,
£
¤
Lv (N ) = L∞ (N ) , Lv1 (N ) θ .
By the complex interpolation method, we can find bounded analytic functions
f2k : S → Lq0 (M) + L∞ (M)
satisfying f2k (θ) = yk for k = 1, 2, . . . , m and such that
n
°
°
°
°
(2.8)
max sup °f2k (z)°L (M) , sup °f2k (z)°L
z∈∂0
q0
z∈∂1
∞ (M)
o
< 1.
Similarly, by Lemma 2.4 we also have bounded analytic functions
f1k , f3k : S → N
2.3. A METRIC STRUCTURE ON THE SOLID K
39
for any k = 1, 2, . . . , m satisfying (f1k (θ), f3k (θ)) = (αk , βk ) and
n°
o
°
°
°
sup max °f1k (z)°L∞ (N ) , °f3k (z)°L∞ (N ) < 1,
z∈∂
n°0 1
o
(2.9)
°
°
1 °
sup max °D u1 f1k (z)°Lu (N ) , °f3k (z)D v1 °Lv (N ) < 1.
1
z∈∂1
1
Given δ > 0, we consider the following functions on the boundary
½
1
if z ∈ ∂0 ,
P
W1 (z) =
δ1 + k λk f1k (z)f1k (z)∗ if z ∈ ∂1 ,
½
1
if z ∈ ∂0 ,
P
W3 (z) =
δ1 + k λk f3k (z)∗ f3k (z) if z ∈ ∂1 .
According to Devinatz’s factorization theorem [44], there exist bounded analytic
functions w1 , w3 : S → N with bounded analytic inverse and satisfying the following
identities on ∂S
w1 (z)w1 (z)∗ = W1 (z),
w3 (z)∗ w3 (z) = W3 (z).
Then we can write
m
m
h
³X
´
i
X
λk xk = w1 (θ) w1−1 (θ)
λk f1k (θ)f2k (θ)f3k (θ) w3−1 (θ) w3 (θ).
k=1
k=1
Let us estimate the norms of the three factors above. First we clearly have
°
°
°
°1/2
sup °w1 (z)°L (N ) = sup °w1 (z)w1 (z)∗ °L (N ) = 1,
∞
z∈∂0
°
°
sup °w3 (z)°
z∈∂0
∞
z∈∂0
L∞ (N )
°
°1/2
sup °w3 (z)∗ w3 (z)°L (N ) = 1.
∞
=
z∈∂0
On the other hand, since Lp (N ) is always a min(p, 1)-normed space
° u1
° 1
° 1
1 °u /2
1
sup °D u1 w1 (z)°L (N ) = sup °D u1 w1 (z)w1 (z)∗ D u1 °L
(N )
u1
z∈∂1
u1 /2
z∈∂1
 ³
´u1 /2
°
1
 Pm λ °
°D u1 f1k (z)°2
+
δ
, if u1 ≥ 2,
k
k=1
≤ sup
°Luu11 (N )
1
Pm
u1 /2 °
u
/2

1
u
°
°
z∈∂1
D 1 f1k (z) L (N ) + δ
, if u1 < 2.
k=1 λk
u1
(
)
m
X
u /2
< max (1 + δ)u1 /2 , δ u1 /2 +
λk 1
.
k=1
Similarly, we have
°
1 °v
1
sup °w3 (z)D v1 °L
v
z∈∂1
(
v1 /2
1 (N )
< max (1 + δ)
,δ
v1 /2
+
m
X
)
v /2
λk1
.
k=1
In summary, we obtain by Kosaki’s interpolation
)
(
m
³
´1/u1
X
√
° 1
°
u
/2
1
u
/2
1
°D u w1 (θ)°
1 + δ, δ
+
λk
,
< max
L (N )
u
°
°
°w3 (θ)D v1 °
L
v (N )
< max
(
√
k=1
m
³
´1/v1
X
v /2
1 + δ, δ v1 /2 +
λk1
k=1
)
.
40
2. AMALGAMATED Lp SPACES
As we have pointed out above, we have 1/u1 + 1/v1 ≤ 1. In particular, u1 /2 and
v1 /2 can not be simultaneously less than 1. Thus, at least one of the two sums
above must be less or equal than 1. This means that taking
n
o
γ = min 1, u1 /2, v1 /2 ,
we deduce
³° 1
°
lim °D u w1 (θ)°L
δ→0
m
m
´ ³X
´1/2γ ³ X
´1/γ
°
°
γ
γ
°w3 (θ)D v1 °
<
λ
≤
λ
.
k
k
(N
)
L
(N
)
u
v
k=1
k=1
In particular, it suffices to see that
m
°
°
³X
´
° −1
°
λk f1k (θ)f2k (θ)f3k (θ) w3−1 (θ)°
°w1 (θ)
Lq (M)
k=1
≤ 1.
Let us consider the bounded analytic function
m
³X
´
g(z) = w1−1 (z)
λk f1k (z)f2k (z)f3k (z) w3−1 (z).
k=1
∗
∗
Since w1 (z)w1 (z) = 1 = w3 (z) w3 (z) for all z ∈ ∂0 , we clearly have
m
°X
°
°
°
°
°
sup °g(z)°L (M) = sup °
λ
f
(z)f
(z)f
(z)
°
k 1k
2k
3k
q
0
z∈∂0
z∈∂0
≤
sup
z∈∂0
Lq0 (M)
k=1
m
X
°
°
λk °f1k (z)°L
∞ (N )
°
°
°f2k (z)°
Lq0 (M)
k=1
°
°
°f3k (z)°
L
∞ (N )
.
Then it follows from (2.8) and (2.9) that the expression above is bounded above by
1. On the other hand, since w1 and w3 are invertible, we can define the functions
h1k , h3k : S → N by the relations
p
p
λk f1k (z) = w1 (z)h1k (z) and
λk f3k (z) = h3k (z)w3 (z).
Then it is easy to check that for any z ∈ ∂1
m
X
(2.10)
h1k (z)h1k (z)∗ ≤ 1 and
k=1
m
X
h3k (z)∗ h3k (z) ≤ 1.
k=1
Therefore, we obtain from (2.8) and (2.10) that
m
°X
°
°
°
°
°
sup °g(z)°L∞ (M) = sup °
h1k (z)f2k (z)h3k (z)°
z∈∂1
z∈∂1
≤
z∈∂1
×
×
L∞ (M)
k=1
m
°³ X
´1/2 °
°
°
sup °
h1k (z)h1k (z)∗
°
k=1
°
°
sup max °f2k (z)°L
z∈∂1 1≤k≤m
m
°³ X
°
sup °
z∈∂1
k=1
L∞ (N )
∞ (M)
´1/2 °
°
h3k (z)∗ h3k (z)
°
L∞ (N )
< 1.
°
°
By Kosaki’s interpolation we have °g(θ)°Lq (M) ≤ 1. This completes the proof.
¤
2.3. A METRIC STRUCTURE ON THE SOLID K
41
Arguing as in the proof of Proposition 2.6 ii), we can see that
¡ any amalgamated
¢
space Lu (N )Lq (M)Lv (N ) is a γ-Banach space and that ju,v Nu Lq (M)Nv (with
ju,v as defined in Proposition 2.6) is a dense subspace. Indeed, we just need to
repeat the two given arguments (using geometric series) conveniently rewritten
with exponents γ everywhere. We leave the details to the reader. Let us state this
result for future reference.
Proposition 2.8. If (1/u, 1/v, 1/q) ∈ K, there exists 0 < γ ≤ 1 such that
i) Lu (N
¡ )Lq (M)Lv (N
¢ ) is a γ-Banach space.
ii) ju,v Nu Lq (M)Nv is a dense subspace of Lu (N )Lq (M)Lv (N ).
Observation 2.9. We do not claim at this moment that ju,v is an isometry.
CHAPTER 3
An interpolation theorem
In this chapter we prove that the solid K is an interpolation family on the
indices (u, q, v). Of course, we need to know a priori that Lu (N )Lq (M)Lv (N ) is
a Banach space for any (1/u, 1/v, 1/q) in the solid K. In fact, as we shall see the
proofs of both results depend on the proof of the other. Indeed, let us consider a
parameter 0 ≤ τ ≤ 1. According to the terminology employed in Figure I, let us
define Kτ to be the intersection of K with the plane Pτ which contains the point
(0, 0, τ ) and is parallel to the upper face ACDF of K. Roughly speaking, we first
prove that
n
o
¯
Kτ = Lu (N )Lq (M)Lv (N ) ¯ (1/u, 1/v, 1/q) ∈ Kτ
is an interpolation family on the indices (u, q, v) for any 0 ≤ τ ≤ 1 and with ending
points lying on ∂∞ K. Note that we already proved in Chapter 2 that the spaces
associated to the points in ∂∞ K are Banach spaces. Then, we use this result to
prove that every point in K corresponds to a Banach space and derive our main
interpolation theorem. More concretely, we first prove the following result.
Lemma 3.1. Let us assume that
i) (1/uj , 1/vj , 1/qj ) ∈ ∂∞ K for j = 0, 1.
ii) 1/u0 + 1/q0 + 1/v0 = 1/u1 + 1/q1 + 1/v1 .
Then Luθ (N )Lqθ (M)Lvθ (N ) is a Banach space isometrically isomorphic to
h
i
Xθ (M) = Lu0 (N )Lq0 (M)Lv0 (N ), Lu1 (N )Lq1 (M)Lv1 (N ) .
θ
We know from Lemma 2.5 and Proposition 2.6 that the interpolation pairs
considered in Lemma 3.1 are made of Banach spaces. After the proof of Lemma 3.1
we shall show that Lu (N )Lq (M)Lv (N ) is a Banach space for any (1/u, 1/v, 1/q)
in the solid K and we shall deduce our main result in this chapter.
Theorem 3.2. The amalgamated space Lu (N )Lq (M)Lv (N ) is a Banach space
for any (1/u, 1/v, 1/q) ∈ K. Moreover, if (1/uj , 1/vj , 1/qj ) ∈ K for j = 0, 1, the
space Luθ (N )Lqθ (M)Lvθ (N ) is isometrically isomorphic to
h
i
Xθ (M) = Lu0 (N )Lq0 (M)Lv0 (N ), Lu1 (N )Lq1 (M)Lv1 (N ) .
θ
Note that the pairs of Banach spaces considered in Lemma 3.1 and Theorem
3.2 are compatible for complex interpolation. Indeed, since the amalgamated space
Luj (N )Lqj (M)Lvj (N ) is continuously injected (by means of Hölder inequality) in
Lpj (M) for j = 0, 1 and 1/pj = 1/uj + 1/qj + 1/vj , any such pair lives in the sum
Lp0 (M) + Lp1 (M). The main difficulty which appears to prove Lemma 3.1 lies in
the fact that the intersection of amalgamated spaces is quite difficult to describe
in the general case. Therefore, any attempt to work on a dense subspace meets
43
44
3. AN INTERPOLATION THEOREM
this obstacle. We shall avoid this difficulty by proving this interpolation result in
the particular case of finite von Neumann algebras whose density is bounded above
and below. Under this assumption, we will be able to find a nice dense subspace to
work with. Then we shall use the Haagerup construction [12] sketched in Section
2.1 (suitably modified to work in the present context) to get the result in the general
case. After that, the proof of Theorem 3.2 follows easily by using similar techniques.
3.1. Finite von Neumann algebras
We begin by proving Lemma 3.1 for a finite von Neumann algebra M equipped
with a n.f. state ϕ with respect to which, the corresponding density D satisfies the
following property for some positive constants 0 < c1 ≤ c2 < ∞
(3.1)
c1 1 ≤ D ≤ c2 1.
Lemma 3.3. Assume that (1/uj , 1/vj , 1/qj ) ∈ ∂∞ K for j = 0, 1. Then, given
any 0 < θ < 1, the space Lq0 (M) ∩ Lq1 (M) is dense in
h
i
Xθ (M) = Lu0 (N )Lq0 (M)Lv0 (N ), Lu1 (N )Lq1 (M)Lv1 (N ) .
θ
Proof. Let us write ∆ for the intersection
Lu0 (N )Lq0 (M)Lv0 (N ) ∩ Lu1 (N )Lq1 (M)Lv1 (N ).
According to the complex interpolation method, we know that ∆ is dense in Xθ (M).
In particular, it suffices to show that we have density in ∆ with respect to the norm
of Xθ (M). Let x be an element in ∆ so that we have decompositions
x =
x =
a0 ỹ0 b0
a1 ỹ1 b1
where
aj ∈ Luj (N ), ỹj ∈ Lqj (M), bj ∈ Lvj (N ).
©
ª
Case I. Let us first assume that max u0 , u1 , v0 , v1 < ∞. Then, we can define
X ¡
¢uj /2
a =
aj a∗j
+ δD,
j=0,1
b =
X ¡
¢vj /2
+ δD.
b∗j bj
j=0,1
Note that, since 2/uj and 2/vj are in (0, 1], we have
)
¡ ∗ ¢uj /2
½
aj aj
≤ a
aj a∗j ≤
⇒
¡ ∗ ¢vj /2
b∗j bj ≤
bj bj
≤ b
a2/uj ,
b2/vj .
Therefore, if we define αj , βj by aj = a1/uj αj and bj = βj b1/vj , we have αj αj∗ ≤ 1
and βj∗ βj ≤ 1. Thus, αj , βj ∈ N for j = 0, 1 and we can set yj = αj ỹj βj in Lqj (M)
so that
(3.2)
x = a1/uj yj b1/vj
for j = 0, 1. Now we consider the spectral projections
en = 1[ n1 ,n] (a)
and fn = 1[ n1 ,n] (b).
Then, since en xfn = (en a1/uj )yj (b1/vj fn ) we find en xfn ∈ Lq0 (M) ∩ Lq1 (M).
Thus, we have to show that en xfn tends to x in the norm of Xθ (M). Let us write
x − en xfn = (1 − en )x + en x(1 − fn ).
3.1. FINITE VON NEUMANN ALGEBRAS
Then it is clear that
°
°
°(1 − en )x°
∆
45
n°
o
1
1 °
¯
max °(1 − en )a uj yj b vj °uj ·qj ·vj ¯ j = 0, 1
n°
o
1 °
° 1° ¯
≤ max °(1 − en )a uj °uj kyj kqj °b vj °vj ¯ j = 0, 1 .
=
The term on the right tends to 0 as n → ∞ since max(u0 , u1 ) < ∞. Therefore,
since the norm of Xθ (M) is controlled by the norm of ∆ when 0 < θ < 1, we obtain
that (1 − en )x tends to 0 in the norm of Xθ (M). The second term en x(1 − fn ) can
be estimated in the same way. This concludes the proof of Case I.
Case II. Now let us assume that there exists some indices among u0 , u1 , v0 , v1
which are infinite. Then we can assume without lost of generality that aj (resp.
bj ) is 1 whenever uj (resp. vj ) is infinite. According to this and assuming the
convention 1∞ = 1, the previous definition of a and b still makes sense. Moreover,
the relations obtained in (3.2) also hold in this case. Therefore, we need to prove
again that en xfn tends to x with respect to the norm of Xθ (M). We only prove
the convergence for the term (1 − en )x since again the second one can be estimated
in the same way. By the three lines lemma, we have
°
°
°
°
°
°
°(1 − en )x°
°(1 − en )x°1−θ
°(1 − en )x°θ
≤
.
X (M)
u0 ·q0 ·v0
u1 ·q1 ·v1
θ
Now, let us recall that both norms on the right hand side are uniformly bounded
on n ≥ 1. In particular, since 0 < θ < 1, it suffices to prove that en x tends to x
with respect to the norm of Luj (N )Lqj (M)Lvj (N ) for either j = 0 or j = 1. There
are only three possible situations:
(a) Assume min(u0 , u1 ) < ∞. Let us suppose (w.l.o.g.) that min(u0 , u1 ) =
u0 . Then, we apply the estimate
°
°
°
° 1°
1 °
°(1 − en )x°
°(1 − en )a u0 °
°b v0 °
≤
ky
k
.
0
L
(M)
q
u0 ·q0 ·v0
Lu (N )
Lv (N )
0
0
0
(b) Assume min(q0 , q1 ) < ∞. Let us suppose (w.l.o.g.) that min(q0 , q1 ) = q0 .
Then, we apply the estimate
°
°
° 1 °
° 1°
°(1 − en )x°
≤ °a u0 °L (N ) k(1 − en )y0 kLq0 (M) °b v0 °L (N ) .
u0 ·q0 ·v0
u
v
0
0
(c) Finally, assume that neither (a) nor (b) hold. Let us suppose (w.l.o.g.)
that min(v0 , v1 ) = v0 . In this case we have u0 = q0 = u1 = q1 = ∞
and ∆ = L∞ (N )L∞ (M)Lv1 (N ). Therefore, M (= Lq0 (M) ∩ Lq1 (M)) is
dense in ∆ (by Proposition 2.6) and thereby in Xθ (M).
¤
Remark 3.4. Note that the finiteness of M is used in the proof of Lemma 3.3
to ensure that en a1/uj and b1/vj fn are in N . In particular, in the case of general
von Neumann algebras we shall need to use a different approach.
Proof of Lemma 3.1. The lower estimate is easy. Indeed, let
T0 : Lu0 (N ) × Lq0 (M) × Lv0 (N ) → Lu0 (N )Lq0 (M)Lv0 (N ),
T1 : Lu1 (N ) × Lq1 (M) × Lv1 (N ) → Lu1 (N )Lq1 (M)Lv1 (N ),
be the multilinear maps given by T0 (a, y, b) = ayb = T1 (a, y, b). It is clear that
both T0 and T1 are contractive. Hence, the lower estimate follows by multilinear
interpolation. For the converse, we proceed in two steps.
46
3. AN INTERPOLATION THEOREM
Step 1. Let x be of the form
x = D1/uθ αyβD1/vθ
with
α, β ∈ N , y ∈ Lqθ (M).
Using the isometry j defined in Proposition 2.6, we have
³
´
x ∈ juθ ,vθ Nuθ Lqθ (M)Nvθ .
We claim that it suffices to see that
|||ju−1
(x)|||uθ ·qθ ·vθ ≤ kxkXθ (M)
θ ,vθ
(3.3)
for all x of the form considered above. Indeed, let us assume that inequality (3.3)
holds. Then, combining this inequality with the lower estimate proved above, we
obtain
(3.4)
|||ju−1
(x)|||uθ ·qθ ·vθ ≤ kxkXθ (M) ≤ kxkuθ ·qθ ·vθ ≤ |||ju−1
(x)|||uθ ·qθ ·vθ .
θ ,vθ
θ ,vθ
Note that, since Xθ (M) is a Banach space, we deduce that Nuθ Lqθ (M)Nvθ satisfies
the triangle inequality. Then, applying again Proposition 2.6 we deduce that the
space Luθ (N )Lqθ (M)Lvθ (N ) is the completion of Nuθ Lqθ (M)Nvθ . Therefore, it
follows that for any x ∈ Luθ (N )Lqθ (M)Lvθ (N ), we have
kxkuθ ·qθ ·vθ = kxkXθ (M) .
In particular, Luθ (N )Lqθ (M)Lvθ (N ) embeds isometrically in Xθ (M). To conclude,
it remains to see that both spaces are the same. However, according to Lemma 3.3,
we deduce that Luθ (N )Lqθ (M)Lvθ (N ) is norm dense in Xθ (M).
Step 2. Now we prove inequality (3.3). We use one more time the interpolation
trick based on Devinatz’s factorization theorem [8]. Again, we refer the reader to
Pisier’s paper [44] for a precise statement of Devinatz’s theorem adapted to our
aims. Let x be an element of the form
x = D1/uθ αyβD1/vθ
with
α, β ∈ N , y ∈ Lqθ (M).
Assume the norm of x in Xθ (M) is less than one. Then, the complex interpolation
method provides a bounded analytic function
f : S −→ Lu0 (N )Lq0 (M)Lv0 (N ) + Lu1 (N )Lq1 (M)Lv1 (N )
satisfying f (θ) = x and
(3.5)
sup kf (z)ku0 ·q0 ·v0
< 1,
sup kf (z)ku1 ·q1 ·v1
< 1.
z∈∂0
z∈∂1
On the other hand, we are assuming that the density D satisfies the boundedness
condition (3.1). In particular, z ∈ S 7→ Dλz ∈ M is a bounded analytic function
for any λ ∈ C. Therefore (multiplying if necessary on the left and on the right by
certain powers of Dz and its inverses), we may assume that f has the form
f (z) = D
1−z
u0
z
z
D u1 f1 (z)D v1 D
1−z
v0
,
where f1 : S → Lq0 (M) + Lq1 (M) is bounded analytic. Hence, we deduce from
the boundary conditions (3.5) and Proposition 2.6 that f can be written on the
boundary ∂S as follows
f (z) = D
1−z
u0
z
z
D u1 g1 (z)g2 (z)g3 (z)D v1 D
1−z
v0
,
3.1. FINITE VON NEUMANN ALGEBRAS
47
where g1 , g3 : ∂S → N and g2 : ∂j → Lqj (M) satisfy the following estimates
n° 1
o
°
°
1 °
sup max °D u0 g1 (z)°u0 , kg2 (z)kq0 , °g3 (z)D v0 °v0
< 1,
z∈∂0
n° 1
o
(3.6)
°
°
1 °
sup max °D u1 g1 (z)°u , kg2 (z)kq1 , °g3 (z)D v1 °v
< 1.
1
z∈∂1
1
Given any δ > 0, we define the following functions on the boundary
W1 : z ∈ ∂S
W3 : z ∈ ∂S
g1 (z)g1 (z)∗ + δ1 ∈ N ,
g3 (z)∗ g3 (z) + δ1 ∈ N .
7→
7→
According to Devinatz’s factorization theorem [44], we can find invertible bounded
analytic functions w1 , w3 : S → N with bounded analytic inverse and satisfying
the following identities on ∂S
w1 (z)w1 (z)∗
w3 (z)∗ w3 (z)
(3.7)
= W1 (z),
= W3 (z).
Then, we consider the factorization
f (z) = h1 (z)h2 (z)h3 (z)
with h2 (z) =
−1
h−1
1 (z)f (z)h3 (z)
and h1 , h3 given by
1−z
u0
z
z
=
D
h3 (z)
=
D v0 D− v1 w3 (z)D v1 D
z
z
D u1 w1 (z)D− u1 D u0 ,
h1 (z)
z
z
1−z
v0
.
Note than our original hypothesis (3.1) implies the boundedness and analyticity of
h1 , h2 , h3 . Then, recalling that Dω is a unitary for any ω ∈ C such that Re ω = 0
and that 2 ≤ uj , vj ≤ ∞ (so that we have triangle inequality on Luj /2 (N ) and
Lvj /2 (N )), we obtain from (3.6) the following estimates for h1 and h3
°
°2
° 1
1 °
sup °h1 (z)°L (N ) = sup °D u0 w1 (z)w1 (z)∗ D u0 °L
(N )
u
0
z∈∂0
≤
v0 (N )
=
°
1 °2
1
sup °h1 (z)D u1 − u0 °Lu
1 (N )
=
z∈∂1
v1 (N )
+ δ < 1 + δ,
° 1
1 °
sup °D v0 w3 (z)∗ w3 (z)D v0 °L
v0 /2 (N )
°
1 °2
sup °g3 (z)D v0 °
Lv0 (N )
z∈∂0
+ δ < 1 + δ,
° 1
1 °
sup °D u1 w1 (z)w1 (z)∗ D u1 °L
u1 /2 (N )
z∈∂1
≤
° 1 1
°2
sup °D v1 − v0 h3 (z)°L
Lu0 (N )
z∈∂0
≤
z∈∂1
° 1
°2
sup °D u0 g1 (z)°
z∈∂0
°
°2
sup °h3 (z)°L
z∈∂0
u0 /2
z∈∂0
°2
° 1
sup °D u1 g1 (z)°Lu
1 (N )
z∈∂1
=
° 1
1 °
sup °D v1 w3 (z)∗ w3 (z)D v1 °L
v1 /2 (N )
z∈∂1
≤
+ δ < 1 + δ,
°
1 °2
sup °g3 (z)D v1 °
Lv1 (N )
z∈∂1
By Kosaki’s interpolation, we obtain
°
θ
θ °
°h1 (θ)D u1 − u0 °
Luθ (N )
° θ−θ
°
°D v1 v0 h3 (θ)°
Lvθ (N )
<
<
√
√
1 + δ,
1 + δ.
+ δ < 1 + δ.
48
3. AN INTERPOLATION THEOREM
On the other hand, using again the unitarity of Dω when Re ω = 0, we have
°
°2
sup °h2 (z)°Lq (M)
z∈∂0
=
0
°
°2
sup °w−1 (z)g1 (z)g2 (z)g3 (z)w−1 (z)°
z∈∂0
1
3
Lq0 (M)
°
°
°
°
°g2 (z)°2
°g3 (z)w−1 (z)°2
≤
3
1
L∞ (N )
Lq0 (M)
L∞ (N )
z∈∂0
°
°
° −1
°
−1
−1
< sup °w1 (z)g1 (z)g1 (z)∗ w1 (z)∗ °L∞ (N ) °w3 (z)∗ g3 (z)∗ g3 (z)w3−1 (z)°L∞ (N ) ,
°
°2
sup °w−1 (z)g1 (z)°
z∈∂0
° 1 1
1
1 °2
sup °D u0 − u1 h2 (z)D v0 − v1 °L
z∈∂1
=
q1 (M)
°
°2
sup °w−1 (z)g1 (z)g2 (z)g3 (z)w−1 (z)°
z∈∂1
1
3
Lq1 (M)
°
°2
sup °w−1 (z)g1 (z)°
°
°
°
°
°g2 (z)°2
°g3 (z)w−1 (z)°2
≤
3
1
L∞ (N )
L∞ (N )
Lq1 (M)
z∈∂1
° −1
°
°
°
< sup °w1 (z)g1 (z)g1 (z)∗ w1−1 (z)∗ °L∞ (N ) °w3−1 (z)∗ g3 (z)∗ g3 (z)w3−1 (z)°L∞ (N ) .
z∈∂1
Then we combine (3.7) with the inequalities
g1 (z)g1 (z)∗ ≤ g1 (z)g1 (z)∗ + δ1,
g3 (z)∗ g3 (z) ≤ g3 (z)∗ g3 (z) + δ1,
to obtain by Kosaki’s interpolation
° θ−θ
θ
θ °
°D u0 u1 h2 (θ)D v0 − v1 °
L
qθ (M)
< 1.
In summary, we have obtained the following factorization
¡ 1
¢¡ θ θ
1 ¢
θ
θ ¢¡
f (θ) = D uθ w1 (θ) D u0 − u1 h2 (θ)D v0 − v1 w3 (θ)D vθ = j1 (θ)j2 (θ)j3 (θ),
with
kj1 (θ)kLuθ (N ) <
√
1 + δ,
kj2 (θ)kLqθ (M) < 1,
Therefore, (3.3) follows by letting δ → 0 in
¯¯¯ −1 ¡
¢¯¯¯
¯¯¯ju ,v f (θ) ¯¯¯
θ θ
u
θ ·qθ ·vθ
kj3 (θ)kLvθ (N ) <
√
1 + δ.
< 1 + δ.
This concludes the proof of Lemma 3.1 for finite von Neumann algebras.
¤
Remark 3.5. Note that condition ii) of Lemma 3.1 is not needed at any point
in the proof given above for finite von Neumann algebras. This will be crucial in
the proof of Theorem 3.2 below.
3.2. Conditional expectations on ∂∞ K
Before the proof of Lemma 3.1 for general von Neumann algebras, we need
some preliminary results in order to adapt Haagerup’s construction to the present
context. We begin with a technical lemma and some auxiliary interpolation results.
3.2. CONDITIONAL EXPECTATIONS ON ∂∞ K
49
Lemma 3.6. Let M be a von Neumann algebra equipped with a n.f. state ϕ
and let D be the associated density. Let us consider a bounded analytic function
f : S → M. Then, given 1 ≤ p < ∞, the following functions are also bounded
analytic
h1 : z ∈ S
7→
D(1−z)/p f (z)Dz/p ∈ Lp (M),
h2 : z ∈ S
7→
Dz/p f (z)D(1−z)/p ∈ Lp (M).
Proof. The arguments to be used hold for both h1 and h2 . Hence, we only
prove the assertion for h1 . The continuity and boundedness of h1 on the closure of
S is trivial. To prove the analyticity of h1 we may clearly assume that the function
f is a finite power series
f (z) =
n
X
xk z k
with
xk ∈ M.
k=1
In particular, it suffices to see that for a fixed element x0 ∈ M, the function
h0 : z ∈ S 7→ D(1−z)/p x0 Dz/p ∈ Lp (M)
is analytic. If x0 ∈ Ma is an analytic element this is clear. Assume that x0 is not
an analytic element. According to Pedersen-Takesaki [42], the net (xγ )γ>0 ⊂ Ma
given by
r Z
γ
xγ =
σt (x0 ) exp(−γt2 ) dt
π R
converges strongly to x0 as γ → ∞. Then Lemma 2.3 in [16] gives that
hγ (z) = D(1−z)/p xγ Dz/p
converges pointwise to h0 (z) in the norm of Lp (M). Now, let us consider a linear
functional ϕ : Lp (M) → C and a cycle Γ in S homologous to zero with respect to
S. Then, since kxγ kM ≤ kx0 kM for all γ > 0, the dominated convergence theorem
gives
Z
Z
ϕ(h0 (z))dz = lim
ϕ(hγ (z))dz = 0.
Γ
γ→∞
Γ
Thus, ϕ(h0 ) is analytic for any linear functional ϕ : Lp (M) → C and so is h0 .
¤
Lemma 3.7. If 2 ≤ u, v ≤ ∞, we have the following isometries
£
¤
Lu (N )L∞ (M), Lu (N )L2 (M) θ = Lu (N )L2/θ (M),
£
¤
L∞ (M)Lv (N ), L2 (M)Lv (N ) θ = L2/θ (M)Lv (N ).
Proof. We shall only prove that
£
¤
Xθ (M) = Lu (N )L∞ (M), Lu (N )L2 (M) θ = Lu (N )L2/θ (M).
The other isometry can be proved in the same way. The lower estimate follows by
multilinear interpolation just as in the proof of Lemma 3.1 for finite von Neumann
algebras. To prove the upper estimate we first note that, according to Proposition
2.6, we have a dense subset
¡
¢
D1/u N L2 (M) ∩ L∞ (M) ⊂ Lu (N )L2/θ (M).
50
3. AN INTERPOLATION THEOREM
Since this subset is also dense in Xθ (M) (note that the intersection in this case is
Lu (N )L∞ (M)) , it suffices to show that for any x of the form D1/u ay with a ∈ N
and y ∈ L2 (M) ∩ L∞ (M), we have the following inequality
(3.8)
kxku· θ2 ≤ kxkXθ (M) .
Assume that the norm of x in Xθ (M) is less than 1. Then we can find a bounded
analytic function f : S → Lu (N )L∞ (M) + Lu (N )L2 (M) satisfying f (θ) = x and
the inequalities
°
°
sup °f (z)°u·∞ < 1,
z∈∂0 °
°
(3.9)
sup °f (z)°u·2 < 1.
z∈∂1
Moreover, by our previous considerations we can assume that f has the form
1
f (z) = D u f1 (z),
where f1 : S → L2 (M) ∩ L∞ (M) is a bounded analytic function. Then we deduce
from the boundary conditions (3.9) and Proposition 2.6 that f can be written on
∂S as follows
1
f (z) = D u g1 (z)g2 (z),
with g1 : ∂S → N and g2 : ∂S → L2 (M) ∩ L∞ (M) satisfying
n
°
° 1
°
°
°
°
max sup °D u g1 (z)°Lu (N ) , sup °g2 (z)°L (M) , sup °g2 (z)°L
z∈∂S
z∈∂0
∞
z∈∂1
o
2 (M)
< 1.
Now we apply Devinatz’s theorem [44] to W : z ∈ ∂S → g1 (z)g1 (z)∗ + δ1, so that
we find an invertible bounded analytic function w : S → N satisfying ww∗ = W on
the boundary. Then we consider the factorization
¡ 1
¢¡
¢
f (z) = D u w(z) w−1 (z)f1 (z) .
Clearly both factors are bounded
°2
° 1
sup °D u w(z)°Lu (N ) =
z∈∂S
°
°
sup °w−1 (z)f1 (z)°L (M) ≤
∞
z∈∂0
° −1
°
sup °w (z)f1 (z)°L (M) ≤
2
z∈∂1
analytic and we have
° 1
1 °
sup °D u w(z)w(z)∗ D u °L (N ) < 1 + δ,
u/2
z∈∂S
° −1
°
°
°
°g2 (z)°
sup °w (z)g1 (z)°
< 1,
z∈∂0
L∞ (N )
L∞ (M)
°
°
°
°
sup °w−1 (z)g1 (z)°L (N ) °g2 (z)°L
∞
z∈∂1
2 (M)
< 1.
Finally, by Kosaki’s interpolation and letting δ → 0, we obtain inequality (3.8).
¤
Lemma 3.8. Assume 1/u + 1/2 = 1/p = 1/2 + 1/v and 1/qθ = (1 − θ)/p + θ/2.
Then, we have the following isometries
£
¤
Lp (M), Lu (N )L2 (M) θ = Lu/θ (N )Lqθ (M),
£
¤
Lp (M), L2 (M)Lv (N ) θ = Lqθ (M)Lv/θ (N ).
Proof. One more time, the lower estimate follows by multilinear interpolation
and we shall only prove the first isometry
£
¤
Xθ (M) = Lp (M), Lu (N )L2 (M) θ = Lu/θ (N )Lqθ (M).
Let us point out that Hölder inequality gives
∆ = Lp (M) ∩ Lu (N )L2 (M) = Lu (N )L2 (M).
3.2. CONDITIONAL EXPECTATIONS ON ∂∞ K
51
On the other hand, it follows from Lemma 1.2 that
D1/p Ma = D1/u Na Ma D1/2 = Dθ/u Na Ma D1/qθ .
Hence D1/p Ma is dense in ∆ and in Lu/θ (N )Lqθ (M) so that it suffices to see
kxk uθ ·qθ ≤ kxkXθ (M)
for any element x of the form D1/p y for some y ∈ Ma . Assume that the norm of x
in Xθ (M) is less than 1. Then, according to the considerations above, we can find
a bounded analytic function f : S → Lp (M) + Lu (N )L2 (M) of the form
f (z) = D1/p f1 (z) with
f1 : S → Ma
bounded analytic,
satisfying f (θ) = x and such that
n
°
°
°
° o
(3.10)
max sup °f (z)°p , sup °f (z)°u·2 < 1.
z∈∂0
z∈∂1
Moreover, since f1 takes values in Ma we can rewrite f as
z
f (z) = D u +
1−z
p
z
f2 (z)D 2
with
f2 : S → Ma
bounded analytic.
In particular, f|∂1 has the form
¡
¢ 1
1
f (1 + it) = D u σ−t/2 f2 (1 + it) D 2 .
According to Proposition 2.6 and the boundary estimate for f on ∂1 , we can write
1
f (1 + it) = D u g1 (1 + it)g2 (1 + it),
with g1 : ∂1 → N and g2 : ∂1 → L2 (M) satisfying
n
o
° 1
°
°
°
(3.11)
max sup °D u g1 (z)°Lu (N ) , sup °g2 (z)°L2 (M) < 1.
z∈∂1
z∈∂1
By Devinatz’s theorem, we can consider an invertible bounded analytic function
w : S → N satisfying w(z)w(z)∗ = W(z) for all z ∈ ∂S where this time the
function W : ∂S → N is given by
½
1,
if z ∈ ∂0 ,
¡
¢
W(z) =
σ−Imz/u g1 (z)g1 (z)∗ + δ1, if z ∈ ∂1 ,
1
with z = Rez + iImz. Then we factorize f as f (z) = h1 (z)D− u h2 (z) where
h1 (z)
h2 (z)
=
=
z
D u w(z)D
z
u
1−z
u
D w−1 (z)D
,
1−z
u
D
1−z
2
z
f2 (z)D 2 .
Note that Lemma 3.6 provides the boundedness and analyticity of h1 and h2 . As
usual, we need to estimate the norms of h1 and h2 on the boundary. We begin with
the estimates for h1 . On ∂0 we have
°
°
¡
¢°
1 °
= sup °σt/u w(it) °
sup °h1 (z)D− u °
z∈∂0
L∞ (N )
t∈R
=
°
°
sup °w(z)°
z∈∂0
=
L∞ (N )
L∞ (N )
°
°1/2
sup °w(z)w(z)∗ °
z∈∂0
L∞ (N )
On ∂1 we apply the boundary condition (3.11)
°
°2
°
¢°2
¡ 1
sup °h1 (z)°L (N ) = sup °σt/u D u w(1 + it) °Lu (N )
u
z∈∂1
t∈R
= 1.
52
3. AN INTERPOLATION THEOREM
° 1
°2
sup °D u w(z)°Lu (N )
=
z∈∂1
° 1
1 °
sup °D u w(z)w(z)∗ D u °L
=
u/2 (N )
z∈∂1
°
¡ 1
1 ¢°
sup °σ−Imz/u D u g1 (z)g1 (z)∗ D u °L
≤
u/2 (N )
z∈∂1
° 1
°2
sup °D u g1 (z)°L
=
u (N )
z∈∂1
+ δ < 1 + δ.
For the estimate of h2 on ∂0 we use (3.10)
°
°
°
¡
¡ 1
¢°
1 ¢
sup °h2 (z)°Lp (M) = sup °σt/u w−1 (it)D u σ−t/2 D 2 f2 (it) °L
t∈R
z∈∂0
+δ
p (M)
°
¡
¢
¡
¡ 1
¢¢°
= sup °σt/u w−1 (it) σt/u σ−t/p D p f2 (it) °Lp (M)
t∈R
°
¢°
¡ 1
= sup °w−1 (it)σ−t/p D p f2 (it) °Lp (M)
t∈R
≤
° 1
°
°¡
¢−1 °1/2
°
°D p f2 (it)°
sup ° w(it)w(it)∗
L∞ (N )
Lp (M)
t∈R
°
¡ 1
¢°
= sup °σ−t/2 D p f2 (it) °L (M)
p
t∈R
°
°
= sup °f (z)°
< 1.
z∈∂0
Lp (M)
Finally, we use again (3.11) to estimate h2 on ∂1
°
°
1
sup °D− u h2 (z)°L (M)
2
z∈∂1
°
¡
¢
¡
1 ¢°
= sup °σt/u w−1 (1 + it) σ−t/2 f2 (1 + it)D 2 °L2 (M)
t∈R
°
°
¡
¢
= sup °σt/u w−1 (1 + it) g1 (1 + it)g2 (1 + it)°L (M)
2
t∈R
°
°
°
°
¡ −1
¢
°
°
°
≤ sup σt/u w (1 + it) g1 (1 + it) L (N ) g2 (1 + it)°L
<
2 (M)
∞
t∈R
°
¡
¢
¡
¢∗ °1/2
sup °σt/u w−1 (1 + it) g1 (1 + it)g1 (1 + it)∗ σt/u w−1 (1 + it) °L (N )
∞
t∈R
°
°1/2
¡
¢
= sup °w−1 (1 + it)σ−t/u g1 (1 + it)g1 (1 + it)∗ w−1 (1 + it)∗ °L∞ (N )
t∈R
≤
°
°1/2
sup °w−1 (z)w(z)w(z)∗ w−1 (z)∗ °L (N ) = 1.
∞
z∈∂1
In summary, by Kosaki’s interpolation we find
√
°
°
°h1 (θ)D− 1−θ
u °
< 1 + δ and
L
(N )
u/θ
° −θ
°
°D u h2 (θ)°
< 1.
Lq
θ
Therefore, since we have
¡
¢
1−θ ¢¡
θ
f (θ) = h1 (θ)D− u D− u h2 (θ) ,
the result follows by letting δ → 0. This completes the proof.
¤
Remark 3.9. For the proof of Lemmas 3.7 and 3.8 it has been essential to have
an explicit description of the intersection ∆ of the interpolation pair. The lack of
this description in the general case is what forces us to use Haagerup’s construction
to extend the result from finite to general von Neumann algebras.
3.2. CONDITIONAL EXPECTATIONS ON ∂∞ K
53
Let us consider a so-called commutative square of conditional expectations.
That is, on one hand we have a von Neumann subalgebra N1 of M1 . On the other,
we consider a von Neumann subalgebra N2 of M2 such that M2 (resp. N2 ) is a
von Neumann subalgebra of M1 (resp. N1 ) so that the diagram in Figure II below
commutes. The interpolation results in Lemmas 3.7 and 3.8 will allow us to show
that the conditional expectation EM extends to a contractive projection on the
amalgamated spaces which correspond to points (1/u, 1/v, 1/q) in the set ∂∞ K.
M1
EM
-M2
E1
?
N1
E2
?
- N2
EN
Figure II: A commutative square.
Observation 3.10. Let us assume that the commutative square above satisfies
that E1 (M2 ) ⊂ N2 . Then we claim that the restriction E1 |M2 of E1 to M2 coincides
with E2 . Indeed, let us consider an element x ∈ M2 . Then, since E1 (x) ∈ N2 , we
have
E1 (x) = EN ◦ E1 (x) = E2 ◦ EM (x) = E2 (x).
Similarly, we have EM |N1 = EN whenever EM (N1 ) ⊂ N2 . In what follows we shall
assume these conditions on the commutative squares we are using. In fact, the
commutative squares obtained from the Haagerup construction defined below will
satisfy these assumptions.
Proposition 3.11. If 2 ≤ u, v ≤ ∞, EM extends to contractions
EM : Lu (N1 )L2 (M1 )
EM : L2 (M1 )Lv (N1 )
→ Lu (N2 )L2 (M2 ),
→ L2 (M2 )Lv (N2 ).
Proof. Let us note that for any index 2 ≤ p ≤ ∞, the natural inclusion
j : Lrp (M2 , E2 ) → Lrp (M1 , E1 )
is an isometry. Indeed, since the space Lp (N2 ) embeds isometrically in Lp (N1 ) and
the conditional expectation E2 is the restriction E1 |M2 of E1 to M2 (see Observation
3.10), the following identity holds for any x ∈ Lrp (M2 , E2 )
°
°
°
°
kj(x)kLrp (M1 ,E1 ) = °E1 (xx∗ )1/2 °L (N ) = °E2 (xx∗ )1/2 °L (N ) = kxkLrp (M2 ,E2 ) .
p
1
p
2
Now let us consider the index 2 ≤ u ≤ ∞ defined by 1/2 = 1/u + 1/p. When u > 2
we have p < ∞ and Proposition 1.11 gives
Lu (Nj )L2 (Mj ) = Lrp (Mj , Ej )∗
for
j = 1, 2.
Therefore, our map EM : Lu (N1 )L2 (M1 ) → Lu (N2 )L2 (M2 ) coincides with the
adjoint of j so that we obtain a contraction. Finally, for u = 2 we just need to note
that Lu (Nj )L2 (Mj ) embeds isometrically in Lrp (Mj , Ej )∗ for j = 0, 1. Indeed, the
first part of the proof of Proposition 1.11 holds even for p = ∞. Therefore, in this
case our map is a restriction of j ∗ to a closed subspace of Lrp (M1 , E1 )∗ and hence
contractive. The proof of the contractivity of the second mapping follows in the
same way after replacing Lrp (M, E) by Lcp (M, E).
¤
54
3. AN INTERPOLATION THEOREM
Proposition 3.12. If 2 ≤ u, v ≤ ∞, EM extends to a contraction
EM : Lu (N1 )L∞ (M1 )Lv (N1 ) → Lu (N2 )L∞ (M2 )Lv (N2 ).
Proof. According
to Proposition
¡
¢ 2.6, it suffices to prove the assertion on the
dense subspace juv N1u L∞ (M1 )N1v . Thus, let x be an element in such subspace
so that it decomposes as x = D1/u αy0 βD1/v with α, β ∈ N1 and y0 ∈ M1 . Let us
define the operators
¡
¢1/2
¡
¢1/2
a = αα∗ + δ1
, b = β ∗ β + δ1
, y = a−1 αy0 βb−1 ,
so that x = D1/u aybD1/v . Then, we proceed as in [27] by writing EM (x) as follows
h
i
EM (x) = D1/u EM (a2 )1/2 EM (a2 )−1/2 EM (ayb)EM (b2 )−1/2 EM (b2 )1/2 D1/v .
We clearly have
° 1
°
°D u EM (a2 )1/2 °
Lu (N2 )
(3.12)
°
1°
°EM (b2 )1/2 D v °
Lv (N2 )
≤
≤
³° 1 °
´1/2
°D u α°2
+δ
,
Lu (N1 )
´
³°
1/2
°
°βD v1 °2
+δ
.
Lv (N1 )
On the other hand, let us consider for a moment a von Neumann subalgebra N
of a given von Neumann algebra M and let E : M → N be the corresponding
conditional expectation. Let us consider a positive element γ ∈ M+ satisfying
γ ≥ δ1. Then we define the map
Λγ : w ∈ M → E(γ 2 )−1/2 E(γwγ)E(γ 2 )−1/2 ∈ N .
According to [27], this is a completely positive map so that kΛγ k = kΛγ (1)k = 1.
Thus, Λγ is a contraction. Then we apply this result to
³ a 0 ´
γ=
.
0 b
More concretely, Λγ : M2 ⊗ M1 → M2 ⊗ M2 is given by
h
³ 2
´i− 12
h
³ 2
´i− 12
eM a 02
eM (γwγ) E
eM a 02
eM = id ⊗ EM .
Λγ (w) = E
E
with E
0 b
0 b
Then we observe that
¶
³ 0 y ´ µ
0 EM (a2 )−1/2 EM (ayb)EM (b2 )−1/2
Λγ
=
.
0 0
0
0
°
°
°
°
In particular, since kykM1 ≤ °a−1 α°N ky0 kM1 °βb−1 °N ≤ ky0 kM1 , we deduce
1
1
°
°
°EM (a2 )−1/2 EM (ayb)EM (b2 )−1/2 °
(3.13)
≤ ky0 kL∞ (M1 ) .
L∞ (M2 )
In summary, according to (3.12) and (3.13)
³°
³° 1 °
´ 21
´ 12
° 1
°
°
°D u EM (αy0 β)D v1 °
°βD v1 °2
°D u α°2
≤
+δ
ky
k
+δ
.
0
L
(M
)
∞
1
u·∞·v
Lu (N1 )
Lv (N1 )
The proof is completed by letting δ → 0 and taking the infimum on the right.
Corollary 3.13. If (1/u, 1/v, 1/q) ∈ ∂∞ K, EM extends to a contraction
EM : Lu (N1 )Lq (M1 )Lv (N1 ) → Lu (N2 )Lq (M2 )Lv (N2 ).
¤
3.3. GENERAL VON NEUMANN ALGEBRAS I
55
Proof. The case 1/q = 0 has already been considered in Proposition 3.12.
Thus, assume (without lost of generality) that 1/v = 0. Then, we know from
Propositions 3.11 and 3.12 that
EM :
EM :
Lu (N1 )L2 (M1 ) →
Lu (N1 )L∞ (M1 ) →
Lu (N2 )L2 (M2 ),
Lu (N2 )L∞ (M2 ),
are contractions. Therefore, it follows from Lemma 3.7 that the same holds for
EM : Lu (N1 )Lq (M1 ) → Lu (N2 )Lq (M2 )
when 2 ≤ q ≤ ∞. Finally, it remains to consider the case 1 ≤ q ≤ 2. Given
1 ≤ q ≤ 2 and 2 ≤ u ≤ ∞ such that 1/u + 1/q ≤ 1, we consider the index defined
by 1/p = 1/u + 1/q. Then, according to Lemma 3.8 we have
£
¤
Lu (N1 )Lq (M1 ) = Lp (M1 ), Lu1 (N1 )L2 (M1 ) θ ,
for some 0 ≤ θ ≤ 1 and some 2 ≤ u1 ≤ ∞. Note that the fact that u1 can be chosen
so that 2 ≤ u1 ≤ ∞ follows by a quick look at Figure I. By Proposition 3.11, EM
is contractive on Lu1 (N1 )L2 (M1 ) and of course on Lp (M1 ). Therefore, the result
follows by complex interpolation.
¤
3.3. General von Neumann algebras I
In this section we prove Lemma 3.1 under the assumption min(q0 , q1 ) < ∞.
However, most of the arguments we are giving here will be useful for the remaining
case. Before starting the proof we fix some notation. As in Lemma 3.1, we are
given a von Neumann subalgebra N of M and we denote by E : M → N the
corresponding conditional expectation. According to the Haagerup construction
sketched in Section 2.1, we have
[
RM = M oσ G =
Mk .
k≥1
As above, we consider the conditional expectation
X
EM :
xg λ(g) ∈ RM 7→ x0 ∈ M
g∈G
and the state ϕ
b = ϕ ◦ EM on RM . Moreover, we have conditional expectations
EMk : RM → Mk for each k ≥ 1 so that (if ϕk denotes the restriction of ϕ
b to Mk
and Dϕk stands for the corresponding density) the family of finite von Neumann
subalgebras (Mk )k≥1 satisfy
(3.14)
c1 (k)1Mk ≤ Dϕk ≤ c2 (k)1Mk .
Moreover, given 0 ≤ η ≤ 1 and 1 ≤ p < ∞, we have
°
¢ η/p °
° (1−η)/p ¡
°
(3.15)
lim °Dϕb
x̂ − Ek (x̂) Dϕb °
k→∞
Lp (RM )
=0
for any x̂ ∈ RM . Clearly, we can consider another Haagerup construction for the
von Neumann subalgebra N so that the analogous properties hold. In summary,
we sketch the situation in Figure III below.
56
3. AN INTERPOLATION THEOREM
EM
M¾
RM
E Mk
- Mk
Ek
E
E
?
N ¾
EN
?
RN
?
- Nk
ENk
Figure III: Haagerup’s construction.
Now let us consider two points (1/u0 , 1/v0 , 1/q0 ) and (1/u1 , 1/v1 , 1/q1 ) in ∂∞ K.
Then it is clear that the natural inclusion mappings
id : Lu0 (N )Lq0 (M)Lv0 (N ) → Lu0 (RN )Lq0 (RM )Lv0 (RN ),
id : Lu1 (N )Lq1 (M)Lv1 (N ) → Lu1 (RN )Lq1 (RM )Lv1 (RN ),
are contractions. In particular, defining for 0 ≤ θ ≤ 1
h
i
Xθ (M) = Lu0 (N )Lq0 (M)Lv0 (N ), Lu1 (N )Lq1 (M)Lv1 (N ) ,
θ
h
i
Xθ (RM ) = Lu0 (RN )Lq0 (RM )Lv0 (RN ), Lu1 (RN )Lq1 (RM )Lv1 (RN ) ,
θ
we obtain by interpolation that id : Xθ (M) → Xθ (RM ) is also a contraction.
Now, considering the commutative square on the right of Figure III, it follows from
Corollary 3.13 that for each k ≥ 1 we have contractions
EMk : Lu0 (RN )Lq0 (RM )Lv0 (RN ) → Lu0 (Nk )Lq0 (Mk )Lv0 (Nk ),
EMk : Lu1 (RN )Lq1 (RM )Lv1 (RN ) → Lu1 (Nk )Lq1 (Mk )Lv1 (Nk ).
Moreover, Mk satisfies (3.14) for each k ≥ 1. In particular, since we have already
proved that Lemma 3.1 holds in this case, we obtain by complex interpolation a
contraction EMk : Xθ (RM ) → Luθ (Nk )Lqθ (Mk )Lvθ (Nk ). In summary, writing
EMk = EMk ◦ id,
we have found a contraction
(3.16)
EMk : Xθ (M) → Luθ (Nk )Lqθ (Mk )Lvθ (Nk ).
On the other hand, regarding the space Luθ (Nk )Lqθ (Mk )Lvθ (Nk ) as a subspace
of Luθ (RN )Lqθ (RM )Lvθ (RN ), we can consider the restriction of the conditional
expectation EM to Mk and define the following map
EM : Luθ (Nk )Lqθ (Mk )Lvθ (Nk ) → Luθ (N )Lqθ (M)Lvθ (N ).
The following technical result will be the key to prove Lemma 3.1.
Proposition 3.14. If k ≥ 1 and 0 ≤ θ ≤ 1, EM extends to a contraction
EM : Luθ (Nk )Lqθ (Mk )Lvθ (Nk ) → Luθ (N )Lqθ (M)Lvθ (N ).
Proof. It clearly suffices to see our assertion on the dense subspace
1/2qθ
θ
θ
Nk,a D1/vθ ,
Mk,a D1/2q
Ak = D1/u
ϕk
ϕk Nk,a Dϕk
where Mk,a and Nk,a denote the ∗-algebras of analytic elements in Mk and Nk
respectively. Thus, let us consider an element x in Ak and assume that the norm
of x in Luθ (Nk )Lqθ (Mk )Lvθ (Nk ) is less that 1. Then, since we know that Lemma
3.3. GENERAL VON NEUMANN ALGEBRAS I
57
3.1 holds for finite von Neumann algebras satisfying (3.14), we can find a bounded
analytic function f : S → Ak satisfying f (θ) = x and the boundary estimate
n
o
°
°
°
°
max sup °f (z)°u0 ·q0 ·v0 , sup °f (z)°u1 ·q1 ·v1 < 1.
z∈∂0
z∈∂1
Then we use the hypothesis 1/uθ + 1/qθ + 1/vθ = 1/u0 + 1/v0 + 1/q0 and Lemma
1.2 to rewrite Ak as follows
0 +1/2q0
0 +1/v0
Ak = D1/u
Mk,a D1/2q
.
ϕk
ϕk
Therefore, multiplying if necessary on the left and on the right by certain powers
of Dzϕk and its inverses, we may assume that f has the following form (recall that
we are using here the property (3.14))
1−z
z
1−z
z
z
1−z
1−z
z
f (z) = Dϕuk0 Dϕuk1 Dϕ2qk0 Dϕ2qk1 f1 (z) Dϕ2qk1 Dϕ2qk0 Dϕv1k Dϕvk0 ,
(3.17)
with f1 : S → Mk,a bounded analytic. In particular, we have
° 1+ 1
°
1
+ 1 °
°
< 1,
sup °Dϕuk0 2q0 f1 (z)Dϕ2qk0 v0 °
z∈∂0
u0 ·q0 ·v0
z∈∂1
u1 ·q1 ·v1
° 1+ 1
°
1
+ 1 °
°
sup °Dϕuk1 2q1 f1 (z)Dϕ2qk1 v1 °
<
1.
Note that here we have used the fact that Dw
ϕk is a unitary for any w ∈ C such that
Re w = 0. Now we apply Corollary 3.13 to the commutative square on the left of
Figure III. In other words, we have contractions
EM : Luj (Nk )Lqj (Mk )Lvj (Nk ) → Luj (N )Lqj (M)Lvj (N )
since (1/uj , 1/vj , 1/qj ) ∈ ∂∞ K for j = 0, 1. Therefore,
° 1+ 1
°
1
+ 1 °
°
sup °Dϕu0 2q0 EM (f1 (z))Dϕ2q0 v0 °
z∈∂0
u0 ·q0 ·v0
z∈∂1
u1 ·q1 ·v1
°
° 1+ 1
1
+ 1 °
°
sup °Dϕu1 2q1 EM (f1 (z))Dϕ2q1 v1 °
< 1,
< 1.
Then, according to Proposition 2.6, we can find functions g1 , g3 : ∂S → N and
g2 : ∂j → Lqj (M) such that
1/2qj
j
Dϕ
EM (f1 (z))D1/2q
= g1 (z)g2 (z)g3 (z) for all
ϕ
and satisfying the following boundary estimates
n° 1
1 °
°
°
°
°
sup °Dϕu0 g1 (z)°L (N ) , °g2 (z)°L (M) , °g3 (z)Dϕv0 °L
z∈∂0
u0
n° 1
°
sup °Dϕu1 g1 (z)°Lu
z∈∂1
q0
1 (N )
°
°
, °g2 (z)°Lq
1 (M)
1 °
°
, °g3 (z)Dϕv1 °L
z ∈ ∂j
o
v0 (N )
<
1,
<
1.
o
v1 (N )
Then we can proceed as in the proof of Lemma 3.1 for finite von Neumann algebras.
Namely, we apply Devinatz’s factorization theorem to the functions
¡
¢
W1 : z ∈ ∂S 7→ σImz/2q1 −Imz/2q0 g1 (z)g1 (z)∗ + δ1 ∈ N ,
¡
¢
W3 : z ∈ ∂S 7→ σImz/2q0 −Imz/2q1 g3 (z)∗ g3 (z) + δ1 ∈ N ,
so that we can find invertible bounded analytic functions w1 , w3 : S → N satisfying
w1 (z)w1 (z)∗
∗
w3 (z) w3 (z)
= W1 (z),
=
W3 (z),
58
3. AN INTERPOLATION THEOREM
for all z ∈ ∂S. Then we consider the factorization
EM (f (z)) = h1 (z)h2 (z)h3 (z)
with h2 (z) =
−1
h−1
1 (z)EM (f (z))h3 (z)
and h1 , h3 given by
1−z
u0
h1 (z)
= Dϕ
h3 (z)
=
z
z
−
z
z
z
z
Dϕu1 w1 (z)Dϕ u1 Dϕu0 ,
−
1−z
Dϕv0 Dϕ v1 w3 (z)Dϕv1 Dϕv0 .
To estimate the norm of hj (θ) for j = 1, 2, 3 in the corresponding Lp space we
proceed as in the proof of Lemma 3.1 for finite von Neumann algebras. That is, we
first show the boundedness and analyticity of h1 , h2 , h3 . This enables us to estimate
the norm on the boundary ∂S and apply Kosaki’s interpolation. Let us start with
the function h1 . To that aim we note that
(a) If u0 ≤ u1 , we can write
1
(
1
h1 (z) = Dϕu1 Dϕu0
− u1 )(1−z)
1
(
1
w1 (z)Dϕu0
− u1 )z
1
.
(b) If u0 ≥ u1 , we can write
³ 1 ( 1 − 1 )z
´ (1−1)
( 1 − 1 )(1−z)
h1 (z) = Dϕu0 Dϕu1 u0 w1 (z)Dϕu1 u0
Dϕu0 u1 .
By Lemma 3.6, h1 is either bounded analytic or can be written as
(
1
h1 (z) = j1 (z)Dϕu0
− u1 )
1
,
with j1 bounded analytic. In any case, Kosaki’s interpolation gives
n
θ °
1 °
θ
1
°
°
°
°
°h1 (θ)Dϕu1 − u0 °
°h1 (z)°
°h1 (z)Dϕu1 − u0 °
≤
max
sup
,
sup
Lu (N )
Lu
L (N )
θ
u0
z∈∂0
1
z∈∂1
o
.
(N )
Moreover, arguing as in the proof of Lemma 3.1 for finite von Neumann algebras
√
°
°
sup °h1 (z)°L (N ) <
1 + δ,
u0
z∈∂0
1
°
− 1 °
sup °h1 (z)Dϕu1 u0 °L
z∈∂1
<
u1 (N )
√
1 + δ.
Indeed, the only significant difference (with respect to the proof for finite von
Neumann algebras) is that we can not assume any longer that Dw
ϕ is a unitary for a
purely imaginary complex number w. However, this difficulty is easily avoided by
using the norm-invariance property of the one-parameter modular automorphism
group. In fact, that is the reason why we used the modular automorphism group
in the definition of W1 and W3 , see also the proof of Lemma 3.8. Our previous
estimates give rise to
θ
θ °
√
°
°h1 (θ)Dϕu1 − u0 °
(3.18)
< 1 + δ.
L (N )
uθ
Similarly, we have
° vθ − vθ
°
°Dϕ1 0 h3 (θ)°
(3.19)
Lvθ (N )
<
√
1 + δ.
Finally, we consider the function h2 . To study the boundedness and analyticity of
h2 , we recall the expression for f obtained in (3.17). Then, we can rewrite h2 in
the following way
z
−
z
1−z
z
z
1−z
−
z
z
h2 (z) = Dϕu1 Dϕ u0 w1−1 (z)Dϕ2q0 Dϕ2q1 EM (f1 (z))Dϕ2q1 Dϕ2q0 w3−1 (z)Dϕ v0 Dϕv1 .
3.3. GENERAL VON NEUMANN ALGEBRAS I
59
Using one more time that 1/u0 + 1/q0 + 1/v0 = 1/u1 + 1/q1 + 1/v1 , we can write
(a) If max(u0 , u1 ) = u0 and max(v0 , v1 ) = v0 , we have
z
h2 (z)
−
1−z
z
− 1−z
Dϕu1 u0 w1−1 (z)Dϕu1 u0
´
³ 1−z − 1−z + 1−z + z
z
+ 1−z − 1−z + 1−z
× EM Dϕuk0 u1 2q0 2q1 f1 (z)Dϕ2qk1 2q0 v1 v0
=
1−z
× Dϕv1
− 1−z
v
0
z
w3−1 (z)Dϕv1
− vz
0
.
The first and third terms on the right hand side are bounded analytic
by Lemma 3.6. The middle term is clearly bounded analytic in Lq1 (M).
Indeed, it follows easily from the fact that Dϕk satisfies (3.14).
(b) If max(u0 , u1 ) = u1 and max(v0 , v1 ) = v1 , we proceed as above with
1
Dϕu0
− u1
1
1
h2 (z)Dϕv0
− v1
1
.
(c) If max(u0 , u1 ) = u0 and max(v0 , v1 ) = v1 , we proceed as above with
1
h2 (z)Dϕv0
− v1
.
1
(d) If max(u0 , u1 ) = u1 and max(v0 , v1 ) = v0 , we proceed as above with
1
Dϕu0
− u1
1
h2 (z).
In any of the possible situations considered, Kosaki’s interpolation provides the
following estimate
θ
θ °
° uθ − uθ
°Dϕ0 1 h2 (θ)Dϕv0 − v1 °
n
≤ max
Lqθ (M)
°
°
sup °h2 (z)°L
q0 (M)
z∈∂0
1
° 1−1
− 1 °
, sup °Dϕu0 u1 h2 (z)Dϕv0 v1 °Lq
1 (M)
z∈∂1
o
.
On the other hand, arguing one more time as in the proof of Lemma 3.1,
°
°
sup °h2 (z)°L (M) < 1,
q
0
z∈∂0
1
° 1−1
− 1 °
sup °Dϕu0 u1 h2 (z)Dϕv0 v1 °Lq
1 (M)
z∈∂1
< 1.
In particular,
(3.20)
θ
θ °
° uθ − uθ
°Dϕ0 1 h2 (θ)Dϕv0 − v1 °
Lqθ (M)
< 1.
In summary, we have found a factorization of EM (x)
θ
θ
θ
θ
¡
¢
− θ ¢¡
− θ
− θ ¢¡
− θ
EM (f (θ)) = h1 (θ)Dϕu1 u0 Dϕu0 u1 h2 (θ)Dϕv0 v1 Dϕv1 v0 h3 (θ)
which, according to (3.18, 3.19, 3.20), provides the estimate
°
°
°EM (x)°
< 1 + δ.
uθ ·qθ ·vθ
Therefore, the proof is concluded by letting δ → 0 in the inequality above.
¤
Corollary 3.15. If k ≥ 1 and 0 ≤ θ ≤ 1, we have
°
°
°EM EM (x)°
≤ kxkXθ (M) for all x ∈ Xθ (M).
k
u ·q ·v
θ
θ
θ
Proof. The result follows automatically from (3.16) and Proposition 3.14.
¤
60
3. AN INTERPOLATION THEOREM
Proof of Lemma 3.1. As in Lemmas 3.7 and 3.8 and also as in the proof
for finite von Neumann algebras, the lower estimate follows by using multilinear
interpolation. This means that we have a contraction
(3.21)
id : Luθ (N )Lqθ (M)Lvθ (N ) → Xθ (M).
To prove the converse, let us consider the space
θ
θ
θ
θ
A = D1/u
N D1/2q
MD1/2q
N D1/v
.
ϕ
ϕ
ϕ
ϕ
According to Proposition 2.8 and Corollary 3.15, given x ∈ A we have
°
°γ
kxkγuθ ·qθ ·vθ ≤ kxkγXθ (M) + lim °x − EM EMk (x)°u ·q ·v .
k→∞
θ
θ
θ
Therefore, we need to see that the limit above is 0. To that aim, we use that x ∈ A
so that we can write
θ
θ
yD1/v
x = D1/u
ϕ
ϕ
with y ∈ Lqθ (M).
Then we use the contractivity of EM on Lqθ (RM ) to obtain
°
°
°
°
¡
¢
θ
θ°
lim °x − EM EMk (x)°u ·q ·v = lim °D1/u
y − EM EMk (y) D1/v
ϕ
ϕ
uθ ·qθ ·vθ
θ θ θ
k→∞
k→∞
°
°
≤ lim °y − EM EMk (y)°Lq (M)
k→∞
θ
°
¡
¢°
= lim °EM y − EMk (y) °L (M)
qθ
k→∞
°
°
≤ lim °y − EMk (y)°L (R ) .
q
k→∞
θ
M
Then, recalling our hypothesis min(q0 , q1 ) < ∞ assumed at the beginning of this
paragraph, we deduce that qθ < ∞ for any 0 < θ < 1. Therefore, according to
(3.15) we conclude that the limit above is 0. In particular, we have
kxkuθ ·qθ ·vθ = kxkXθ (M)
for all
x ∈ A.
Now, using the density of A in Luθ (N )Lqθ (M)Lvθ (N ) and (3.21) we deduce that
the same holds for any x ∈ Luθ (N )Lqθ (M)Lvθ (N ). Hence, it remains to see that
Luθ (N )Lqθ (M)Lvθ (N ) is dense in Xθ (M). To that aim, it suffices to prove that
the subspace
EM EMk (Xθ (M)) ⊂ Luθ (N )Lqθ (M)Lvθ (N )
is dense in Xθ (M). Moreover, since the intersection space
∆ = Lu0 (N )Lq0 (M)Lv0 (N ) ∩ Lu1 (N )Lq1 (M)Lv1 (N )
is dense in Xθ (M) for any 0 < θ < 1, we just need to approximate any element x
in ∆ by an element in EM EMk (Xθ (M)) with respect to the norm of Xθ (M). Using
one more time that min(q0 , q1 ) < ∞ we assume (without lost of generality) that
q0 < ∞. Then, applying the three lines lemma
°
°
°
°1−θ
°
°θ
°x − EM EM (x)°
≤ °x − EM EMk (x)°u0 ·q0 ·v0 °x − EM EMk (x)°u1 ·q1 ·v1 ,
k
X (M)
θ
we just need to show that the first term on the right tends to 0 as k → ∞ while
the second term is uniformly bounded on k. The uniform boundedness follows from
Corollary 3.15 since
°
°
°
°
°x − EM EM (x)°
≤ kxku1 ·q1 ·v1 + °EM EM (x)°
≤ 2kxku1 ·q1 ·v1 .
k
u1 ·q1 ·v1
k
For the first term, we pick y ∈ Lq0 (M) so that
°
°
0
0°
°x − D1/u
yD1/v
ϕ
ϕ
u ·q
0
0 ·v0
≤ δ.
u1 ·q1 ·v1
3.4. GENERAL VON NEUMANN ALGEBRAS II
Then we have
°
°
°x − EM EM (x)°
k
u0 ·q0 ·v0
≤
+
+
61
°
°
0
0°
°x − D1/u
yD1/v
ϕ
ϕ
u0 ·q0 ·v0
° 1/u ¡
°
¢
0°
°Dϕ 0 y − EM EM (y) D1/v
ϕ
k
u0 ·q0 ·v0
° 1/u
°
0
°Dϕ 0 EM EM (y)D1/v
°
−
E
E
M M (x)
ϕ
k
u0 ·q0 ·v0
k
.
In particular, according to Corollary 3.15
°
°
°
°
¡
¢
0
0°
°x − EM EM (x)°
≤ 2δ + °D1/u
y − EM EMk (y) D1/v
ϕ
ϕ
k
u0 ·q0 ·v0
u0 ·q0 ·v0
°
°
≤ 2δ + °y − EM EMk (y)°Lq (M)
°
¡
¢°0
= 2δ + °EM y − EMk (y) °Lq (M)
0
°
°
≤ 2δ + °y − EMk (y)°Lq (R ) .
0
M
Finally, since q0 < ∞ by hypothesis, we know that the second term on the right
tends to 0 as k → ∞. Then, we let δ → 0. This completes the proof of Lemma 3.1
for general von Neumann algebras with min(q0 , q1 ) < ∞.
¤
3.4. General von Neumann algebras II
To complete the proof of Lemma 3.1 we have to study the case min(q0 , q1 ) = ∞.
Note that the proof above fails in this case since (3.15) does not hold for p = ∞.
However, Corollary 3.15 is still valid in this case so that it suffices to see that
°
°
(3.22)
kxku ·∞·v ≤ sup °EM EM (x)°
for all x ∈ Xθ (M).
θ
θ
k
k≥1
uθ ·∞·vθ
Indeed, since the lower estimate follows one more time by multilinear interpolation,
inequality (3.22) and Corollary 3.15 are enough to conclude the proof of Lemma
3.1. In order to prove (3.22) we shall need to consider the spaces
nX
o
¯
Lrp (M, E) ⊗M Lcq (M, E) =
w1k w2k ¯ w1k ∈ Lrp (M, E), w2k ∈ Lcq (M, E)
k
for 2 ≤ p, q ≤ ∞ and equipped with
½°³ X
´1/2 °
°
°
∗
E(w1k w1k
)
kykrp ·cq = inf °
°
k
Lp (N )
°³ X
´1/2 °
°
°
∗
E(w2k
w2k )
°
°
k
¾
Lq (N )
where the infimum runs over all possible decompositions
X
y=
w1k w2k .
k
It is not hard to check that Lrp (M, E) ⊗M Lcq (M, E) is a normed space, see e.g.
Lemma 3.5 in [16] for a similar result. The notation ⊗M is motivated by the fact
that the norm given above comes from an amalgamated Haagerup tensor product,
we refer the reader to Chapter 6 below for a more detailed explanation. The
following result is the key to conclude the proof of Lemma 3.1. We use the well
known Grothendieck-Pietsch version of the Hahn-Banach theorem, see [43] for more
on this topic.
Theorem 3.16. Let 2 < p, q, u, v < ∞ related by
1/u + 1/p = 1/2 = 1/v + 1/q.
¡
¢
Then, we have the following isometry via the anti-linear bracket hx, yi = tr x∗ y
¡
¢∗
Lu (N )L∞ (M)Lv (N ) = Lrp (M, E) ⊗M Lcq (M, E) .
62
3. AN INTERPOLATION THEOREM
Proof. Given x = ayb in Lu (N )L∞ (M)Lv (N ), Hölder inequality gives
¯ ³ X
´¯
¯
¯
w1k w2k ¯
¯tr x∗
k
¯ ³ hX
ih X
i´¯
¯
¯
= ¯tr y ∗
a∗ w1k ⊗ e1k
w2k b∗ ⊗ ek1 ¯
k
k
° ³X
°³ X
´°
´ °
°
°
°
°
≤ °a∗
w1k ⊗ e1k ° kyk∞ °
w2k ⊗ ek1 b∗ °
k
k
2
2
i´1/2
³h X
i ´1/2
³
hX
∗
∗
∗
kyk∞ tr
E(w2k w2k ) b∗ b
= tr aa
E(w1k w1k )
k
k
°³ X
´1/2 °
´1/2 ° °³ X
°
° °
°
∗
∗
w2k )
E(w2k
≤ kaku kyk∞ kbkv °
E(w1k w1k )
° .
° °
k
k
p
q
Thus, taking the infimum on the right, we have a contraction
¡
¢∗
x ∈ Lu (N )L∞ (M)Lv (N ) 7→ tr(x∗ ·) ∈ Lrp (M, E) ⊗M Lcq (M, E) .
To prove the converse, we take a norm one functional ϕ on Lrp (M, E) ⊗M Lcq (M, E).
If 1/s = 1/p + 1/q it is clear that (see Remark 1.7)
Ls (M) = Lp (M)Lq (M) → Lrp (M, E) ⊗M Lcq (M, E)
is a dense contractive inclusion. In particular, we can assume that there exists
x ∈ Ls0 (M) = Lu (M)Lv (M)
satisfying
ϕ(y) = ϕx (y) = tr(x∗ y).
(3.23)
To conclude, it suffices to see that kxku·∞·v ≤ 1. Let us consider a finite family
y1 , y2 , . . . , ym in the dense subspace Lp (M)Lq (M) with decompositions
yk = w1k w2k .
Since ϕx has norm one
¯X
°1/2
¡
¢¯¯ °
¯
°X
∗ °
ϕx w1k w2k ¯ ≤ °
)°
E(w1k w1k
¯
k
k
Lp/2 (N )
°X
°1/2
°
°
∗
w2k )°
E(w2k
°
k
Lq/2 (N )
.
Moreover, since the right hand side remains unchanged under multiplication with
unimodular complex numbers zk ∈ T, we have the following inequality
°1/2
°X
°1/2
X ¯ ¡
¢¯ ° X
°
°
∗ °
∗
¯ϕx w1k w2k ¯ ≤ °
E(w1k w1k
)°
E(w2k
w2k )°
.
°
°
k
k
Lp/2 (N )
k
Lq/2 (N )
Now we consider the unit balls in Lu (N ) and Lv (N )
n
o
¯
B1 =
α ∈ Lu (N ) ¯ kαkLu (N ) ≤ 1 ,
n
o
¯
B2 =
β ∈ Lv (N ) ¯ kβkLv (N ) ≤ 1 .
By the arithmetic-geometric mean inequality
X ¯ ¡
¢¯
¯ϕx w1k w2k ¯
(3.24)
k
X ¡
X ¡
¢
¢´
1³
∗
∗
sup
tr αE(w1k w1k
)α∗ + sup
tr β ∗ E(w2k
w2k )β
≤
k
k
2 α∈B1
β∈B2
³
´
X
X
°
°
°
°
1
°αw1k °2
°w2k β °2
=
sup
+
sup
.
L
(M)
L
(M)
2
2
k
k
2 α∈B1
β∈B2
3.4. GENERAL VON NEUMANN ALGEBRAS II
63
Note that B1 and B2 are compact when equipped with the σ(Lu (N ), Lu0 (N ))
and the σ(Lv (N ), Lv0 (N )) topologies respectively. Now, labelling (w1k )k≥1 and
(w2k )k≥1 by w1 and w2 , we consider fw1 w2 : B1 × B2 → R defined by
X °
X °
X ¯ ¡
°
°
¢¯
°αw1k °2
°w2k β °2
¯ϕx w1k w2k ¯.
fw1 w2 (α, β) =
+
−2
L2 (M)
L2 (M)
k
k
k
This gives rise to the cone
n
o
¯
C+ = fw1 w2 ∈ C(B1 × B2 ) ¯ w1k w2k ∈ Lp (M)Lq (M) .
Then we consider the open cone
n
o
¯
C− = f ∈ C(B1 × B2 ) ¯ sup f < 0 .
According to (3.24), the cones C+ and C− are disjoint. Therefore, the geometric
Hahn-Banach theorem provides a norm one functional ξ : C(B1 ×B2 ) → R satisfying
ξ(f− ) < ρ ≤ ξ(f+ )
for some ρ ∈ R and all (f+ , f− ) ∈ C+ × C− . Moreover, since we are dealing with
cones, it turns out that ρ = 0 and ξ is a positive functional. Then, according to
Riesz representation theorem, there exists a unique (positive) Radon measure µξ
on B1 × B2 satisfying
Z
(3.25)
ξ(f ) =
f dµξ for all f ∈ C(B1 × B2 ).
B1 ×B2
In fact, since ξ is a norm one positive functional, µξ is a probability measure. Now
we use that ξ|C+ takes values in R+ and (3.25) to obtain the following inequality
X ¯
X Z
¯
¡
¢
∗
¯ϕx (w1k w2k )¯ ≤
2
tr w1k w1k
α∗ α dµξ (α, β)
k
k B ×B
1
2
X Z
¡ ∗
¢
+
tr w2k
w2k ββ ∗ dµξ (α, β)
k
B1 ×B2
X °
°
°α0 w1k °2
L
=
k
2 (M)
+
X °
°
°w2k β0 °2
L
k
2 (M)
,
where (α0 , β0 ) ∈ B1 × B2 are given by
³Z
´1/2
α0 =
α∗ α dµξ (α, β)
∈ B1 ,
B1 ×B2
³Z
´1/2
β0 =
ββ ∗ dµξ (α, β)
∈ B2 .
B1 ×B2
Then, using the identity 2rs = inf γ>0 (γr)2 + (s/γ)2 , we conclude
´1/2 ³ X °
´1/2
X ¯
¯ ³X °
°
°
¯ϕx (w1k w2k )¯ ≤
°α0 w1k °2
°w2k β0 °2
.
L2 (M)
L2 (M)
k
k
k
In particular, given any pair (w1 , w2 ) ∈ Lp (M) × Lq (M), we have
¯
¯ °
°
°
°
¯ϕx (w1 w2 )¯ ≤ °α0 w1 °
°w2 β0 °
(3.26)
.
L2 (M)
L2 (M)
Let us write qα0 and qβ0 for the support projections of α0 and β0 . Then we define
dα0
=
α0u + (1 − qα0 )D(1 − qα0 ),
dβ0
=
β0v + (1 − qβ0 )D(1 − qβ0 ).
64
3. AN INTERPOLATION THEOREM
Note that φα0 = tr(dα0 ·) and φβ0 = tr(dβ0 ·) are n.f. finite weights on M. In
particular, by Theorem 1.3 we know that
d1/2
α0 M → L2 (M) and
1/2
Mdβ0 → L2 (M)
are dense inclusions. Therefore, since
1/2
qα0 dα0 M =
1/2
Mdβ0 qβ0 =
u/2
α0 M =
v/2
Mβ0
=
u/p
α0 α0 M ⊂
v/q
Mβ0 β0 ⊂
α0 Lp (M),
Lq (M)β0 ,
it follows that α0 Lp (M) (resp. Lq (M)β0 ) is dense in qα0 L2 (M) (resp. L2 (M)qβ0 ).
Hence we can consider the linear map
Tx : qα0 L2 (M) → qβ0 L2 (M)
determined by the relation
­
®
¡
¢
β0 w2∗ , Tx (α0 w1 ) = tr w2 β0 Tx (α0 w1 ) = ϕx (w1 w2 ).
According to (3.26), Tx is contractive. Moreover, Tx is clearly a right M module
map so that it commutes with the right action on M. This means that there exists
a contraction m ∈ M satisfying Tx (α0 w1 ) = mα0 w1 . Finally, applying (3.23) we
deduce the following identity
¡
¡
¢
¡
¢
tr x∗ w1 w2 ) = ϕx (w1 w2 ) = tr Tx (α0 w1 )w2 β0 = tr β0 mα0 w1 w2 ,
which holds for any pair (w1 , w2 ) ∈ Lp (M) × Lq (M). Therefore, by the density of
Lp (M)Lq (M) in Lrp (M, E) ⊗M Lcq (M, E) we have
x = α0∗ m∗ β0∗ .
Then, since (α0 , β0 ) ∈ B1 × B2 and m is contractive, we have kxku·∞·v ≤ 1.
¤
Observation 3.17. With a slight change in the arguments used, we can see
that Theorem 3.16 holds for any (u, v) ∈ [2, ∞] × [2, ∞] such that max(u, v) > 2.
Indeed, by symmetry it suffices to see that
¡
¢∗
(a) Lu (N )L∞ (M)L2 (N ) = Lrp (M, E) ⊗M Lc∞ (M, E) for any 2 < u ≤ ∞.
¢∗
¡ r
(b) Lu (N )L∞ (M)L∞ (N ) = Lp (M, E) ⊗M Lc2 (M, E) for any 2 < u ≤ ∞.
Since the proofs are similar, we only prove (a). Recalling that Lp (M)L∞ (M) is
norm dense in Lrp (M, E) ⊗M Lc∞ (M, E), we deduce that every norm one functional
ϕ : Lrp (M, E) ⊗M Lc∞ (M, E) → C is given by ϕ(y) = ϕx (y) = tr(x∗ y) for some
x ∈ Lp0 (M). Using one more time the Grothendieck-Pietsch separation trick we
get
¯
¯ °
° ¡
¢
¯tr(x∗ w1 w2 )¯ ≤ °α0 w1 ° ψ E(w2∗ w2 ) 1/2
2
for some α0 in the unit ball of Lu (N ) and ψ ∈ N ∗ . Let (eα ) be a net such that
eα → 1 strongly so that limα ϕ(eα y) = ϕn (y) gives the normal part. Now replace
y by yeα and we get in the limit
¯
¯ °
°
¡
¢
¯tr(x∗ w1 w2 )¯ ≤ °α0 w1 ° ϕn E(w2∗ w2 ) 1/2 .
2
Thus |tr(x∗ w1 w2 )| ≤ kα0 w1 k2 kw2 β0 k2 and we may continue as in Theorem 3.16.
Proof of Lemma 3.1. As we already pointed out at the beginning of this
section, we have to prove inequality (3.22). Before doing it we recall that the indices
(uθ , vθ ) satisfy
(3.27)
2 < max(uθ , vθ )
and
min(uθ , vθ ) < ∞.
3.4. GENERAL VON NEUMANN ALGEBRAS II
65
for any 0 < θ < 1. Indeed, otherwise we would have uθ = vθ = 2 or uθ = vθ = ∞.
However, (1/2, 1/2, 0) and (0, 0, 0) are extreme points of K ∩ {z = 0} and this is not
possible for 0 < θ < 1. The first inequality in (3.27) allows us to apply Theorem
3.16 after Observation 3.17, while the second inequality will be used below. Now,
let x be an element of Luθ (N )L∞ (M)Lvθ (N ). According to Theorem 3.16, we
know that we can find a finite family (w1k , w2k ) ∈ Lpθ (M) × Lqθ (M) with
1/uθ + 1/pθ = 1/2 = 1/vθ + 1/qθ
so that
(3.28) ½
°³ X
´1/2 °
°
°
∗
)
E(w1k w1k
max °
°
k
and
Lpθ (N )
¾
°³ X
´1/2 °
°
°
∗
E(w2k
w2k )
,°
°
k
Lqθ (N )
≤ 1,
¯ ³ X
´¯
¯
¯
w1k w2k ¯ + δ.
kxkuθ ·∞·vθ ≤ ¯tr x∗
k
Moreover, given 1/s = 1/uθ + 1/vθ , we have
°
¯ ³£
´¯
°
° °
¤∗ X
¯
°X
°
¯
w1j w2j ° 0 .
w1j w2j ¯ ≤ °x − EM EMk (x)°s °
¯tr x − EM EMk (x)
j
j
s
According to (3.27) we have 1 < s < ∞. Then, it follows from (3.15) that the first
factor on the right hand side tends to 0 as k → ∞ while the second factor belongs
to Ls0 (M). In conclusion, we obtain the following estimate
¯ ³
´¯
X
¯
¯
kxkuθ ·∞·vθ ≤ lim ¯tr EM EMk (x)∗
w1k w2k ¯ + δ.
k
k→∞
Using (3.28) and applying Theorem 3.16 one more time
°
°
kxku ·∞·v ≤ sup °EM EM (x)°
θ
θ
k≥1
k
uθ ·∞·vθ
+ δ.
Thus, (3.22) follows for x ∈ Luθ (N )L∞ (M)Lvθ (N ) by letting δ → 0. Finally,
as in the case min(q0 , q1 ) < ∞, it remains to see that Luθ (N )L∞ (M)Lvθ (N ) is
dense in Xθ (M). Here we also need a different argument. Let us keep the notation
1/u0 + 1/v0 = 1/s = 1/u1 + 1/v1 . Then, we may assume that (u0 , v0 ) = (s, ∞)
and (u1 , v1 ) = (∞, s). Indeed, if we conclude the proof in this particular case,
the general case follows from the reiteration theorem for complex interpolation, see
e.g. [2]. This can be justified by means of Figure I, since the segment joining the
points (1/u0 , 1/v0 , 0) and (1/u1 , 1/v1 , 0) is always contained in the segment joining
(1/s, 0, 0) and (0, 1/s, 0). Thus, we assume in what follows that
h
i
Xθ (M) = Ls (N )L∞ (M)L∞ (N ), L∞ (N )L∞ (M)Ls (N )
θ
so that 1/uθ = (1 − θ)/s and 1/vθ = θ/s. By the density of
∆ = Ls (N )L∞ (M)L∞ (N ) ∩ L∞ (N )L∞ (M)Ls (N )
in Xθ (M), it suffices to approximate any element x ∈ ∆. In particular, we can
write x = a0 b0 and x = b1 a1 where a0 , a1 ∈ Ls (N ) and b0 , b1 ∈ M. Moreover, we
can assume that a0 = a1 . Indeed, taking
¡
¢1/2
a = a0 a∗0 + a∗1 a1 + δD2/s
ϕ
we have x = aa−1 a0 b0 = ac0 and x = b1 a1 a−1 a = c1 a with
kcj kM ≤ kbj kM
for j = 0, 1.
66
3. AN INTERPOLATION THEOREM
Then c1 = ac0 a−1 and we claim that aθ c0 a−θ is in M for all 0 < θ < 1. Indeed, let
φ be the n.f. finite weight φ(·) = tr(as ·) and let M oσφ R be the crossed product
with respect to the modular automorphism group associated to φ. Let us consider
the spectral projection pn = 1[1/n,n] (a). Then, for a fixed integer n the function
fn (z) = pn az c0 a−z pn = pn az pn c0 pn a−z pn
is analytic so that
° φ
°1−θ ° φ
°θ
kfn (θ)k ≤ sup °σt/s
(c0 )° °σt/s
(ac0 a−1 )° = kc0 k1−θ kac0 a−1 kθ = kc0 k1−θ kc1 kθ .
t∈R
Sending n to infinity, we deduce that aθ c0 a−θ is a bounded element of M oσφ R.
Moreover, using the dual action with respect to φ, we find that aθ c0 a−θ belongs to
M. Therefore, we obtain
x = a1−θ aθ c0 a−θ aθ ∈ Luθ (N )L∞ (M)Lvθ (N ).
We have seen that the intersection space ∆ is included in Luθ (N )L∞ (M)Lvθ (N ).
Hence, the result follows since ∆ is dense in Xθ (M). The proof of Lemma 3.1 (for
any von Neumann algebra) is therefore completed.
¤
3.5. Proof of the main interpolation theorem
To prove Theorem 3.2, we need to know a priori that Lu (N )Lq (M)Lv (N ) is
a Banach space for any indices (u, q, v) associated to a point (1/u, 1/v, 1/q) ∈ K.
This is a simple consequence of Lemma 3.1. Indeed, according to Lemma 2.5 and
Proposition 2.6, we know that our assertion is true for any (1/u, 1/v, 1/q) ∈ ∂∞ K.
In particular, it follows from Lemma 3.1 that
Luθ (N )Lqθ (M)Lvθ (N )
is a Banach space for any 0 ≤ θ ≤ 1 whenever (uj , qj , vj ) ∈ ∂∞ K for j = 0, 1
and 1/u0 + 1/q0 + 1/v0 = 1/u1 + 1/q1 + 1/v1 . In other words, according to the
notation introduced at the beginning of this chapter, this condition holds whenever
(1/uj , 1/vj , 1/qj ) ∈ Kτ ∩ ∂∞ K for j = 0, 1 and some 0 ≤ τ ≤ 1. Therefore, it
suffices to see that Kτ is the convex hull of Kτ ∩ ∂∞ K for any 0 ≤ τ ≤ 1. However,
this follows easily from Figure I. Note that Kτ is either a point (τ = 0), a triangle
(0 < τ ≤ 1/2), a pentagon (1/2 < τ < 1) or a parallelogram (τ = 1).
Observation 3.18. In fact, in the case of finite von Neumann algebras, Lemma
3.1 provides more information. Namely, according to (3.4) and the fact that Kτ
is the convex hull of Kτ ∩ ∂∞ K, we deduce that Nu Lq (M)Nv is a normed space
when equipped with ||| |||u·q·v for any (1/u, 1/v, 1/q) ∈ K. Moreover, Nu Lq (M)Nv
embeds isometrically in Lu (N )Lq (M)Lv (N ) as a dense subspace, something we
did not know up to now (see Observation 2.9). This means that Lemma 2.5 and
Proposition 2.6 hold for any point in K in the case of finite von Neumann algebras.
Proof of Theorem 3.2. Now we are ready to prove Theorem 3.2. The
arguments to be used follow the same strategy used for the proof of Lemma 3.1. In
particular, we only need to point out how to proceed. In first place, as usual, the
lower estimate follows by multilinear interpolation.
Step 1. Let us show the validity of Theorem 3.2 for finite von Neumann algebras
satisfying the boundedness condition (3.1). First we note that, once we know that
Lu (N )Lq (M)Lv (N ) is always a Banach space, the proof of Lemma 3.3 is still
3.5. PROOF OF THE MAIN INTERPOLATION THEOREM
67
valid for ending points (1/uj , 1/vj , 1/qj ) lying on K \ ∂∞ K. Then we follow the
proof of Lemma 3.1 for finite von Neumann algebras verbatim to deduce Theorem
3.2 in this case. Here is essential to observe (as we did in Remark 3.5) that the
proof of Lemma 3.1 for finite von Neumann algebras does not use at any point the
restriction 1/u0 + 1/q0 + 1/v0 = 1/u1 + 1/q1 + 1/v1 . Note also that, in order to
obtain the boundary estimates (3.6), Proposition 2.6 is needed. Here is where we
apply Observation 3.18. This proves Theorem 3.2 for finite von Neumann algebras.
Step 2. The next goal is to show that the corresponding conditional expectations
are contractive. First we observe that, according to Lemma 3.1, we can extend the
validity of Corollary 3.13 to any point
(1/u, ¢1/v, 1/q) ∈ K by complex interpolation.
¡
Here we use again that Kτ = conv Kτ ∩∂∞ K . Then, it is straightforward to see that
Corollary 3.15 also holds in this case. Indeed, first we apply complex interpolation
to obtain a contraction
EMk : Xθ (M) → Xθ (RM ) → Luθ (Nk )Lqθ (Mk )Lvθ (Nk ).
Second, the contractivity of
EM : Luθ (Nk )Lqθ (Mk )Lvθ (Nk ) → Luθ (N )Lqθ (M)Lvθ (N )
follows since, as we have seen, Corollary 3.13 holds for any point (1/u, 1/v, 1/q) ∈ K.
Step 3. We now prove Theorem 3.2 in the case min(q0 , q1 ) < ∞. It follows
easily from Step 2. Indeed, recalling again that the restriction 1/u0 + 1/q0 + 1/v0 =
1/u1 + 1/q1 + 1/v1 is not used in the proof of Lemma 3.1 (once we know the validity
of Corollary 3.15), the proof follows verbatim.
Step 4. Finally, we consider the case min(q0 , q1 ) = ∞. First we observe that the
first half of the proof of Lemma 3.1 for this case holds for any two ending points
pj = (1/uj , 1/vj , 0) in the square K ∩ {z = 0}. Thus it only remains to check that
Luθ (N )L∞ (M)Lvθ (N ) is dense in Xθ (M). Applying the reiteration theorem as we
did in the proof of Lemma 3.1, we may assume that p0 and p1 are in the boundary of
K ∩ {z = 0}. We have three possible situations. First we assume that p0 and p1 live
in the same edge of K ∩ {z = 0}. Let ∆ be the intersection of the interpolation pair.
In this case, we have ∆ = Lu0 (N )L∞ (M)Lv0 (N ) or ∆ = Lu1 (N )L∞ (M)Lv1 (N )
since the points of any edge of K ∩ {z = 0} are directed by inclusion. In particular,
we deduce ∆ ⊂ Luθ (N )L∞ (M)Lvθ (N ) from which the result follows. If p0 and
p1 live in consecutive edges of K ∩ {z = 0}, we have four choices according to the
common vertex v of the corresponding (consecutive) edges. Following Figure I we
may have v = 0, E, F, G. When v = E, G we are back to the situation above (one
endpoint is contained in the other) and there is nothing to prove. When v = 0, we
may assume w.l.o.g. that
Lu0 (N )L∞ (M)Lv0 (N ) =
Lu1 (N )L∞ (M)Lv1 (N ) =
Ls0 (N )L∞ (M),
L∞ (M)Ls1 (N ),
for some 2 ≤ s0 , s1 ≤ ∞. Moreover, we may also assume w.l.o.g. that s0 ≤ s1 .
This allows us to write 1/s0 = 1/s1 + 1/r for some index 2 ≤ r ≤ ∞. In particular,
Ls0 (N )L∞ (M) = (Ls1 (N )Lr (N ))L∞ (M). Let x ∈ ∆ so that
x = αγm0 = m1 β
with α, β ∈ Ls1 (N ), γ ∈ Lr (N ) and m0 , m1 ∈ M. Taking
¢1/2
¡
,
a = αα∗ + β ∗ β + δD2/s1
68
3. AN INTERPOLATION THEOREM
we may write x = ac0 = c1 a (so that c1 = ac0 a−1 ) with
c0 = (a−1 αγ)m0 ∈ Lr (N )L∞ (M) and
c1 = m1 βa−1 ∈ L∞ (M).
On the other hand, (1/r, 0, 0) and (0, 0, 0) are in the same edge of K∩{z = 0}. Thus
[Lr (N )L∞ (M), L∞ (M)]θ = Lrθ (N )L∞ (M) with 1/rθ = (1 − θ)/r. Using this
interpolation result and arguing as in the proof of Lemma 3.1 for min(q0 , q1 ) = ∞,
we easily obtain that aθ c0 a−θ ∈ Lrθ (N )L∞ (M). Thus we deduce
x = a1−θ aθ c0 a−θ aθ ∈ (Ls1 /(1−θ) (N )Lrθ (N ))L∞ (M)Ls1 /θ (N ).
Then, since the latter space is Luθ (N )L∞ (M)Lvθ (N ), we have seen that ∆ is
included in this space. This completes the proof for v = 0. When v = F, we may
assume w.l.o.g. that
Lu0 (N )L∞ (M)Lv0 (N ) = Ls0 (N )L∞ (M)L2 (N ),
Lu1 (N )L∞ (M)Lv1 (N ) = L2 (N )L∞ (M)Ls1 (N ).
Writing 1/2 = 1/s0 + 1/r0 = 1/s1 + 1/r1 for some 2 ≤ r0 , r1 ≤ ∞ we have
Ls0 (N )L∞ (M)L2 (N ) = Ls0 (N )L∞ (M)Lr1 (N )Ls1 (N ),
L2 (N )L∞ (M)Ls1 (N ) = Ls0 (N )Lr0 (N )L∞ (M)Ls1 (N ).
Thus, using our result for v = 0 we find
¡
¢
∆ = Ls0 (N ) L∞ (M)Lr1 (N ) ∩ Lr0 (N )L∞ (M) Ls1 (N )
¡
¢
⊂ Ls0 (N ) Lr0 /θ (N )L∞ (M)Lr1 /(1−θ) (N ) Ls1 (N )
⊂
Ls0 (θ) (N )L∞ (M)Ls1 (θ) (N ),
with 1/s0 (θ) = (1 − θ)/s0 + θ/2 and 1/s1 (θ) = (1 − θ)/2 + θ/s1 . This completes
the proof for consecutive edges. Finally, we assume that p0 and p1 live in opposite
edges of K ∩ {z = 0}. Since the two possible situations are symmetric, we only
consider the case
Lu0 (N )L∞ (M)Lv0 (N ) = Ls0 (N )L∞ (M),
Lu1 (N )L∞ (M)Lv1 (N ) = Ls1 (N )L∞ (M)L2 (N ).
If s0 ≥ s1 we clearly have
∆ = Lu0 (N )L∞ (M)Lv0 (N ) ⊂ Luθ (N )L∞ (M)Lvθ (N )
and there is nothing to prove. If s0 < s1 we have 1/s0 = 1/s1 + 1/r so that
¡
¢
∆ = Ls1 (N ) Lr (N )L∞ (M) ∩ L∞ (M)L2 (N )
¡
¢
⊂ Ls1 (N ) Lr/(1−θ) (N )L∞ (M)L2/θ (N )
=
Luθ (N )L∞ (M)Lvθ (N ).
This proves the assertion for opposite edges and so the space Luθ (N )L∞ (M)Lvθ (N )
is always dense in Xθ (M). The proof of Theorem 3.2 is completed.
¤
Remark 3.19. The key points to see that the proof of Lemma 3.1 applies
whenever we start with any two ending points (1/uj , 1/vj , 1/qj ) lying on K are the
following:
(a) Luj (N )Lqj (M)Lvj (N ) is a Banach space.
(b) Lemma 2.5 holds on K for finite von Neumann algebras.
(c) Corollary 3.15 also holds with ending points in K \ ∂∞ K.
3.5. PROOF OF THE MAIN INTERPOLATION THEOREM
69
Thus, it suffices to see that Lemma 3.1 gives (a), (b), (c) by complex interpolation.
On the other hand, it is worthy to explain with some more details why restriction
1/u0 + 1/q0 + 1/v0 = 1/u1 + 1/q1 + 1/v1 can be dropped. The only two points
where this restriction is needed (apart from the case min(q0 , q1 ) = ∞ which has
been discussed in Step 4 above) are in the proofs of Lemma 3.8 and Proposition
3.14. However, Lemma 3.8 is only needed to obtain Corollary 3.13 (which we have
auto-improved in Step 2 above by using Lemma 3.1). Moreover, as we also pointed
out in Step 2, Proposition 3.14 now follows from our improvement of Corollary 3.13.
Therefore, restriction 1/u0 + 1/q0 + 1/v0 = 1/u1 + 1/q1 + 1/v1 can be ignored.
CHAPTER 4
Conditional Lp spaces
We conclude the first part of this paper by studying the duals of amalgamated
Lp spaces and the subsequent applications of Theorem 3.2. Let us consider a von
Neumann algebra M equipped with a n.f. state ϕ and a von Neumann subalgebra
N of M. Let E : M → N denote the corresponding conditional expectation. We
consider any three indices (u, p, v) such that (1/u, 1/v, 1/p) belongs to K and we
define 1 ≤ s ≤ ∞ by 1/s = 1/u + 1/p + 1/v. Then, the conditional Lp space
Lp(u,v) (M, E)
is defined as the completion of Lp (M) with respect to the norm
n
o
¯
kxkLp(u,v) (M,E) = sup kαxβkLs (M) ¯ kαkLu (N ) , kβkLv (N ) ≤ 1 .
We shall show below that amalgamated and conditional Lp spaces are related
by duality. According to our main result in Chapter 3, this immediately provides
interpolation isometries of the form
h
i
p0
(4.1)
L(u
(M, E), Lp(u1 1 ,v1 ) (M, E) = Lp(uθ θ ,vθ ) (M, E).
0 ,v0 )
θ
Our aim now is to explore these identities, since they will be useful in the sequel.
Example 4.1. As in Chapter 2, several noncommutative function spaces arise
as particular cases of our notion of conditional Lp space. Let us mention four
particularly relevant examples:
(a) The noncommutative Lp spaces arise as
Lp (M) = Lp(∞,∞) (M, E).
(b) If p ≥ q and 1/r = 1/q − 1/p, the spaces Lp (N1 ; Lq (N2 )) arise as
¯ 2 , E),
Lp(2r,2r) (N1 ⊗N
¯ 2 → N1 is E = 1N1 ⊗ ϕN2 .
where the conditional expectation E : N1 ⊗N
(c) If 2 ≤ p ≤ ∞ and 1/p + 1/q = 1/2, Lemma 1.8 gives
Lrp (M, E)
= Lp(q,∞) (M, E),
Lcp (M, E)
= Lp(∞,q) (M, E).
As we shall see below, Lp (M; Rpn ) and Lp (M; Cpn ) are particular cases.
(d) In Chapter 7 we will also identify certain asymmetric noncommutative Lp
spaces as particular cases of conditional Lp spaces. We prefer in this case
to leave the details for Chapter 7.
71
72
4. CONDITIONAL Lp SPACES
4.1. Duality
Note that given (1/u, 1/v, 1/q) ∈ K, we usually take 1/p = 1/u + 1/q + 1/v. In
the following it will be more convenient to replace p by p0 , the index conjugate to
p. Let us consider the following restriction of (2.1)
(4.2)
1<q<∞
and 2 < u, v ≤ ∞
and 0 <
1 1 1
1
+ + = 0 < 1.
u q
v
p
Theorem 4.2. Let E : M → N denote the conditional expectation of M onto
N and let 1 < p < ∞ given by 1/q 0 = 1/u + 1/p + 1/v, where the indices (u, q, v)
satisfy (4.2) and q 0 is conjugate to q. Then, the following isometric isomorphisms
hold via the anti-linear duality bracket hx, yi = tr(x∗ y)
¡
¢∗
Lu (N )Lq (M)Lv (N ) = Lp(u,v) (M, E),
¡ p
¢∗
L(u,v) (M, E) = Lu (N )Lq (M)Lv (N ).
Proof. Let us consider the map
¡
¢ ¡
¢∗
Λp : x ∈ Lp(u,v) (M, E) 7→ tr x∗ · ∈ Lu (N )Lq (M)Lv (N ) .
We first show that Λp is an isometry
n¯
o
°
°
¯¯
°Λp (x)°
= sup ¯tr(x∗ y)¯ ¯ inf kαkLu (N ) kzkLq (M) kβkLv (N ) ≤ 1
(u·q·v)∗
y=αzβ
n¯
o
¯
¯
= sup ¯tr(βx∗ αz)¯ ¯ kαkLu (N ) , kzkLq (M) , kβkLv (N ) ≤ 1
n°
o
°
¯
= sup °βx∗ α°L 0 (M) ¯ kαkLu (N ) ≤ 1, kβkLv (N ) ≤ 1
q
n°
o
°
¯
= sup °α∗ xβ ∗ °L 0 (M) ¯ kαkLu (N ) ≤ 1, kβkLv (N ) ≤ 1
q
= kxkLp
(u,v)
(M,E) .
It remains to see that Λp is surjective. To that aim we use again the solid K
in Figure I. Note that, since the case u = v = ∞ is clear, we may assume that
min(u, v) < ∞. In that case any point (1/u, 1/v, 1/q) with (u, q, v) satisfying (4.2)
lies in the interior of a segment S contained in K and satisfying
(a) One end point of S lies in the open interval (0, A).
(b) The segment S belongs to a plane parallel to ACDF.
According to Theorem 3.2, this means that
h
i
(4.3)
Lu (N )Lq (M)Lv (N ) = Lp0 (M), Lu1 (N )Lq1 (M)Lv1 (N ) ,
θ
for some 0 < θ < 1 and some (1/u1 , 1/v1 , 1/q1 ) ∈ K. Recalling that 1 < p < ∞, we
know that Lp0 (M) is reflexive. In particular, the same holds for the interpolation
space in (4.3) and we obtain the following isometric isomorphism
¡
¢∗ h
¡
¢∗ i
Lu (N )Lq (M)Lv (N ) = Lp (M), Lu1 (N )Lq1 (M)Lv1 (N )
.
θ
Moreover, since 0 < θ < 1, we know that the intersection
¡
¢∗
Lp (M) ∩ Lu1 (N )Lq1 (M)Lv1 (N )
4.2. CONDITIONAL L∞ SPACES
73
¡
¢∗
is norm dense in the space Lu (N )Lq (M)Lv (N ) . On the other hand, recalling
that 1/u1 + 1/q1 + 1/v1 = 1/p0 , we know from the definition of amalgamated spaces
that
Lp0 (M) = Lp0 (M) + Lu1 (N )Lq1 (M)Lv1 (N ).
¡
¢∗
Hence, Lp (M) = Lp (M) ∩ Lu1 (N )Lq1 (M)Lv1 (N ) is norm dense in
¡
¢∗
Lu (N )Lq (M)Lv (N ) .
Therefore, Λp has dense range since
Lp (M) ⊂ Lp(u,v) (M, E).
For the second part, we use from (4.3) that Lu (N )Lq (M)Lv (N ) is reflexive.
¤
Remark 4.3. The first part of the proof of Theorem 4.2 holds for any point
(1/u, 1/v, 1/q) in the solid K. In particular, we always have an isometric embedding
¡
¢∗
Lp(u,v) (M, E) −→ Lu (N )Lq (M)Lv (N ) .
Remark 4.4. Note that the indices excluded in Theorem 4.2 by the restriction
imposed by property (4.2) are the natural ones. For instance, the last restriction
0 < 1/u + 1/q + 1/v < 1 only affects conditional/amalgamated L1 and L∞ spaces,
which are not expected to be reflexive. Moreover, the spaces Lu (N )L∞ (M)Lv (N )
are not reflexive in general. Indeed, let us consider the particular case in which N
is the complex field. These spaces collapse into L∞ (M) which is not reflexive.
4.2. Conditional L∞ spaces
Among the non-reflexive conditional spaces, we concentrate on some properties
of conditional L∞ spaces that will be needed in the second half of this paper. Note
that, given indices (u, q, v) satisfying (2.1) with 1/u+1/q+1/v = 1, we have defined
the space
L∞
(u,v) (M, E)
as the completion of L∞ (M) with respect to the norm
n
o
¯
¯ kαkL (N ) , kβkL (N ) ≤ 1 .
kxkL∞
=
sup
kαxβk
(M,E)
L
(M)
0
u
v
q
(u,v)
According to Remark 4.3, we know that this space embeds isometrically in
¡
¢∗
L∞
(u,v) (M, E) = Lu (N )Lq (M)Lv (N ) .
Proposition 4.5. The following properties hold:
0
i) L∞
(u,v) (M, E) is contractively included in Lq (M).
∗
∞
ii) L∞ (M) and L∞
(u,v) (M, E) are weak dense subspaces of L(u,v) (M, E).
Proof. Let us consider the map
¡ 1
1¢
u
v
∈ Lq0 (M).
j : ϕ ∈ L∞
(u,v) (M, E) → ϕ D · D
By Proposition 2.6, the map j is clearly injective. On the other hand,
° ¡ 1
¢°
°ϕ D u · D v1 °
Lq0 (M)
n¯ ¡ 1
o
1 ¢¯ ¯
= sup ¯ϕ D u yD v ¯ ¯ kykq ≤ 1
o
n¯ ¡ 1
°
°
1 ¢¯ ¯ °
1
1°
≤ sup ¯ϕ D u αyβD v ¯ ¯ °D u α°Lu (N ) , kykq , °βD v °Lv (N ) ≤ 1
74
4. CONDITIONAL Lp SPACES
=
kϕkL∞
(M,E) .
(u,v)
The last identity follows from Proposition 2.6. This shows that j is a contraction.
For the second part, it suffices to see that L∞ (M) is weak∗ dense. However, it is
clear from the definition of amalgamated spaces that the inclusion map
Lu (N )Lq (M)Lv (N ) −→ L1 (M)
is injective. In particular, taking adjoints we get the announced weak∗ density. ¤
4.3. Interpolation results and applications
In this last section we consider some interesting particular cases of the dual
version (4.1) of Theorem 3.2. One of the applications we shall consider generalizes
Pisier’s interpolation result [44] and Xu’s recent extension [69].
Theorem 4.6. Let N be a von Neumann subalgebra of M and let E : M → N
be the corresponding conditional expectation. Assume that (uj , qj , vj ) satisfy (4.2)
for j = 0, 1 and that 1/uj + 1/qj + 1/vj = 1/p0j . Then, if pj denotes the conjugate
index to p0j , the following isometric isomorphism holds
h
i
p0
p1
L(u
(M,
E),
L
= Lp(uθ θ ,vθ ) (M, E).
(M,
E)
(u1 ,v1 )
0 ,v0 )
θ
Moreover, if 2 ≤ uj , vj ≤ ∞ for j = 0, 1, we also have
h
iθ
∞
L∞
= L∞
(u0 ,v0 ) (M, E), L(u1 ,v1 ) (M, E)
(uθ ,vθ ) (M, E).
Proof. The first part follows automatically from Theorem 3.2 and Theorem
4.2. The second part follows from Theorem 3.2 and the duality properties which
link the complex interpolation brackets [ , ]θ and [ , ]θ , see e.g. [2].
¤
Now we study some consequences of Theorem 4.6. We shall content ourselves
by exploring only the case p0 = p1 . This restriction is motivated by the applications
we are using in the successive chapters. The last part of the following result requires
to introduce some notation. As usual, we shall write Rpn (resp. Cpn ) to denote the
interpolation space [Rn , Cn ]1/p (resp. [Cn , Rn ]1/p ), where Rn and Cn denote the
n-dimensional row and column Hilbert spaces. Alternatively, we may define Rpn
and Cpn as the first row and column subspaces of the Schatten class Spn . On the
other hand, given an element x0 in a von Neumann algebra M, we shall consider
the mappings Lx0 and Rx0 on M defined respectively as follows
Lx0 (x) = x0 x
and
Rx0 (x) = xx0 .
Corollary 4.7. Let N be a von Neumann subalgebra of M and let E : M → N
be the corresponding conditional expectation of M onto N . Then, we have the
following isometric isomorphisms:
i) If 2 ≤ p < ∞ and 2 < q ≤ ∞ are such that 1/2 = 1/p + 1/q, we have
£
¤
L (M), Lrp (M, E) θ = Lp(s,∞) (M, E),
1
θ
£ p
¤
with
= .
Lp (M), Lcp (M, E) θ = Lp(∞,s) (M, E),
s
q
In the case p = ∞, we obtain
£
¤
L∞ (M), Lr∞ (M, E) θ
£
¤
L∞ (M), Lc∞ (M, E) θ
=
=
L∞
(2/θ,∞) (M, E),
∞
L(∞,2/θ) (M, E).
4.3. INTERPOLATION RESULTS AND APPLICATIONS
75
ii) If 2 ≤ p < ∞ and 2 < q ≤ ∞ are such that 1/2 = 1/p + 1/q, we have
¤
£ c
Lp (M, E), Lrp (M, E) θ = Lp(u,v) (M, E)
with
(1/u, 1/v) = (θ/q, (1 − θ)/q).
iii) Let us define for 2 ≤ p < ∞
£
¤
Xθ (M) = Lp (M; Cpn ), Lp (M; Rpn ) θ .
Then, if 1/w = 1/p + 1/v with 1/v = (1 − θ)/q, we deduce
n
n
°X
°2
°X
°
°
°
°
°
xk ⊗ δk °
=°
Lxk Rx∗k : Lv/2 (M) → Lw/2 (M)°.
°
Xθ (M)
k=1
k=1
Proof. The assertions in the first part of i) and ii) follow from Theorem 4.6
after the obvious identifications, see Example 4.1. For the last part of i) we only
prove the first identity since the second one follows in the same way. According to
Remark 4.3, Lr∞ (M, E) is the closure of L∞ (M) in L∞
(2,∞) (M, E). Then, applying
a well-known property of the complex method (see e.g. [2, Theorem 4.2.2]) we find
¤
£
¤
£
for 0 < θ < 1.
L∞ (M), Lr∞ (M, E) θ = L∞ (M), L∞
(2,∞) (M, E) θ
By Berg’s theorem, we have an isometric inclusion
£
¤
£
¤θ
∞
L∞ (M), L∞
(2,∞) (M, E) θ ⊂ L∞ (M), L(2,∞) (M, E) .
Therefore given x ∈ L∞ (M), Theorem 4.6 gives
kxk[L∞ (M),Lr∞ (M,E)]θ
=
kxk[L∞ (M),L∞
(M,E)]θ
(2,∞)
=
kxk[L∞ (M),L∞
(M,E)]θ
(2,∞)
=
kxkL∞
(M,E) .
(2/θ,∞)
The assertion then follows by a simple density argument
£
¤
L∞ (M), Lr∞ (M, E) θ = L∞
(2/θ,∞) (M, E).
Finally, for part iii) we consider the direct sum M⊕n = M ⊕ M ⊕
P· · · ⊕ M with
n terms and equipped with the n.f. state ϕn (x1 , x2 , · · · , xn ) = n1 k ϕ(xk ). The
natural conditional expectation is given by
En :
n
X
n
xk ⊗ δk ∈ M⊕n 7→
k=1
1X
xk ∈ M.
n
k=1
It is clear that we have the isometries
Lp (M; Rpn )
=
Lp (M; Cpn )
=
√
√
n Lrp (M⊕n , En ),
n Lcp (M⊕n , En ).
According to ii) and the definition of the norm in Lp(u,v) (M⊕n , En ), we have
n
°2
°X
°
°
xk ⊗ δk °
°
k=1
(
Xθ (M)
n
°2
°X
°
°
αxk β ⊗ δk °
= sup n°
k=1
L2 (M⊕n )
)
¯
¯ kαkL (M) , kβkL (M) ≤ 1
u
v
76
4. CONDITIONAL Lp SPACES
(
n
X
¡
¢¯
= sup
tr αxk ββ ∗ x∗k α∗ ¯ kαkLu (M) , kβkLv (M) ≤ 1
(
k=1
n
°X
°
sup °
=
(
k=1
n
°X
°
sup °
=
Recalling that
P
k
)
°
°
Lxk Rx∗k (ββ )°
∗
L(u/2)0 (M)
°
°
Lxk Rx∗k (γ)°
L(u/2)0 (M)
k=1
)
¯
¯ kβkL (M) ≤ 1
v
)
¯
¯ γ ≥ 0, kγkL (M) ≤ 1 .
v/2
Lxk Rx∗k is positive and that (u/2)0 = w/2, we conclude.
¤
Observation 4.8. Arguing as in Corollary 4.7, we easily obtain
¤
£
∞
L∞ (M), L∞
(u,v) (M, E) θ = L(u/θ,v/θ) (M, E).
These results shall be frequently used in the successive chapters. Moreover, let us
also mention that Corollary 4.7 i) is needed in [21] to study the noncommutative
John-Nirenberg theorem.
At the time of this writing we do not know whether Corollary 4.7 ii) extends to
p = ∞ in full generality. We will now show that the equality holds when restricted
to elements in M. This result will play a very important role in the sequel.
Lemma 4.9. If 1 < p, q < ∞ and z ∈ M, we have
n
o
¯
¯
∞
inf kxkL∞
kyk
z
=
xy,
x,
y
∈
M
≤ kzkL∞
(M,E)
L(∞,2q) (M,E)
(M,E) .
(2p,∞)
(2p,2q)
Proof. On M we define the norm
kzkh = inf kxkL∞
(M,E) kykL∞
(M,E)
(2p,∞)
(∞,2q)
z=xy
where the infimum is taken over x, y ∈ L∞ (M). It is easy to check that we do not
need to consider sums here. Let us assume that kzkh = 1. By the Hahn-Banach
theorem there exists a linear functional φ : M → C such that φ(z) = 1 and
¯X
¯
°X
°1/2
°X
°1/2
¯
¯
°
°
°
°
(4.4)
φ(xk yk )¯ ≤ sup °
axk x∗k a∗ °
sup °
b∗ yk∗ yk b° .
¯
k
k
kak2p ≤1
p
k
kbk2q ≤1
q
Note that kzkh ≤ kzk∞ and thus φ is continuous. It follows immediately that we
may move the absolute values in (4.4) inside. Thus, we get
³X
´1/2
³X
´1/2
X
b∗ yk∗ yk bd
.
axk x∗k a∗ c
sup tr
|φ(xk yk )| ≤ sup tr
k
k
a,c
k
b,d
Here we take the supremum over (a, c, b, d) in
BL2p (N ) × B+
L
p0 (M)
× BL2q (N ) × B+
L 0 (M) ,
q
∗
all equipped with the weak topology. Using the standard Grothendieck-Pietsch
separation argument as in Theorem 3.16 we obtain two probability measures µ1
and µ2 such that
³Z
´1/2 ³ Z
´1/2
|φ(xy)| ≤
tr(axx∗ a∗ c) dµ1 (a, c)
tr(b∗ y ∗ ybd) dµ2 (b, d)
.
Since L2p (N )Lp0 (M)L2p (N ) and L2q (N )Lq0 (M)L2q (N ) are Banach spaces,
Z
Z
∗
α = a ca dµ1 (a, c) and β = bdb∗ dµ2 (b, d)
4.3. INTERPOLATION RESULTS AND APPLICATIONS
77
are positive elements respectively in the unit balls of
L2p (N )Lp0 (M)L2p (N )
and L2q (N )Lq0 (M)L2q (N ).
Therefore we find a1 , a2 ∈ BL2p (N ) and c1 , c2 ∈ BL2p0 (M) such that α = a1 c1 c2 a2 .
We deduce from the Cauchy-Schwartz inequality and the arithmetic-geometric
mean inequality that
¯
¯
tr(xx∗ α) = ¯tr(xx∗ a1 c1 c2 a2 )¯
¯
¯
= ¯tr(c2 a2 xx∗ a1 c1 )¯
¡
¢1/2 ¡ ∗ ∗ ∗
¢1/2
≤ tr c2 a2 xx∗ a∗2 c∗2
tr c1 a1 xx a1 c1
³
a1 c1 c∗1 a∗1 + a∗2 c∗2 c2 a2 ´
≤ tr xx∗
.
2
We could consider a = (a∗1 a1 + a∗2 a2 )1/2 to deduce that
a1 c1 c∗1 a∗1 + a∗2 c∗2 c2 a2 −1
a
2
is a positive element in Lp0 (M) of norm ≤ 1. This is not enough for our purposes.
However, following the proof of the triangle inequality in Lemma 2.5, we may apply
Devinatz’s theorem one more time to find an operator a ∈ (1 + ε)BL2p (N ) with full
support and c ∈ BLp0 (M) such that
a−1
a1 c1 c∗1 a∗1 + a∗2 c∗2 c2 a2
= a∗ ca.
2
We leave the details to the interested reader. This implies that c is positive and
tr(xx∗ α) ≤ tr(xx∗ a∗ ca) = kc1/2 axk22 .
The same argument for β gives b ∈ (1 + ε)BL2q (N ) and d ∈ B+
L 0 (M) such that
q
tr(y ∗ yβ) ≤ tr(y ∗ ybdb∗ ) = kybd1/2 k22 .
This yields
|φ(xy)| ≤ kc1/2 axk2 kybd1/2 k2 .
From this it is easy to find a contraction u ∈ M such that
1
1
φ(xy) = tr(uc 2 axybd 2 ).
If 1/r = 1/2p + 1/2q, we deduce from Hölder’s inequality that
¯
1
1 ¯
kzkh = |φ(z)| = ¯tr(uc 2 azbd 2 )¯ ≤ kazbkLr (M) ≤ (1 + ε)2 kzkL∞
(M,E) .
(2p,2q)
Finally, recalling that ε > 0 is arbitrary, the assertion follows by taking ε → 0.
Corollary 4.10. Assume that 1 ≤ p0 , p1 , q0 , q1 ≤ ∞ satisfy
max(p0 , p1 ), max(q0 , q1 ) > 1
and
min(p0 , p1 ), min(q0 , q1 ) < ∞.
Then, given x ∈ M we have
kxkL∞
(2p
θ ,2qθ )
(M,E)
= kxk[L∞
(2p
0 ,2q0 )
(M,E),L∞
(2p
1 ,2q1 )
(M,E)]θ .
In particular, for θ = 1/q we obtain
n
o
¯
kxk[Lc∞ (M,E),Lr∞ (M,E)]θ = sup kaxbkL2 (M) ¯ kakL2q (N ) , kbkL2q0 (N ) ≤ 1 .
¤
78
4. CONDITIONAL Lp SPACES
Proof. The upper estimate is an easy application of trilinear interpolation.
For the converse we apply Lemma 4.9 so that for any ε > 0 we can always find a
factorization x = x1 x2 satisfying
kx1 kL∞
(2p
θ ,∞)
(M,E) kx2 kL∞
(∞,2q
θ)
(M,E)
≤ (1 + ε)kxkL∞
(2p
θ ,2qθ )
(M,E) .
According to Corollary 4.7 i) we know that
£
¤
r
L∞
(2pθ ,∞) (M, E) = L∞ (M), L∞ (M, E) 1/p .
θ
Taking θ0 = 1/p0 and θ1 = 1/p1 , the reiteration theorem implies that
¤
¤
£
£
L∞ (M), Lr∞ (M, E) 1/p = [L∞ (M), Lr∞ (M, E)]θ0 , [L∞ (M), Lr∞ (M, E)]θ1 θ .
θ
In particular,
£
¤
£
¤
∞
L∞ (M), Lr∞ (M, E) 1/p = L∞
(2p0 ,∞) (M, E), L(2p1 ,∞) (M, E) θ .
θ
Therefore, we get
kx1 k[L∞
(2p
0 ,∞)
(M,E),L∞
(2p
1 ,∞)
(M,E)]θ
= kx1 kL∞
(2p
(M,E)]θ
= kx2 kL∞
(∞,2q
θ ,∞)
(M,E) .
Similarly, we have
kx2 k[L∞
(∞,2q
0)
(M,E),L∞
(∞,2q
1)
θ)
(M,E) .
On the other hand the inequality
kxykL∞
(M,E) ≤ kxkL∞
(M,E) kykL∞
(M,E)
(2p,2q)
(2p,∞)
(∞,2q)
holds for all 1 ≤ p, q ≤ ∞. Thus, by bilinear interpolation we deduce
kxk[L∞
(2p
0 ,2q0 )
≤ kx1 k
(M,E),L∞
(2p
1 ,2q1 )
(M,E)]θ
[L∞
(M,E),L∞
(M,E)]θ
(2p0 ,∞)
(2p1 ,∞)
kx2 k[L∞
(∞,2q
0)
(M,E),L∞
(∞,2q
1)
(M,E)]θ .
Combining the previous estimates and taking ε → 0 we obtain the assertion.
¤
Remark 4.11. If M is dense in the intersection Lr∞ (M, E) ∩ Lc∞ (M, E), then
Corollary 4.7 ii) extends to p = ∞. This holds for instance when the inclusion
¯ with A finite-dimensional. However,
N ⊂ M has finite index or when M = A⊗N
at the time of this writing it is not clear whether the density assumption is satisfied
for general conditional expectations. On the other hand, we note that the validity
of Corollary 4.7 ii) for p = ∞ can be understood as a conditional version of Pisier’s
interpolation result [44]. Indeed, taking θ = 1/q we easily find that
n
o
¯
kxk[Lc∞ (M,E),Lr∞ (M,E)]θ = sup kx∗ axkLq (M) ¯ kakLq (N ) ≤ 1
n
o
¯
= sup kxbx∗ kLq0 (M) ¯ kbkLq0 (N ) ≤ 1 .
Furthermore, according to Theorem 4.6, this also applies for 2 ≤ p ≤ ∞ (we
leave the details to the reader). In particular, we also find a conditional version of
Xu’s interpolation result [69]. Moreover, when 1 ≤ p ≤ 2 this result follows from
Theorem 3.2 instead of Theorem 4.6.
CHAPTER 5
Intersections of Lp spaces
In a second part of this paper, we study intersections of certain generalized
Lp spaces. Our main goal is to prove a noncommutative version of (Σpq ), see the
Introduction. In this chapter we begin by proving certain interpolation results
for intersections. As usual we consider a von Neumann subalgebra N of M with
corresponding conditional expectation E : M → N . Then given a positive integer
n, 1 ≤ q ≤ p ≤ ∞ and 1/r = 1/q − 1/p, we define the following intersection spaces
1
1
1
1
Rn2p,q (M, E) = n 2p L2p (M) ∩ n 2q L2p
(2r,∞) (M, E),
n
C2p,q
(M, E)
= n 2p L2p (M) ∩ n 2q L2p
(∞,2r) (M, E).
Our main result in this chapter shows that the two families of intersection spaces
considered above are interpolation scales in the index q. That is, the intersections
commute with the complex interpolation functor. Indeed, we obtain the following
isomorphisms with relevant constants independent of n
¤
£ n
R2p,1 (M, E), Rn2p,p (M, E) θ ' Rn2p,q (M, E),
(5.1)
¤
£ n
n
n
(M, E),
(M, E) θ ' C2p,q
C2p,1 (M, E), C2p,p
and with 1/q = 1 − θ + θ/p. Moreover, we shall also prove that
\
£ n
¤
1−θ
1
θ
n
n u + 2p + v L2p
(5.2)
R2p,1 (M, E), C2p,1
(M, E) θ '
( u
u,v∈{2p0 ,∞}
v
1−θ , θ )
(M, E).
5.1. Free Rosenthal inequalities
Our aim in this section is to present the free analogue given in [26] of Rosenthal
inequalities [56], where mean-zero independent random variables are replaced by
free random variables. This will be one of the key tools needed for the proof of
the isomorphisms (5.1) and (5.2). For the sake of completeness we first recall the
construction of reduced amalgamated free products.
5.1.1. Amalgamated free products. The basics of free products without
amalgamation can be found in [67]. Let A be a von Neumann algebra equipped
with a n.f. state φ and let N be a von Neumann subalgebra of A. Let EN : A → N
be the corresponding conditional expectation onto N . A family A1 , A2 , . . . , An of
von Neumann subalgebras of A, having N as a common subalgebra, is called freely
independent over EN if
EN (a1 a2 · · · am ) = 0
whenever EN (ak ) = 0 for all 1 ≤ k ≤ m and ak ∈ Ajk with j1 6= j2 6= · · · 6= jm . As in
the scalar-valued case, operator-valued freeness admits a Fock space representation.
79
80
5. INTERSECTIONS OF Lp SPACES
We first assume that A1 , A2 , . . . , An are C∗ -algebras having N as a common
C -subalgebra and that Ek = EN |Ak are faithful conditional expectations. Let us
consider the mean-zero subspaces
n
o
¯
◦
Ak = ak ∈ Ak ¯ Ek (ak ) = 0 .
∗
◦
◦
◦
Following [9, 65], we consider the Hilbert N -module Aj1 ⊗ Aj2 ⊗ · · · Ajm (where the
tensor products are amalgamated over the von Neumann subalgebra N ) equipped
with the N -valued inner product
­
®
¡
¢
a1 ⊗ · · · ⊗ am , b1 ⊗ · · · ⊗ bm = Ejm a∗m · · · Ej2 (a∗2 Ej1 (a∗1 b1 ) b2 ) · · · bm .
Then, the usual Fock space is replaced by the Hilbert N -module
M
M
◦
◦
◦
HN = N ⊕
Aj1 ⊗ Aj2 ⊗ · · · ⊗ Ajm .
m≥1 j1 6=j2 6=···6=jm
The direct sums above are assumed to be N -orthogonal. Let L(HN ) stand for the
algebra of adjointable maps on HN . A linear right N -module map T : HN → HN
is called adjointable whenever there exists S : HN → HN such that
hx, Tyi = hSx, yi
for all
x, y ∈ HN .
Let us recall how elements in Ak act on HN . We decompose any ak ∈ Ak as
◦
ak = ak + Ek (ak ).
Any element in N acts on HN by left multiplication. Therefore, it suffices to define
the action of mean-zero elements. The ∗-homomorphism πk : Ak → L(HN ) is then
defined as follows
 ◦

ak ⊗ bj1 ⊗ · · · ⊗ bjm ,
if k 6= j1



◦
πk (ak )(bj1 ⊗ · · · ⊗ bjm ) =
◦

Ek (ak bj1 ) bj2 ⊗ · · · ⊗ bjm ⊕


¢
◦
 ¡◦
ak bj1 − Ek (ak bj1 ) ⊗ bj2 ⊗ · · · ⊗ bjm , if k = j1 .
This definition also applies for the empty word. Now, since the algebra L(HN )
is a C∗ -algebra (c.f. [34]), we can define the reduced N -amalgamated free product
C∗ (∗N Ak ) as the C∗ -closure of linear combinations of operators of the form
πj1 (a1 )πj2 (a2 ) · · · πjm (am ).
Now we consider the case in which N and A1 , A2 , . . . , An are von Neumann
algebras. Let ϕ : N → C be a n.f. state on N . This provides us with the induced
states ϕk : Ak → C given by ϕk = ϕ ◦ Ek . The Hilbert space
¢
¡◦
◦
◦
L2 Aj1 ⊗ Aj2 ⊗ · · · ⊗ Ajm , ϕ
◦
◦
◦
is obtained from Aj1 ⊗ Aj2 ⊗ · · · ⊗ Ajm by considering the inner product
³­
­
®
®´
a1 ⊗ · · · ⊗ am , b1 ⊗ · · · ⊗ bm ϕ = ϕ a1 ⊗ · · · ⊗ am , b1 ⊗ · · · ⊗ bm .
Then we define the orthogonal direct sum
M
M
¢
¡◦
◦
◦
Hϕ = L2 (N ) ⊕
L2 Aj1 ⊗ Aj2 ⊗ · · · ⊗ Ajm , ϕ .
m≥1 j1 6=j2 6=···6=jm
5.1. FREE ROSENTHAL INEQUALITIES
81
Let us consider the ∗-representation λ : L(HN ) → B(Hϕ ) defined by λ(T)x = Tx.
The faithfulness of λ is implied by the fact that ϕ is also faithful. Let M(HN )
be the von Neumann algebra generated in B(Hϕ ) by L(HN ). Then, we define the
reduced N -amalgamated free product ∗N Ak as the weak∗ closure of C∗ (∗N Ak ) in
M(HN ). After decomposing
◦
ak = ak + Ek (ak )
¡
◦
◦ ¢
and identifying Ak with λ πk (Ak ) , we can think of ∗N Ak as
³
´00
M
M
◦ ◦
◦
∗N Ak = N ⊕
Aj1 Aj2 · · · Ajm .
m≥1 j1 6=j2 6=···6=jm
Let us consider the orthogonal projections
Q∅ : H ϕ
→
Qj1 ···jm : Hϕ
→
L2 (N ),
¡◦
¢
◦
◦
L2 Aj1 ⊗ Aj2 ⊗ · · · ⊗ Ajm , ϕ .
If we also consider the projection QAk = Q∅ + Qk , the following mappings
EN : x ∈ ∗N Ak
EAk : x ∈ ∗N Ak
7→ Q∅ xQ∅ ∈ N ,
7
→
QAk xQAk ∈ Ak ,
are faithful conditional expectations. Then, it turns out that A1 , A2 , . . . , An are
von Neumann subalgebras of ∗N Ak freely independent over EN . Reciprocally, if
A1 , A2 , . . . , An is a collection of von Neumann subalgebras of A freely independent
over EN : A → N and generating A, then A is isomorphic to ∗N Ak .
Remark 5.1. Let N1 and N2 be von Neumann algebras and assume that N2
is equipped with a n.f. state ϕ2 . A relevant example of the construction outlined
above is the following. Let A = N1 ⊗ N2 and let us consider the conditional
expectation EN1 : A → N1 defined by
EN1 (x1 ⊗ x2 ) = x1 ⊗ ϕ2 (x2 )1.
Assume that A1 , A2 , . . . , An are freely independent subalgebras of N2 over ϕ2 . Then,
it is well-known that N1 ⊗A1 , N1 ⊗A2 , . . . , N1 ⊗An is a family of freely independent
subalgebras of A over EN1 , see e.g. Section 7 of [17]. In particular, if A1 , A2 , . . . , An
generate N2 , we obtain
¡ n
¢
(5.3)
A = N1 ⊗ ∗ Ak = ∗N1 (N1 ⊗ Ak ).
k=1
5.1.2. A Rosenthal/Voiculescu type inequality. In this paragraph we
recall the free analogue [26] of Rosenthal inequalities [56] for mean-zero random
variables and prove a simple consequence of it. Let A1 , A2 , . . . , An be a family of von
Neumann algebras having N as a common von Neumann subalgebra and let ∗N Ak
be the corresponding amalgamated free product. Given a family a1 , a2 , . . . , an in
∗N Ak we consider the row and column conditional square functions
n
³X
´1/2
r
Scond
(a) =
EN (ak a∗k )
,
k=1
c
Scond
(a)
=
n
³X
k=1
´1/2
EN (a∗k ak )
.
82
5. INTERSECTIONS OF Lp SPACES
◦
Theorem 5.2. If 2 ≤ p ≤ ∞ and a1 , a2 , . . . , an ∈ Lp (∗N Ak ) with ak ∈ Lp (Ak )
for 1 ≤ k ≤ n, the following equivalence of norms holds with relevant constants
independent of p or n
n
n
°
°X
´1/p °
³X
°
° c
°
°
°
r
a
∼
kak kpp
+ °Scond
(a)°p + °Scond
(a)°p .
°
k°
k=1
p
k=1
This is the operator-valued/free analogue of Rosenthal’s original result. On the
other hand, a noncommutative analogue was obtained in [30] for general algebras
(non necessarily free products), see also [68] for the notion of noncommutative
independence (called order independence) employed in it. Recalling that freeness
implies order independence, Theorem 5.2 follows from [30] for 2 ≤ p < ∞. However,
the constants there are not uniformly bounded as p → ∞, in sharp contrast with
the situation in Theorem 5.2. This is another example of an Lp inequality involving
independent random variables which only holds in the limit case as p → ∞ when
considering their free analogue. This constitutes a significant difference in this
paper. Theorem 5.2 for p = ∞ was proved in [17] and constitutes the operator
valued extension of Voiculescu’s inequality [66]. Finally, we refer the reader to [26]
for a generalization of Theorem 5.2, where a1 , a2 , . . . , an are replaced by certain
words of a fixed degree d ≥ 1.
The following result is a standard application for positive random variables.
Corollary 5.3. If 2 ≤ p ≤ ∞ and a1 , . . . , an ∈ Lp (∗N Ak ) with ak ∈ Lp (Ak )
for 1 ≤ k ≤ n, the following equivalence of norms holds with relevant constants
independent of p or n
n
n
°¡ X
³X
´1/p °
°
¢ °
°
∗ 1/2 °
r
a
a
∼
kak kpp
+ °Scond
(a)°p ,
°
°
k k
°¡
°
°
k=1
n
X
p
¢1/2 °
°
a∗k ak
°
p
k=1
k=1
n
³X
∼
kak kpp
´1/p
° c
°
+ °Scond
(a)°p .
k=1
Proof. We we clearly have
n
n
³X
´1/p °¡ X
¢1/2 °
°
°
kak kpp
≤°
ak a∗k
° .
k=1
p
k=1
Indeed, our claim follows by complex interpolation from the trivial case p = ∞ and
the case p = 2, where the equality clearly holds. This, together with the fact that
EN is a contraction on Lp/2 (∗N Ak ), proves the lower estimate with constant 2. For
the upper estimate, we begin with the triangle inequality in Lp (∗N Ak ; Rpn )
n
n
n
°¡ X
°
°X
°
°X
¢1/2 °
◦
°
°
°
°
°
°
ak ⊗ e1k ° = A + B.
ak a∗k
EN (ak ) ⊗ e1k ° + °
°
° ≤°
k=1
p
p
k=1
p
k=1
To estimate A, we apply Kadison’s inequality (see Lemma 1.4 i))
n
n
°³ X
°³ X
´1/2 °
´1/2 °
° r
°
°
°
°
°
A=°
EN (ak )EN (ak )∗
EN (ak a∗k )
(a)°p .
° ≤°
° = °Scond
k=1
p
p
k=1
◦
On the other hand, according to (5.3) we can regard ak ⊗ e1k as an element of
³
´
¡
¢
n
n (N ) S
Spn Lp (∗N Ak ) = Lp ∗S∞
∞ (Ak ) ,
5.2. ESTIMATES FOR BMO TYPE NORMS
83
◦
where EN is replaced by 1Mn ⊗ EN . Then, writing xk for ak ⊗ e1k and EN for
1Mn ⊗ EN , we estimate B using the free Rosenthal inequalities in this bigger space.
Indeed, using kxk kp = kak kp we obtain
n
n
n
°³ X
´1/p °³ X
³X
¡
¢´1/2 °
¡
¢´1/2 °
°
°
°
°
∗
p
EN x∗k xk
EN xk xk
B∼
kak kp
+°
° +°
° .
k=1
Let us note that
n
X
¡
¢
EN xk x∗k
k=1
n
X
p
k=1
¡
EN x∗k xk
¢
n
X
=
¡◦ ◦∗ ¢
e11 ⊗ EN ak ak
k=1
n
X
=
k=1
p
k=1
≤
¡◦∗ ◦ ¢
ekk ⊗ EN ak ak
≤
k=1
n
X
¡
¢
e11 ⊗ EN ak a∗k ,
k=1
n
X
¡
¢
ekk ⊗ EN a∗k ak .
k=1
This implies
n
n
°³ X
°³ X
´1/2 °
¡
¢´1/2 °
°
°
°
°
EN (ak a∗k )
EN xk x∗k
° .
° ≤°
°
p
k=1
p
k=1
On the other hand, the third term is controlled by
n
n
°³ X
°X
¡
¢´1/2 °
¡
¢°
°
°
°
°1/2
EN x∗k xk
EN x∗k xk °
°
° = °
k=1
p
p/2
k=1
n
°X
¡
¢°
°
°1/2
≤ °
ekk ⊗ EN a∗k ak °
=
k=1
n
³X
p/2
n
´1/p
° ¡ ∗ ¢°p/2 ´1/p ³ X
p
°EN ak ak °
≤
ka
k
.
k
p
p/2
k=1
k=1
Combining the inequalities above we have the upper estimate with constant 2.
¤
5.2. Estimates for BMO type norms
Apart from the free Rosenthal inequalities, our second key tool in the proof of
(5.1) and (5.2) will be certain estimates that we develop in this section. Let us recall
the definition of several noncommutative Hardy spaces. Xu’s survey [68] contains a
systematic exposition of these notions. Let M be a von Neumann algebra equipped
with a n.f. state ϕ. Let M1 , M2 , . . . be an increasing filtration of von Neumann
subalgebras of M which are invariant under the modular automorphism group
on M associated to ϕ. This allows us to consider the corresponding conditional
expectations En : M → Mn and noncommutative Lp martingales x = (xn )n≥1
with martingale differences dx1 , dx2 , . . . adapted to this filtration. The row and
column Hardy spaces Hpr (M) and Hpc (M) are defined respectively as the closure of
the space of finite Lp martingales with respect to the following norms
°¡ X
¢1 °
°
°
kxkHrp (M) = °
dxk dx∗k 2 ° ,
k
p
°¡ X
¢ 12 °
°
°
∗
kxkHcp (M) = °
dxk dxk ° .
k
p
On the other hand, the space Hpp (M) measures the p-variation
³X
´1/p
kxkHpp (M) =
kdxk kpp
.
k
84
5. INTERSECTIONS OF Lp SPACES
Finally, the conditional row and column Hardy spaces for martingales hrp (M) and
hcp (M) are defined as the closure of the space of finite Lp martingales with respect
to the following norms
°¡ X
¢1 °
°
°
kxkhrp (M) = °
Ek−1 (dxk dx∗k ) 2 ° ,
k
p
°¡ X
¢ 12 °
°
°
∗
kxkhcp (M) = °
Ek−1 (dxk dxk ) ° .
k
p
A further tool are maximal functions. The notion of maximal function was
introduced in [16] via the spaces Lp (M; `∞ ). Here we are using variations of these
from Musat’s paper [38]. Given 2 ≤ p ≤ ∞ we define the spaces Lrp (M; `∞ )
and Lcp (M; `∞ ) as the space of bounded sequences in Lp (M) with respect to the
following norms
°
°1/2
k(xn )kLrp (M;`∞ ) = ° sup xn x∗n °p/2 ,
n≥1
k(xn )kLcp (M;`∞ )
=
°
°
° sup x∗n xn °1/2 .
p/2
n≥1
This definition requires some explanation. Indeed, in the noncommutative setting
there is no obvious analogue for the pointwise supremum of a family of positive
operators. Therefore, the above is to be understood in the sense of the suggestive notation introduced in [16]. Among several characterizations of the norm in
Lp (M; `∞ ) of a sequence (zn )n≥1 of positive operators, we outline the following
obtained by duality
°X
°
nX
o
°
°
¯
°
°
° sup zn ° = sup
(5.4)
tr(zn wn ) ¯ wn ≥ 0, °
wn ° ≤ 1 .
p
0
n≥1
n
n
p
The reader is also referred to [29] for a rather complete exposition. One of the
fundamental properties obtained in [38] of the spaces Lrp (M; `∞ ) and Lcp (M; `∞ )
is that they form interpolation families. Indeed, given 2 ≤ p0 , p1 ≤ ∞, we have the
following isometric isomorphisms for 1/pθ = (1 − θ)/p0 + θ/p1
£ r
¤
Lp0 (M; `∞ ), Lrp1 (M; `∞ ) θ = Lrpθ (M; `∞ ),
(5.5)
£ c
¤
Lp0 (M; `∞ ), Lcp1 (M; `∞ ) θ = Lcpθ (M; `∞ ).
Here it is also interesting to point out that these spaces can easily be identified as
one-sided amalgamated Lp spaces. In particular, the interpolation formula (5.5)
follows from Theorem 3.2.
5.2.1. One-sided estimates. We will simplify our arguments considerably
by assuming that M is finite and the density D associated to the state ϕ satisfies
c1 1M ≤ D ≤ c2 1M . The general case will follow one more time from Haagerup’s
approximation theorem.
Lemma 5.4. Let 1 ≤ p ≤ 2 ≤ q ≤ ∞ be such that 1/p = 1/2 + 1/q. Given
0 < θ < 1, let x be a norm one element in [Hpp (M), Hpr (M)]θ and let us consider
the indices 1 ≤ u ≤ 2 ≤ v ≤ ∞ defined as follows
1/u = 1/p − θ/q
and
1/v = θ/q.
5.2. ESTIMATES FOR BMO TYPE NORMS
85
Then we may find a positive element a ∈ Lv/2 (M) and bk ∈ Lu (Mk ) such that
n
³X
´1/u o √
1
dxk = Ek (a) 2 bk and max kakv/2 ,
kbk kuu
≤ 2.
k≥1
1
If x belongs to [Hpp (M), Hpc (M)]θ , the same conclusion holds with dxk = b∗k Ek (a) 2 .
Proof. The last assertion follows from the first part of the statement by taking
adjoints. To prove the first assertion, we may assume by approximation that x is a
finite martingale in Lp (Mm ) for some m ≥ 1. Therefore, since x is of norm 1 there
exists an analytic function f : S → Lp (Mm ) of the form
f (z) =
m
X
dk (z),
k=1
which satisfies f (θ) = x and the estimate
(
)
m
m
°¡ X
°
³X
´ p1
1
¢
°
°
max sup
kdk (z)kpp , sup °
dk (z)dk (z)∗ 2 °
≤ 1.
z∈∂0
z∈∂1
k=1
p
k=1
Note that we are assuming that the dk ’s are also analytic where d1 (z), d2 (z), . . .
denote the martingale differences of f (z). Now we consider the following functions
on the strip for 1 ≤ k ≤ m
(
1,
if z ∈ ∂0 ,
¡ Pk
2 ¢p
gk (z) =
∗
q
p
d
(z)d
(z)
+
δD
,
if z ∈ ∂1 .
j
j=1 j
According to our original assumption on the finiteness of M and the invertibility
of D, we are in position to apply Devinatz’s theorem. Indeed, here we need Xu’s
modification, which can be found in Section 8 of Pisier and Xu’s survey [52]. This
provides us with analytic function hk with analytic inverse and such that
hk (z)hk (z)∗ = gk (z) for all
(5.6)
Step 1. We claim that
m
°X
°
°
°
(5.7)
δk ⊗ hk (θ)°
°
k=1
Lrv (M;`∞ )
z ∈ ∂S.
¡
p ¢θ
≤ 1+δ2 q,
where δ appears in the definition of gk . Indeed, according to the interpolation
isometries (5.5) and the three lines lemma, it suffices to see that the following
estimates hold
m
°X
°
°
°
(5.8)
sup °
δk ⊗ hk (z)°
≤ 1,
r
z∈∂0
k=1
m
°X
°
sup °
(5.9)
z∈∂1
k=1
L∞ (M;`∞ )
°
°
δk ⊗ hk (z)°
Lrq (M;`∞ )
≤
¡
p ¢1
1+δ2 q.
To prove (5.8) we first recall from (5.6) and the definition of gk that hk (z) is a
unitary for any z ∈ ∂0 and any 1 ≤ k ≤ m. Therefore, since Fubini’s theorem gives
Lr∞ (M; `∞ ) = L∞ (`∞ (M)), we conclude
m
°X
°
°
°
sup °
δk ⊗ hk (z)° r
= sup sup khk (z)k∞ = 1.
z∈∂0
k=1
L∞ (M;`∞ )
z∈∂0 1≤k≤m
86
5. INTERSECTIONS OF Lp SPACES
On the other hand, if z ∈ ∂1 we have the following estimate from (5.6)
m
k
°q
°
°X
³X
´p/q °q/2
2
°
°
°
°
δk ⊗ hk (z)° r
= ° sup
dj (z)dj (z)∗ + δD p
°
°
k=1
Lq (M;`∞ )
1≤k≤m
q/2
j=1
m
°³ X
´p/q °q/2
2
°
°
≤ °
dj (z)dj (z)∗ + δD p
°
q/2
j=1
m
°X
°
p
2 °p/2
≤ 1+δ2,
dj (z)dj (z)∗ + δD p °
°
= °
p/2
j=1
where the last inequality follows from the fact that Lp/2 (M) is a p/2-normed space
and also from the right boundary estimate for the function f given above. Thus,
inequality (5.9) follows and the proof of (5.7) is completed.
Step 2. Let us consider the functions wk (z) = hk (z)−1 dk (z). Now we claim that
m
³X
´1/u √ θ ¡
p ¢θ/2
.
(5.10)
≤ 2 1+δ2
kwk (θ)kuu
k=1
1
According to (5.6), we can write hk (z) = gk (z) 2 uk (z) for some unitary uk (z). Thus,
we deduce that for z ∈ ∂0 we have wk (z) = uk (z)∗ dk (z). This and the left boundary
estimate for f yield
m
³X
´1/p
sup
kwk (z)kpp
≤ 1.
z∈∂0
k=1
The interesting argument, based on the classical Fefferman-Stein duality theorem,
1
appears for z ∈ ∂1 . In that case we have hk (z)−1 = uk (z)∗ gk (z)− 2 and we find the
following estimate for any z ∈ ∂1
m
m
³
´
X
X
2
kwk (z)k2 =
tr hk (z)−1 dk (z)dk (z)∗ hk (z)−1∗
k=1
=
=
k=1
m
X
k=1
m
X
³
´
1
1
tr uk (z)∗ gk (z)− 2 dk (z)dk (z)∗ gk (z)− 2 uk (z)
³
´
1
1
tr gk (z)− 2 dk (z)dk (z)∗ gk (z)− 2 .
k=1
Now we define the positive operators
³ k−1
´ p2
X
2
αk =
dj (z)dj (z)∗ + δD p ,
j=1
βk
=
k
³X
2
dj (z)dj (z)∗ + δD p
´ p2
,
j=1
and the indices (s, t) = (2/q, 2/p). Lemma 7.2 in [28] gives
³
´
¡
¢
−s/2
−s/2
tr βk (βkt − αkt )βk
≤ 2 tr βk − αk .
According to our choice of (s, t), we may rewrite this inequality as follows
³
´
¡
¢
1
1
tr gk (z)− 2 dk (z)dk (z)∗ gk (z)− 2 ≤ 2 tr βk − βk−1 .
5.2. ESTIMATES FOR BMO TYPE NORMS
87
Summing up, we deduce that
m
m
°X
°
X
¡
p¢
2 °p/2
°
2
kwk (z)k2 ≤ 2 °
dj (z)dj (z)∗ + δD p °
≤ 2 1+δ2 .
p/2
j=1
k=1
Finally, recalling that the functions wk : S → Lp (M) are analytic since are products
of analytic functions, inequality (5.10) follows from the estimates given above and
the three lines lemma.
Step 3. For the moment, we have seen that
dxk = hk (θ)wk (θ).
Now, recalling that
gk : ∂S → Lq/2 (Mk ) for each 1 ≤ k ≤ m,
we conclude that h(θ) = (h1 (θ), h2 (θ), . . . , hm (θ)) is an adapted sequence. That is,
we have hk ∈ Lv (Mk ) for all 1 ≤ k ≤ m. On the other hand, according to the
definition of the space Lrv (M; `∞ ), we may find for any δ > 0 a positive operator a
such that hk (θ)hk (θ)∗ ≤ a for 1 ≤ k ≤ m and
(5.11)
m
°X
°
°
°
1/2
kakv/2 ≤ (1 + δ)°
δk ⊗ hk (θ)°
Lrv (M;`∞ )
k=1
¡
p ¢1+ θ
q
.
≤ 1+δ2
Moreover, since h(θ) is adapted, we have hk (θ)hk (θ)∗ = Ek (hk (θ)hk (θ)∗ ) ≤ Ek (a).
1
This gives a contraction γk ∈ Mk with hk (θ) = Ek (a) 2 γk . In particular, we deduce
1
(5.12)
1
dxk = Ek (a) 2 γk wk (θ) = Ek (a) 2 bk .
Finally, since γk is a contraction
m
m
³X
´1/u ³ X
´1/u √ θ ¡
p ¢θ/2
.
(5.13)
kbk kuu
≤
kwk (θ)kuu
≤ 2 1+δ2
k=1
k=1
Therefore, the assertion follows from (5.11), (5.12) and (5.13) by letting δ → 0+ .
¤
Applying the anti-linear duality bracket hx, yi = tr(x∗ y), we consider in the
following result an immediate application of Lemma 5.4 for the following dual spaces
£
¤∗
Zpr (M, θ) = Hpp (M), Hpr (M) θ ,
¤∗
£
Zpc (M, θ) = Hpp (M), Hpc (M) θ .
Lemma 5.5. If p, q, u, v are as above and 1/u + 1/s = 1, we have
³X °
° ´1/s
°Ek (a) 12 dxk °s
kxkZpr (M,θ) ≤ 2 sup
,
s
kakv/2 ≤1
a≥0
kxkZpc (M,θ)
≤
2
sup
kakv/2 ≤1
a≥0
k
³X °
° ´1/s
°dxk Ek (a) 12 °s
.
s
k
Proof. There exists y in the unit ball of [Hpp (M), Hpr (M)]θ such that
¯
¯
kxkZpr (M,θ) = ¯tr(xy ∗ )¯.
88
5. INTERSECTIONS OF Lp SPACES
On the other hand, according to Lemma 5.4 and using homogeneity, we may write
dyk = Ek (a)1/2 bk with a being a positive operator in the unit ball of Lv/2 (M) and
b1 , b2 , . . . satisfying
³X
´1/u
(5.14)
kbk kuu
≤ 2.
k≥1
With this decomposition we obtain the following estimate
¯X ¡
¢¯¯
¯
tr dxk dyk∗ ¯
kxkZpr (M,θ) = ¯
¯ Xk ¡
¯
1 ¢¯
¯
= ¯
tr dxk b∗k Ek (a) 2 ¯
k
³X
´1/u ³ X °
° ´1/s
°Ek (a) 12 dxk °s
≤
kbk kuu
.
s
k
k
The assertion follows from (5.14). The proof of the second estimate is similar.
¤
In the following result and in the rest of this paper we shall write A . B to
denote the existence of an absolute constant c such that A ≤ c B holds.
Lemma 5.6. If p, q, u, v are as above and 1/u + 1/s = 1, we have
³X °
° ´1/s
°Ek (a) 12 dxk °s
sup
. kxk[Hp0 (M),Hr (M)] ,
s
kakv/2 ≤1
a≥0
sup
kakv/2 ≤1
a≥0
k
³X °
° ´1/s
°dxk Ek (a) 21 °s
s
.
k
θ
p0
p0
kxk[Hp0 (M),Hc
(M)]θ
p0
p0
.
Proof. We may find an analytic function
0
f : S → Hpp0 (M) + Hpr 0 (M)
such that f (θ) = x and
o
n
max sup kf (z)kHp0 (M) , sup kf (z)kHr 0 (M) . kxk[Hp0 (M),Hr
p0
z∈∂0
p
z∈∂1
p0
(M)]θ
p0
.
By homogeneity, let us assume that the right hand side above is 1. Now we take
positive element a in the unit ball of Lv/2 (M) so that we may consider an analytic
function g : S → L∞ (M) + Lq/2 (M) satisfying g(θ) = a and
n
o
max sup kg(z)k∞ , sup kg(z)kq/2 ≤ 1.
z∈∂0
z∈∂1
Then we construct the analytic function
X
¡
¢
h(z) =
δk ⊗ dk (f (z))∗ Ek (g(z))dk (f (z)) ∈ `∞ L1 (M) .
k
For z ∈ ∂0 we have
´ 0
³X
´ 0
°
° ³X
0 1/p
0 1/p
sup °Ek (g(z))°∞
kh(z)kp0 /2 ≤
kdk (f (z))kpp0
kdk (f (z))kpp0
. 1.
k≥1
k≥1
k≥1
In the case z ∈ ∂1 , we choose a factorization g(z) = g1 (z)g2 (z) such that
q
kg1 (z)kq = kg2 (z)kq = kg(z)kq/2 ≤ 1.
5.2. ESTIMATES FOR BMO TYPE NORMS
89
Then, Hölder inequality provides the following estimate
X° ¡
¢°
°Ek dk (f (z))∗ g1 (z)g2 (z)dk (f (z)) °
kh(z)k1 =
1
k≥1
≤
³X ¡
¢´ 12
tr dk (f (z))∗ g1 (z)g1 (z)∗ dk (f (z))
k≥1
³X ¡
¢´ 12
×
tr dk (f (z))∗ g2 (z)∗ g2 (z)dk (f (z))
k≥1
°1/2 ° X
°1/2
°X
°
°
°
°
dk (f (z))dk (f (z))∗ ° 0 . 1.
≤ °
dk (f (z))dk (f (z))∗ ° 0 °
p /2
k≥1
p /2
k≥1
Indeed, in order to apply Hölder inequality in the first inequality above, we factorize
the conditional expectation Ek−1 (a∗ b) as the product uk−1 (a)∗ uk−1 (b) by a right
Mk−1 -module map uk−1 : M → C∞ (Mk−1 ), see e.g. [16, 26]. By complex
interpolation (2/s = 2(1 − θ)/p0 + θ), we conclude that
³X °
° ´1/s
1/2
°Ek (a) 21 dxk °s
= kh(θ)ks/2 . 1.
s
k
The column version of this inequality follows by taking adjoints.
¤
Remark 5.7. When 1 < p ≤ 2 we have
³X °
° ´1/s
°Ek (a) 12 dxk °s
kxkZpr (M,θ) ∼
sup
,
s
kakv/2 ≤1
a≥0
kxkZpc (M,θ)
∼
sup
kakv/2 ≤1
a≥0
k
³X °
° ´1/s
°dxk Ek (a) 12 °s
.
s
k
Indeed, since anti-linear duality is compatible with complex interpolation via the
analytic function tr(f (z)∗ g(z)), we find by reflexivity in the case 1 < p ≤ 2 the
following isomorphisms
(5.15)
0
Zpr (M, θ) '
[Hpp0 (M), Hpr 0 (M)]θ ,
Zpc (M, θ) '
[Hpp0 (M), Hpc 0 (M)]θ .
0
Therefore, the result follows from Lemma 5.5 and Lemma 5.6.
Remark 5.8. Lemmas 5.4, 5.5, 5.6 and Remark 5.7 immediately generalize for
the row and column conditional Hardy spaces. Indeed, if we replace the row and
column Hardy spaces Hpr (M) and Hpc (M) by their conditional analogues hrp (M)
and hcp (M) and the conditional expectation Ek by Ek−1 , it can be easily checked
that the same arguments can be adapted to obtain the conditioned results.
We shall also need to generalize the norms of hrp (M) and hcp (M) for arbitrary
(non-necessarily adapted) sequences z1 , z2 , . . . in Lp (M) as follows
°X
°
°¡ X
¢1 °
°
°
°
∗ 2°
δk ⊗ zk ° p
=
E
(z
z
)
°
° ,
°
k−1 k k
k
k
Lcond (M;`r2 )
p
°X
°
°¡ X
°
1
¢
°
°
°
°
δk ⊗ z k ° p
= °
Ek−1 (zk∗ zk ) 2 ° .
°
c
k
Lcond (M;`2 )
k
p
90
5. INTERSECTIONS OF Lp SPACES
Lemma 5.9. Let p, q be as above and let us consider the indices (s, t) given
by 1/s = (1 − η)/p0 + η/2 and 1/t = η/q for some parameter 0 < η < 1. Then,
the following estimates hold for every martingale difference sequence dx1 , dx2 , . . .
in Lp0 (M)
°X
°
³X
´1/t
°
°
t
δk ⊗ dxk ak ° s
≤
ka
k
kxk[hr (M),Hp0 (M)] ,
°
k
t
r
η
k
Lcond (M;`2 )
°X
°
°
°
δk ⊗ ak dxk °
°
k
Lscond (M;`c2 )
p0
k≥1
≤
³X
kak ktt
´1/t
kxk[hc
p0
k≥1
Proof. We recall that
£ p0
¤
1
Lcond (M; `r2 ), Lpcond
(M; `r2 ) θ
£ p0
¤
1
Lcond (M; `c2 ), Lpcond
(M; `c2 ) θ
p0
0
(M),Hp
(M)]η
p0
⊂
θ
Lpcond
(M; `r2 ),
⊂
θ
Lpcond
(M; `c2 ),
.
hold isometrically. Indeed, we recall the factorization
Ek−1 (a∗ b) = uk−1 (a)∗ uk−1 (b)
used in the proof of Lemma 5.6 and the resulting isometric embeddings
¡
¢
Lpcond (M; `r2 ) ⊂ Lp M; Rp (N2 ) ,
¡
¢
Lpcond (M; `c2 ) ⊂ Lp M; Cp (N2 ) .
We refer the reader to [16] for a more detailed explanation. According to our claim
and by bilinear interpolation, it suffices to prove the assertion for the extremal
cases. When η = 0 we have (s, t) = (p0 , ∞) and the following estimate holds
°¡ X
°¡ X
¡
¢¢ 1 °
¡
¢¢ 1 °
°
°
°
°
Ek−1 dxk ak a∗k dx∗k 2 ° 0 ≤ sup kak k∞ °
Ek−1 dxk dx∗k 2 ° 0 .
°
k
p
k
k≥1
p
On the other hand, for η = 1 we have (s, t) = (2, q) so that
°¡ X
³X ¡
¢´1/2
¡
¢¢ 1 °
°
°
tr ak a∗k dx∗k dxk
Ek−1 dxk ak a∗k dx∗k 2 ° =
°
k
k
2
³X
´1/2
≤
kak k2q kdxk k2p0
k
´1/q ³ X
´ 0
³X
0 1/p
.
kdxk kpp0
≤
kak kqq
k
k
This proves the first inequality. The second one follows by taking adjoints.
The following is the main result of this paragraph.
Proposition 5.10. If p, q, u, v are as above and 1/u + 1/s = 1, we have
o
n
kxkZpr (M,θ) ≤ c(p, θ) max kxkHp0 (M) , kxk[Hp0 (M),hr (M)] ,
θ
p0
p0
p0
o
n
kxkZpc (M,θ) ≤ c(p, θ) max kxkHp0 (M) , kxk[Hp0 (M),hc (M)] .
p0
p0
p0
Here the constant c(p, θ) satisfies c(p, θ) ∼ 1 as v → ∞ and
´−1/2 r 2
³
p
−1
=
as v → 2.
c(p, θ) ∼
(2 − p)θ
v−2
θ
¤
5.2. ESTIMATES FOR BMO TYPE NORMS
91
Proof. According to Lemma 5.5 we have
³X °
° ´1/s
°Ek (a) 12 dxk °s
.
kxkZpr (M,θ) ≤ 2 sup
s
k
kakv/2 ≤1
a≥0
However, since 2 ≤ s ≤ ∞ we have
³X °
° ´1/s
°Ek (a) 12 dxk °s
(5.16)
s
k
³X °
° ´1/s
¡
¢
°dx∗k dak + Ek−1 (a) dxk °s/2
=
s/2
k
´1/s ³ X °
³X °
°
° ´1/s
s/2
°dx∗k Ek−1 (a)dxk °s/2
°dx∗k dak dxk °
+
.
≤
s/2
s/2
k
k
As we pointed above, Lemma 5.6 and Remark 5.7 generalize to conditional Hardy
spaces after replacing Ek by Ek−1 . In other words, the inequality below holds with
absolute constants for 1 ≤ p ≤ 2
³X °
° ´1/s
°dx∗k Ek−1 (a)dxk °s/2
(5.17)
sup
. kxk[Hp0 (M),hr (M)] .
s/2
kakv/2 ≤1
a≥0
k
p0
p0
θ
Therefore, it suffices to estimate the first term on the right of (5.16).
Step 1. We first assume 4 ≤ v ≤ ∞. Since 1/s = 1/v + 1/p0
´1/p0
³X °
´1/v ³ X
° ´1/s ³ X
v/2
p0
°dxk dak dxk °s/2
kdx
k
.
kda
k
≤
0
k
k
p
v/2
s/2
k
k
k
Then, complex interpolation gives
´1/v √
³X
√
v/2
1/2
kdak kv/2
≤ 2 kakv/2 ≤ 2
k
for
4 ≤ v ≤ ∞.
Indeed, our claim is trivial for the extremal cases.
Step 2. The case 2 < v < 4 is a little more complicated. By the noncommutative
Burkholder inequality [28], we may find a decomposition dak = dαk + dβk + dγk
into three martingales satisfying the following estimates
³X
´2/v
v/2
kdαk kv/2
≤ cv kakv/2 ,
k
°¡ X
°
¢1 °
°
Ek−1 (dβk dβk∗ ) 2 °
(5.18)
≤ cv kakv/2 ,
°
k
v/2
°
°¡ X
¢1 °
°
Ek−1 (dγk∗ dγk ) 2 °
≤ cv kakv/2 ,
°
k
v/2
where we know from [54] that
1
for 2 ≤ v ≤ 4.
v−2
Since we need to estimate the first term on the right of (5.16), we decompose it into
three terms according to the martingale decomposition above. The resulting term
associated to α can be estimated as in Step 1. For the second term (associated to
β) we may find a norm one element (bk ) in L(s/2)0 (M; `(s/2)0 ) such that
r¯
³X °
°s/2 ´1/s
¢¯¯
¯X ¡
∗
°
°
(5.19)
tr bk dx∗k dβk dxk ¯.
dxk dβk dxk s/2
= ¯
cv .
k
k
92
5. INTERSECTIONS OF Lp SPACES
However, we have
¯X ¡
¢¯¯
¯
tr bk dx∗k dβk dxk ¯
¯
=
k
=
≤
×
≤
¯X ¡
¢¯¯
¯
tr dβk dxk bk dx∗k ¯
¯
¯ Xk ¡
¢¯¯
¯
tr Ek−1 (dβk dxk bk dx∗k ) ¯
¯
k
°¡ X
¢1 °
°
°
Ek−1 (dβk dβk∗ ) 2 °
°
k
v/2
°¡ X
¢1 °
°
°
∗
∗
Ek−1 (dxk bk dxk dxk bk dx∗k ) 2 °
°
k
(v/2)0
°¡ X
¢1 °
°
°
∗
∗
cv kakv/2 °
Ek−1 (dxk bk dxk dxk bk dx∗k ) 2 °
k
(v/2)0
.
Indeed, the first inequality above follows from Hölder inequality after representing
Ek−1 (a∗ b) as uk−1 (a)∗ uk−1 (b) via the right module map uk−1 considered in the
proof of Lemma 5.6. The estimate for the term associated to γ is similar and yields
the same term with b∗k and bk exchanged. Now we have to estimate the term
°¡ X
¢1 °
°
°
Ek−1 (dxk b∗k dx∗k dxk bk dx∗k ) 2 °
.
°
0
k
(v/2)
Writing each bk as a linear combination
¡
¢
¡
¢
bk = bk1 − bk2 + i bk3 − bk4
of positive elements and allowing an additional constant 2, we may assume that
the operators b1 , b2 , . . . are positive. This consideration allows us to construct the
1
positive elements zk = (dxk bk dx∗k ) 2 . Then, recalling our assumption 2 < v < 4, we
have 2 < (v/2)0 < ∞ and Lemma 5.2 of [28] gives for t = 21 (v/2)0
1
°X
°
°X
° 2(t−1)
³
´ 2t−1
°
°
°
° 2t−1 X
.
Ek−1 (zk4 )° ≤ °
kzk k4t
Ek−1 (zk2 )°
°
4t
k
k
t
2t
k
In our situation, this implies
°¡ X
¢1 °
°
°
(5.20) °
Ek−1 (dxk b∗k dx∗k dxk bk dx∗k ) 2 °
k
≤
Using
1
(v/2)0
−
(v/2)0
t−1 ³
1
°X
° 2t−1
X °
0´
°
°
°
°dxk bk dx∗k °(v/2)0 4t−2 .
Ek−1 (dxk bk dx∗k )°
°
(v/2)
0
k
1
(s/2)0
=
(v/2)
2
s
−
2
v
=
2
p0 ,
k
we find
1
´2
³X
³X °
0´
°
p0 p0
°dxk bk dx∗k °(v/2)0 (v/2)0 ≤ kdxkp0 kbk(s/2)0 kdx∗ kp0 ≤
kdx
k
.
0
k
p
(v/2)
k
k
In other words, we have
1
³X °
4t
0´
°
°dxk bk dx∗k °(v/2)0 4t−2 ≤ kxk 4t−2
p0
(v/2)
Hp0 (M)
k
.
For the first term on the right of (5.20) we use Lemma 5.9 with
´
¡
¢ ³
η, s, t = 1 − θ, 2(v/2)0 , 2(s/2)0 .
This yields
°X
°
°
°
Ek−1 (dxk bk dx∗k )°
°
k
(v/2)0
=
°X
°
1 2
°
°
δk ⊗ dxk bk2 ° 2(v/2)0
°
k
Lcond
(M;`r2 )
5.2. ESTIMATES FOR BMO TYPE NORMS
≤
³X
k
(s/2)0
´
kbk k(s/2)0
1
(s/2)0
93
kxk2
0
[Hp
(M),hrp0 (M)]θ
p0
.
Hence, since (bk ) is in the unit ball of L(s/2)0 (M; `(s/2)0 ), we conclude
t−1
°X
° 2t−1
4t−4
°
°
4t−2
Ek−1 (dxk bk dx∗k )°
≤
kxk
°
p0
r
0
k
(v/2)
[Hp0 (M),hp0 (M)]θ
The estimates above and (5.20) give rise to
°¡ X
n
¢1 °
°
°
Ek−1 (dxk b∗k dx∗k dxk bk dx∗k ) 2 °
≤max
kxk2 p0
, kxk2 p0
°
0
H (M)
[H (M),hr
k
(v/2)
p0
p0
p0
(M)]θ
o
.
Taking square roots as imposed by (5.19) and keeping track of the constants, we
obtain the assertion for Zpr (M, θ). The column case follows by taking adjoints. ¤
At the beginning of this paragraph, we assumed that the von Neumann algebra
M was finite and equipped with a n.f. state ϕ with respect to which the associated
density satisfied c1 1M ≤ D ≤ c2 1M . This assumption was only needed in Lemma
5.4 (and its conditional version) in order to apply a variation of Devinatz’s theorem.
On the other hand, this result has been only applied to prove Lemma 5.5. Therefore,
if we are able to show that Lemma 5.5 (and its conditional version) holds for
arbitrary σ-finite von Neumann algebras, the same will hold for Proposition 5.10.
As we shall see, this is relevant for our aims since we shall use these results in the
context of free products. Let us indicate how to derive Lemma 5.5 for σ-finite von
Neumann algebras. As expected, we apply Haagerup’s construction and consider
R = M oσ G for the discrete group
[
G=
2−n Z.
n∈N
The crossed product R is a direct limit of a family of finite von Neumann algebras
R1 , R2 , . . . (we change our usual notation here since in this chapter M1 , M2 , . . .
stand for a filtration of M) which are obtained as centralizers constructed from the
modular action for ϕ ◦ EM , where
X
EM :
xg λ(g) ∈ R 7→ x0 ∈ M
g∈G
denotes the natural conditional expectation onto M. The trace in Rn is given by
τn (x) = ϕ ◦ EM (dn x) where c1n 1Rn ≤ dn ≤ c2n 1Rn . It is then easily checked that
M̂n
= Mn oσ G,
M̂n (m)
= M̂n ∩ Rm ,
are increasing filtrations in R and Rm respectively. Thus, Lemma 5.5 (and its
conditional version) holds for Rm and the filtration M̂1 (m), M̂2 (m), . . . for fixed
m ≥ 1. Moreover, according to (2.3) and a simple density argument, Lemma 5.5
remains valid for R and the filtration M̂1 , M̂2 , . . . Finally, it remains to see that
EM extends to a contraction on Hps (R) and hsp (R) where s ∈ {r, c, p}. This is
obvious for s = p, see e.g. [28] for the convention hpp (R) = Hpp (R). For s = r, c we
recall that EM and Ek−1 commute. In the case 2 ≤ p ≤ ∞, this implies
°X
°X
¡
¢°
¡
¢°
°
°
°
°
Ek−1 dk (EM (x))dk (EM (x))∗ °
= °
Ek−1 EM (dxk )EM (dx∗k ) °
°
k
k
p/2
p/2
°X
¡
¢°
°
∗ °
≤ °
Ek−1 EM (dxk dxk ) °
k
p/2
94
5. INTERSECTIONS OF Lp SPACES
°
¡X
¢°
°
°
= ° EM
Ek−1 (dxk dx∗k ) °
k
p/2
°X
°
°
∗ °
Ek−1 (dxk dxk )° .
≤ °
k
p/2
Hpr (R),
The arguments for
p ≥ 2 (as well as the analogues for the column spaces)
are the same. For 1 ≤ p < 2 we have to argue differently. By duality, it suffices to
prove the assertion for Lrq MO with 2 < q ≤ ∞, see Theorem 4.1 of [28] for this
duality result. Indeed, the norm in that space is given by
° 12
°
X
°
°
En (dxk dx∗k )° .
kxkLrq MO = ° sup
n≥1
Therefore, using the inequality
°
°
° sup EM (zn zn∗ )°
q/2
n≥1
q/2
k≥n
°
°
≤ ° sup zn zn∗ °q/2
n≥1
which follows easily from (5.4), we see that the same argument above applies. On
the other hand, it is easily checked (as in [28], Theorem 4.1) that the corresponding
dual in the conditional case is given by
°
° 12
X
°
°
kxkLrq mo = ° sup
En (dxk dx∗k )° .
n≥1
q/2
k>n
5.2.2. Two-sided estimates. Now we will perform a similar task considering
two-sided terms. Since most of the arguments are the same, we shall only sketch
the main ideas in the proofs. Again, we begin by assuming that M is finite and
the density D associated to the state ϕ satisfies c1 1M ≤ D ≤ c2 1M . The above
argument via Haagerup’s construction leads to the σ-finite case.
Lemma 5.11. Let 1 ≤ p ≤ 2 ≤ q ≤ ∞ be such that 1/p = 1/2 + 1/q. Given
0 < θ < 1, let x be a norm one element in [Hpr (M), Hpc (M)]θ and let us consider
the indices 2 ≤ wr , wc ≤ ∞ defined as follows
1/wr = (1 − θ)/q
and
1/wc = θ/q.
Then there exists (ar , ac ) ∈ Lwr /2 (M)+ × Lwc /2 (M)+ and bk ∈ L2 (Mk ) such that
n
³X
´ 12
o
1
1
kbk k22 , kac kwc /2 ≤ 2.
dxk = Ek (ar ) 2 bk Ek (ac ) 2 and max kar kwr /2 ,
k≥1
Proof. Assume by approximation that x is a finite martingale in Lp (Mm )
for some integer m ≥ 1. Therefore, since x is of norm 1 there exists an analytic
function f : S → Lp (Mm ) of the form
f (z) =
m
X
dk (z),
k=1
which satisfies f (θ) = x and the estimate
(
)
m
m
°¡ X
°¡ X
¢1 °
¢ 12 °
°
°
°
∗ 2°
∗
max sup °
dk (z)dk (z)
dk (z) dk (z) °
≤ 1.
° , sup °
z∈∂0
Now we define
gkr (z)
p z∈∂1
k=1
( ¡P
k
=
1,
j=1
p
k=1
2
dj (z)dj (z)∗ + δD p
¢ pq
,
if z ∈ ∂0 ,
if z ∈ ∂1 .
5.2. ESTIMATES FOR BMO TYPE NORMS
(
gkc (z)
=
1,
¡ Pk
j=1
∗
dj (z) dj (z) + δD
2
p
¢ pq
95
if z ∈ ∂0 ,
,
if z ∈ ∂1 .
As in Lemma 5.4, we are now in position to apply Devinatz’s theorem. This provides
us with two analytic functions hrk and hck with analytic inverses and satisfying the
following relations
(5.21)
hrk (z)hrk (z)∗ = gkr (z) for all
z ∈ ∂S,
hck (z)∗ hck (z)
z ∈ ∂S.
=
gkc (z)
for all
According to the argument in Step 1 of the proof of Lemma 5.4, we have
m
°X
°
¡
p ¢ 1−θ
°
°
≤ 1+δ2 q ,
δk ⊗ hrk (θ)° r
°
(5.22)
k=1
Lwr (M;`∞ )
k=1
Lcwc (M;`∞ )
m
°X
°
°
°
δk ⊗ hck (θ)°
°
≤
¡
p ¢θ
1+δ2 q.
Now we consider the functions
wk (z) = hrk (z)−1 dk (z)hck (z)−1 .
Note that we have unitaries urk (z) and uck (z) for which
1
1
hrk (z) = gkr (z) 2 urk (z) and hrk (z) = uck (z)gkc (z) 2 .
Thus, wk can be rewritten as follows on ∂S
1
wk (z)
= urk (z)∗ gkr (z)− 2 dk (z)uck (z)∗
on ∂0 ,
wk (z)
1
urk (z)∗ dk (z)gkc (z)− 2 uck (z)∗
on ∂1 .
=
In particular, the same argument as in Lemma 5.4 (second part of Step 2) yields
m
q ¡
³X
´ 12
p¢
≤ 2 1+δ2 .
sup
kwk (z)k22
z∈∂S
k=1
Therefore, the same bound holds for z = θ by the three lines lemma. For the
moment, we have seen that dxk = hrk (θ)wk (θ)hck (θ). Now, recalling that gkr and gkc
take values in Lq/2 (Mk ), we deduce that the sequence hr1 (θ), hr2 (θ), . . . as well as
hc1 (θ), hc2 (θ), . . . are adapted. In particular, the argument in Step 3 of Lemma 5.4
gives rise to
1
1
hrk (θ) = Ek (ar ) 2 γkr and hck (θ) = γkc Ek (ac ) 2
for some contractions γkr , γkc ∈ Mk and some positive elements ar , ac satisfying
n
o ¡
p ¢1+ 1
q
max kar kwr /2 , kac kwc /2 ≤ 1 + δ 2
.
The proof is completed by taking the elements bk to be γkr wk γkc for 1 ≤ k ≤ m.
¤
Exactly as we did in Lemma 5.5, the following result is a direct application of
Lemma 5.11 for the dual space
£
¤∗
Zp (M, θ) = Hpr (M), Hpc (M) θ .
Lemma 5.12. If p, q, wr , wc are as above, we have
³X °
° ´1/2
°Ek (ar ) 12 dxk Ek (ac ) 12 °2
kxkZp (M,θ) ≤ 8
sup
.
2
kar kwr /2 ,kac kwc /2 ≤1
ar ,ac ≥0
k
96
5. INTERSECTIONS OF Lp SPACES
In Proposition 5.10 we found the constant c(p, θ). Now we define
n
o
c0 (p, θ) = max c(p, θ), c(p, 1 − θ) .
Proposition 5.13. If p, q, wr , wc are as above, we have
n
kxkZp (M,θ) ≤ c0 (p, θ) max kxkHp0 , kxk[hr ,Hp0 ] , kxk[Hp0 ,hc
p0
p0
p0 θ
p0
]
p0 θ
o
, kxk[hrp0 ,hcp0 ]θ .
Proof. According to Lemma 5.12 we have to estimate the term
³X °
° ´1/2
°Ek (ar ) 12 dxk Ek (ac ) 12 °2
A=
2
k
for any pair (ar , ac ) of positive elements satisfying kar kwr /2 ≤ 1 and kac kwc /2 ≤ 1.
To that aim, we decompose it into the following three terms
X ¡
¢
(5.23) A2 =
tr dxk Ek (ac )dx∗k Ek (ar )
¯ Xk
¡
¢¯¯
¯
≤ ¯
tr dxk dk (ac )dx∗k dk (ar ) ¯
¯ Xk ¡
¢¯¯
¯
+ ¯
tr dxk Ek−1 (ac )dx∗k dk (ar ) ¯
¯ Xk ¡
¢¯¯
¯
+ ¯
tr dxk dk (ac )dx∗k Ek−1 (ar ) ¯
¯ Xk ¡
¢¯¯
¯
+ ¯
tr dxk Ek−1 (ac )dx∗k Ek−1 (ar ) ¯ = A21 + A22 + A23 + A24 .
k
In particular, we have A ≤ A1 + A2 + A3 + A4 . The estimate for A4 is rather
simple. Indeed, arguing as in Remark 5.8, the conditional version of Lemma 5.12
follows after replacing Ek by Ek−1 . Moreover, as in (5.17) the argument in Lemma
5.6 gives
³X °
° ´1/2
°Ek−1 (ar ) 21 dxk Ek−1 (ac ) 12 °2
sup
. kxk[hrp0 (M),hcp0 (M)]θ ,
2
kar kwr /2 ,kac kwc /2 ≤1
ar ,ac ≥0
k
with absolute constants. Therefore, we find
³X °
° ´1/2
°Ek−1 (ar ) 12 dxk Ek−1 (ac ) 12 °2
A4 =
. kxk[hr 0 (M),hc 0 (M)]θ .
2
k
p
p
It remains to estimate the terms A1 , A2 and A3 .
Step 1. We first estimate the term A1 . Recalling that 1/wr + 1/wc = 1/q ≤ 1/2,
we must have 4 ≤ max(wr , wc ) ≤ ∞. Moreover, since both cases can be argued in
the same way, we assume without lost of generality that 4 ≤ wr ≤ ∞. In this case
we have
¯X ¡
¢¯¯
¯
A21 = ¯
tr dxk dk (ac )dx∗k dk (ar ) ¯
k
0
³X
´2/wr ³ X °
0 ´1/(wr /2)
°
w /2
°dxk dk (ac )dx∗k °(wr /2)0
≤
kdk (ar )kwrr /2
.
(w /2)
k
k
r
The first term on the right is controlled by kar kwr /2 since we are assuming that
wr ≥ 4. For the second term on the right, we observe that 2(wr /2)0 is determined
by the following relation
1
1
1
1 1−θ
1 θ
1
1
= −
= −
=1− + =1− = .
2(wr /2)0
2 wr
2
q
p q
u
s
5.2. ESTIMATES FOR BMO TYPE NORMS
97
Thus, this term is estimated as follows
³X °
³X °
° ´2/s
° ´2/s
°dxk Ek (ac ) 12 °s
°dxk dk (ac )dx∗k °s/2
≤
s/2
s
k
k
³X °
´
1 °s 2/s
°dxk Ek−1 (ac ) 2 °
+
= A211 + A212 .
s
k
By (the proof of) Proposition 5.10 we conclude that
n
A11 ≤ c(p, θ) max kxkHp0 (M) , kxk[Hp0 (M),hc
p0
(M)]θ
p0
p0
o
.
On the other hand, (5.17) yields the estimate
A12 . kxk[Hp0 (M),hc
p0
(M)]θ
p0
.
The case 4 ≤ wc ≤ ∞ is similar and yields
n
A1 ≤ c(p, θ) max kxkHp0 (M) , kxk[hr
0
(M),Hp
(M)]θ
p0
p0
p0
o
.
Therefore, in the general case we conclude
n
A1 ≤ c(p, θ) max kxkHp0 (M) , kxk[hr (M),Hp0 (M)] , kxk[Hp0 (M),hc
θ
p0
p0
p0
p0
(M)]θ
p0
o
.
Step 2. The same arguments as in Step 1 yield the right estimate for A2 in the
case 4 ≤ wr ≤ ∞ and for A3 in the case 4 ≤ wc ≤ ∞. Of course, there is an obvious
symmetry between both cases so that we only prove the estimate for A3 in the case
4 ≤ wc ≤ ∞. We have
0
³X
´2/wc ³ X °
0 ´1/(wc /2)
°
w /2
°Ek−1 (ar ) 12 dxk °2(wc /2)0
A23 ≤
kdk (ac )kwcc /2
.
2(w /2)
k
k
c
The first term on the right is controlled by kac kwc /2 while
1
1
1
1 θ
1 1−θ
= −
= − =1− +
.
2(wc /2)0
2 wc
2 q
p
q
That is, the roles of θ and 1 − θ have exchanged with respect to the situation in
Step 1 above. Therefore, according to the equivalence (5.17) we easily conclude
that
A3 . kxk[hr (M),Hp0 (M)] .
p0
p0
θ
When 4 ≤ wr ≤ ∞ we obtain the estimate
A2 . kxk[Hp0 (M),hc
p0
p0
(M)]θ
.
Step 3. Now we estimate A2 for 2 < wr < 4 and A3 for 2 < wc < 4. Again by
symmetry, we only prove the estimate for A2 . The proof of this estimate follows the
argument given in Step 2 of Proposition 5.10. By the noncommutative Burkholder
inequality, we may find a decomposition dk (ar ) = dk (αr ) + dk (βr ) + dk (γr ) into
three martingales satisfying (5.18) with (α, β, γ) à (αr , βr , γr ) and v à wr . Then
we have A22 ≤ A2 (α)2 + A2 (β)2 + A2 (γ)2 with
¯X ¡
¢¯¯
¯
A2 (α)2 = ¯
tr dxk Ek−1 (ac )dx∗k dk (αr ) ¯,
¯ Xk ¡
¢¯¯
¯
2
A2 (β) = ¯
tr dxk Ek−1 (ac )dx∗k dk (βr ) ¯,
¯ Xk ¡
¢¯¯
¯
2
A2 (γ) = ¯
tr dxk Ek−1 (ac )dx∗k dk (γr ) ¯.
k
98
5. INTERSECTIONS OF Lp SPACES
The term A2 (α) is estimated as in Step 2, due to the first inequality in (5.18). The
terms A2 (β) and A2 (γ) are estimated in the same way so that we only show how
to estimate A2 (β). We proceed as in Proposition 5.10 again and obtain
¯X ¡
¢¯¯
¯
A2 (β)2 = ¯
tr Ek−1 (dk (βr )dxk Ek−1 (ac )dx∗k ) ¯
k
°¡ X
¢1 °
°
°
≤ °
Ek−1 (dk (βr )dk (βr )∗ ) 2 °
k
wr /2
°¡ X
¢1 °
°
∗ 2 2°
× °
Ek−1 (dxk Ek−1 (ac )dxk )
°
k
(wr /2)0
°¡ X
¢1 °
°
°
Ek−1 (dxk Ek−1 (ac )dx∗k )2 2 °
≤ cwr kar kwr /2 °
.
0
k
(wr /2)
1
To estimate the last term on the right we define zk = (dxk Ek−1 (ac )dx∗k ) 2 . Recalling,
our assumption 2 < wr < 4, we have 2 < (wr /2)0 < ∞ and Lemma 5.2 of [28] gives
for t = 12 (wr /2)0
1
°X
°
°X
° 2(t−1)
³
´ 2t−1
°
°
°
° 2t−1 X
Ek−1 (zk4 )° ≤ °
Ek−1 (zk2 )°
kzk k4t
.
°
4t
k
k
t
In our situation, this implies
°¡ X
¢1 °
°
°
Ek−1 (dxk Ek−1 (ac )dx∗k )2 2 °
°
k
( w2r )0
k
2t
t−1
°X
¡
¢°
°
° 2t−1
≤ °
Ek−1 dxk Ek−1 (ac )dx∗k °
k
(wr /2)0
1
³X °
0´
°
°dxk Ek−1 (ac )dx∗k °(wr /2)0 4t−2 .
×
(wr /2)
k
If we take
n
Mθ = max kxkHp0 , kxk[hr
0
]
,Hp
p0 θ
p0
p0
, kxk[Hp0 ,hc
p0
]
p0 θ
o
, kxk[hr 0 ,hc 0 ]θ ,
p
p
we have already seen in Step 1 above that
1
³X °
0´
°
°dxk Ek−1 (ac )dx∗k °(wr /2)0 4t . Mθ .
(wr /2)
k
We claim that
°X
¡
¢°
°
°
(5.24)
Ek−1 dxk Ek−1 (ac )dx∗k °
°
k
(wr /2)0
≤ kxk2[hr 0 ,hc 0 ]θ ≤ M2θ .
p
p
If we prove (5.24) then it is easy to see that A2 (β) . M and the estimate for A2
will be completed. Arguing as in Lemma 5.6, it is not difficult to check that the
left hand side of (5.24) does interpolate. Hence, it suffices to estimate the extremal
cases. When θ = 0, we have (wr , wc ) = (q, ∞) and (wr /2)0 = p0 /2. Consequently,
we find
°X
°X
°
¡
¢°
°
°
°
∗ °
Ek−1 dxk Ek−1 (ac )dx∗k °
≤
ka
E
≤ kxk2hr 0 (M) .
°
c k∞ °
k−1 (dxk dxk )°
0
0
k
p /2
k
p /2
p
When θ = 1 we have (wr , wc ) = (∞, q) and (wr /2)0 = 1 so that
°X
X ¡
¡
¢°
¢
°
°
Ek−1 dxk Ek−1 (ac )dx∗k ° =
tr Ek−1 (ac )dx∗k dxk
°
k
k
1
X ¡
¢
=
tr ac Ek−1 (dx∗k dxk )
k
°X
°
°
°
Ek−1 (dx∗k dxk )° p0 ≤ kxk2hc 0 (M) .
≤ kac k q2 °
k
2
p
5.3. INTERPOLATION OF 2-TERM INTERSECTIONS
99
Note that the first identity assumes that ac is positive and this is not necessarily true
on the boundary. However, decomposing into a linear combination of four positive
elements and allowing an additional constant 2, we may and do assume positivity.
Therefore, (5.24) follows from the tree lines lemma. A detailed reading of the proof
gives now the constant c0 (p, θ) stated above. This completes the proof.
¤
5.3. Interpolation of 2-term intersections
Let us fix some notation which will be used in the sequel. As usual, we begin
by fixing a von Neumann algebra M and a von Neumann subalgebra N with
conditional expectation E : M → N . Given 1 ≤ q ≤ p ≤ ∞ and a positive integer
n ≥ 1, the main spaces in this paragraph will be the following
Rn2p,q (M, E) =
n
C2p,q
(M, E) =
1
1
1
2p
1
2q
n 2p L2p (M) ∩ n 2q L2p
(M, E),
( 2pq ,∞)
p−q
n
L2p (M) ∩ n
L2p
2pq (M, E).
(∞, p−q
)
In order to study these spaces we need to introduce some terminology. We set Ak
to be M ⊕ M for 1 ≤ k ≤ n. Then we consider the reduced amalgamated free
product A = ∗N Ak where the conditional expectation EN : A → N , defined in
Paragraph 5.1.1 as EN (a) = Q∅ aQ∅ , has the following form when restricted to Ak
¢
1¡
EN (x1 , x2 ) = E(x1 ) + E(x2 ) .
2
Given a n.f. state ϕ : N → C, let ϕ2 : M ⊕ M → C be the n.f. state
¢
¡
¢
1¡
ϕ2 (x1 , x2 ) = ϕ(E(x1 )) + ϕ(E(x2 )) = ϕ EN (x1 , x2 ) .
2
We shall write A⊕n for the direct sum A ⊕ A ⊕ . . . ⊕ A with n terms. If φ stands
for the free product state on A, we consider the n.f. state φn : A⊕n → C and the
conditional expectation En : A⊕n → A given by
n
n
n
n
¡X
¢
¡X
¢
1X
1X
φn
xk ⊗ δk =
φ(xk )
and
E⊕n
xk ⊗ δk =
xk .
n
n
k=1
k=1
k=1
k=1
Let πk : Ak → A denote the embedding of Ak into A as defined at the beginning
of this chapter. Moreover, given x ∈ M we shall write xk as an abbreviation of
πk (x, −x). Note that xk is a mean-zero element for 1 ≤ k ≤ n. In the following we
shall use with no further comment the identities
EN (xk x∗k ) = E(xx∗ ) and
EN (x∗k xk ) = E(x∗ x).
Let us consider the following map
(5.25)
u : x ∈ M 7→
n
X
xk ⊗ δk ∈ A⊕n .
k=1
Moreover, if dϕb denotes the density associated to the n.f. state ϕ
b = ϕ ◦ E on M and
dφn stands for the density associated to the n.f. state φn on A⊕n , we may extend
the definition of u to other indices by taking
1
1
¡ 1 ¢
¡ 1¢
u dϕbp x = dφpn u(x) and u xdϕbp = u(x)dφpn .
In the following we shall consider the filtration on the von Neumann algebra A
given by Ak = A1 ∗N A2 ∗N · · · ∗N Ak . In particular, for any x ∈ Lp (M)P
we obtain
that u(x) = (x1 , x2 , . . . , xn ) is the sequence of martingale differences of k xk .
100
5. INTERSECTIONS OF Lp SPACES
Lemma 5.14. If 1 ≤ p < ∞, the following mappings are isomorphisms onto
complemented subspaces
u:
Rn2p,1 (M, E) →
r
H2p
(A),
u:
n
C2p,1
(M, E) →
c
H2p
(A).
Moreover, the constants are independent of n and remain bounded as p → 1.
Proof. Let us observe that
1
√
1
√
Rn2p,1 (M, E) = n 2p L2p (M) ∩
n
C2p,1
(M, E)
= n 2p L2p (M) ∩
nLr2p (M, E),
nLc2p (M, E).
Given x ∈ Rn2p,1 (M, E), Corollary 5.3 gives
°
°
°u(x)°
Hr2p (A)
n
°¡ X
¢1/2 °
°
°
=°
xk x∗k
°
2p
k=1
In other words, we have
°
°
°u(x)° r
H
∼
n
³X
kxk k2p
2p
n
°¡ X
¢1 °
°
°
+°
EN (xk x∗k ) 2 ° .
k=1
k=1
1
∼ n 2p kxkL2p (M) +
2p (A)
´2p
√
2p
n kxkLr2p (M,E) .
r
(A) is an isomorphism onto its image
This proves that u : Rn2p,1 (M, E) → H2p
with relevant constants independent of p, n. A similar argument yields to the same
conclusion for the column case. To prove the complementation, we recall that
r
r
H2p
(A)∗ ' H(2p)
for
0 (A)
1≤p<∞
(with relevant constants which remain bounded as p → 1) and consider the map
ω:x∈
n
1X
1 r
r
L(2p)0 (M, E) 7−→
xk ⊗ δk ∈ H(2p)
0 (A).
1 L(2p)0 (M) + √
n
n
2p
n
1
k=1
Let dϕb be the density associated to ϕ ◦ E. Assume by approximation that
1/(2p)0
x = αdϕb
a
for some (α, a) ∈ N × M. Then, taking dφ to be the density associated to the state
φ on A and defining ak = πk (a, −a), the following estimate holds by Theorem 7.1
in [28]
°1/2
³ n
´
°
°
1°
° 1/(2p)0 X
1/(2p)0 ∗ °
∗
°ω(x)° r
=
a
a
d
α
°αd
°
k
k
φ
φ
H(2p)0 (A)
n
L(2p)0 /2 (A)
k=1
≤
°1/2
³ n
¡
¢´
0
1°
° 1/(2p)0 X
∗°
EN ak a∗k d1/(2p)
α
.
°αdϕ
°
ϕ
n
L(2p)0 /2 (N )
k=1
This gives
(5.26)
°
°
°ω(x)°
Hr(2p)0 (A)
On the other hand, the inequality
°
°
°ω(x)° r
(5.27)
H
(2p)0
1
≤ √ kxkLr(2p)0 (M,E) .
n
(A)
≤
1
1
n 2p
kxkL(2p)0 (M)
5.3. INTERPOLATION OF 2-TERM INTERSECTIONS
101
follows by the complex interpolation method between the (trivial) extremal cases
for p = 1 and p = ∞. The estimates (5.26) and (5.27) show that the map ω is a
contraction. Note also that
n
n
­
®
¡ ∗
¢
1X
1X
u(x1 ), ω(x2 ) =
trA x1k x2k =
trM (x∗1 x2 ) = hx1 , x2 i.
n
n
k=1
In particular, since we have
Rn2p,1 (M, E) =
k=1
³ 1
´∗
1 r
√
0
L
(M)
+
L
(M,
E)
,
0
(2p)
1
n (2p)
n 2p
it turns out that the map ω ∗ u is the identity on Rn2p,1 (M, E) and uω ∗ is a bounded
projection onto the image of u with constants independent of n and bounded for
p ∼ 1. This completes the proof in the row case. The column case is the same. ¤
Before proving our interpolation result, we need to consider a variation of
r
Lemma 5.14. Namely, we know from [28] that H2p
(A) ' Lr2p MO(A) for 1 < p < ∞
and with constants depending on p which diverge as p → ∞. We claim however
that Lemma 5.14 still holds in this setting with bounded constants as p → ∞.
Lemma 5.15. If 1 < p ≤ ∞, the following mappings are isomorphisms onto
complemented subspaces
u : Rn2p,1 (M, E)
→
Lr2p MO(A),
n
(M, E)
C2p,1
→
Lc2p MO(A).
u:
Moreover, the constants are independent of n and remain bounded as p → ∞.
Proof. The noncommutative Doob’s inequality [16] gives
n
n
°X
°1
°
°
°
¡X
¢° 1
°
°
∗° 2
∗ °2
°u(x)° r
≤
γ
x
x
=
sup
E
x
x
°
° .
°
°
p
k
m
k
k
k
L MO
2p
1≤m≤n
k=m
p
k=1
p
Note that γp → ∞ as p → 1 but γp ≤ 2 for p ≥ 2. On the other hand, we may
estimate the term on the right by using the free Rosenthal inequality (see Corollary
5.3) one more time
n
n
n
°X
° 21
³X
´2p °¡ X
¢ 12 °
°
°
°
°
∗
xk x∗k ° ∼
kxk k2p
+
E
(x
x
)
°
° = kxkRn2p,1 (M,E) .
°
N
k k
2p
k=1
p
k=1
Rn2p,1 (M, E)
k=1
2p
This shows that u :
→ Lr2p MO(A) is bounded with constant γp . To
prove complementation and the boundedness of the inverse we proceed by duality
as in Lemma 5.14. Indeed, using the map w one more time and recalling that
r
∗
Lr2p MO(A) ' H(2p)
0 (A)
with constants which remain bounded as p → ∞ (c.f. Theorem 4.1 in [28]), we
may follows verbatim the proof of Lemma 5.14 to conclude that the map ω ∗ u is the
identity on Rn2p,1 (M, E) and uω ∗ is a bounded projection onto the image of u with
constants independent of n and bounded for p ∼ ∞. This completes the proof in
the row case. The column case follows in the same way.
¤
Theorem 5.16. If 1 ≤ p ≤ ∞ and 1/q = 1 − θ + θ/p, we have
£ n
¤
R2p,1 (M, E), Rn2p,p (M, E) θ ' Rn2p,q (M, E),
£ n
¤
n
n
C2p,1 (M, E), C2p,p
(M, E) θ ' C2p,q
(M, E),
102
5. INTERSECTIONS OF Lp SPACES
isomorphically with relevant constant c(p, q) independent of n and such that
r
p−q
c(p, q) .
as (p, q) → (∞, 1).
pq + q − p
Proof. By Corollary 4.7 i), we have contractive inclusions
£ n
¤
R2p,1 (M, E), Rn2p,p (M, E) θ ⊂ Rn2p,q (M, E),
£ n
¤
n
n
C2p,1 (M, E), C2p,p
(M, E) θ ⊂ C2p,q
(M, E).
To prove the converse, we consider again the map given in (5.25). It is clear that
n
³X
´1
°
°
1
2p 2p
°u(x)° 2p
=
kx
k
= n 2p kxk2p .
k
2p
H (A)
2p
k=1
1
2p
2p
This shows that u : n L2p (M) → H2p
(A) is an isometric isomorphism. Moreover,
arguing as in the proof of Lemma 5.14 we easily obtain that the image of u is
contractively complemented. This observation together with Lemmas 5.14 and
5.15 give rise to the following equivalences
°
°
for small p,
kxk[Rn2p,1 (M,E),Rn2p,p (M,E)]θ ∼ °u(x)°[Hr (A),H2p (A)]
θ
2p
°
° 2p
°
°
kxk[Rn2p,1 (M,E),Rn2p,p (M,E)]θ ∼
u(x) [Lr MO(A),H2p (A)] for large p,
2p
2p
θ
with constants independent of p, q, n. On the other hand, Berg’s theorem gives
isometric inclusions
£ r
¤
£ r
¤θ
2p
2p
H2p (A), H2p
(A) θ ⊂ H2p
(A), H2p
(A) ,
£ r
¤
£
¤θ
2p
2p
L2p MO(A), H2p
(A) θ ⊂ Lr2p MO(A), H2p
(A) .
Now we can use duality and obtain
¤∗
£ r
¤θ
£ r
(2p)0
2p
H2p (A), H2p
(A)
' H(2p)
0 (A), H
(2p)0 (A) θ
£ r
¤θ
£ r
¤∗
(2p)0
2p
L2p MO(A), H2p
(A)
' H(2p)
0 (A), H
(2p)0 (A) θ
for 1 ≤ p < ∞,
for 1 < p ≤ ∞,
where the constants in the first isomorphism remain bounded as p → 1 and the
constants in the second one remain bounded as p → ∞. Therefore, recalling the
terminology used in the previous section
¤∗
£ r
(2p)0
r
H(2p)0 (A), H(2p)0 (A) θ = Z(2p)
0 (A, 1 − θ)
and taking adjoints, we obtain
kxk[Rn2p,1 (M,E),Rn2p,p (M,E)]θ
∼
ku(x)kZ r
= Arp (A, θ),
n
n
kxk[ C2p,1
(M,E) , C2p,p
(M,E) ]θ
∼
ku(x)kZ c
= Acp (A, θ),
(A,1−θ)
(2p)0
(A,1−θ)
(2p)0
with constants independent of p, q, n. According to Proposition 5.10 we have
o
n
Arp (A, θ) ≤ c(p, q) max ku(x)kH2p (A) , ku(x)k[H2p (A),hr (A)]1−θ .
2p
2p
2p
Let us estimate the two terms on the right
n
1
³X
´ 2p
1
ku(x)kH2p (A) =
kxk k2p
= n 2p kxk2p ≤ kxkRn2p,q (M,E) .
2p
2p
k=1
5.4. INTERPOLATION OF 4-TERM INTERSECTIONS
For the second term we observe that
n
°³ X
´ 12 °
°
°
Ek−1 (xk x∗k ) °
ku(x)khr2p (A) = °
k=1
n
°³ X
°
= °
k=1
n
°³ X
°
= °
103
2p
´ 12 °
°
EN (xk x∗k ) °
2p
´ °
°
E(xx∗ ) °
1
2
2p
k=1
=
°
√ °
n °E(xx∗ )°2p ,
where the second inequality follows by freeness. By complex interpolation and
Corollary 4.7 we conclude
ku(x)k[hr
2p
2p (A),H2p (A)]θ
≤
n
1−θ
θ
2 + 2p
kxkL2p
(
=
(M,E)
2p0
,∞)
1−θ
1
n 2q kxkL2p
(
(M,E)
2pq
,∞)
p−q
≤ kxkRn2p,q (M,E) .
In summary, we have proved that
kxkRn2p,q (M,E) ≤ kxk[Rn2p,1 (M,E),Rn2p,p (M,E)]θ ≤ c(p, q) kxkRn2p,q (M,E)
where (recalling that p à (2p)0 and θ à 1 − θ), it follows from Proposition 5.10
that
r
p−q
c(p, q) ∼
as (p, q) → (∞, 1).
pq + q − p
This proves the assertion for rows. The column case follows by taking adjoints. ¤
Remark 5.17. At the time of this writing, we do not know whether or not
the relevant constants in Theorem 5.16 are uniformly bounded in p and q. Our
constants are not uniformly bounded due to fact that we use the noncommutative
Burkholder inequality from [28] in our approach. We take this opportunity to pose
this question as a problem for the interested reader. The same question applies to
Theorems 5.18 and 7.2 below.
5.4. Interpolation of 4-term intersections
In this section we study the interpolation spaces between Rn2p,1 (M, E) and
Of course, as it is to be expected, our main tools will be the free
Rosenthal inequalities and the two-sided estimates for BMO type norms. We shall
use below the constant c0 (p, θ) in Proposition 5.13.
n
(M, E).
C2p,1
Theorem 5.18. If 1 ≤ p ≤ ∞, we have
\
£ n
¤
n
R2p,1 (M, E), C2p,1
(M, E) θ '
n
1−θ
1
θ
u + 2p + v
L2p
( u
v
1−θ , θ )
u,v∈{2p0 ,∞}
(M, E)
isomorphically with relevant constant controlled by c0 (p, θ) and independent of n.
Proof. According to Corollary 4.7, we have
¤
£√ r
1−θ
1
θ
nL2p (M, E), n 2p L2p (M) θ = n 2 + 2p L2p2p0
£ 1
¤
√
n 2p L2p (M), nLc2p (M, E) θ
( 1−θ ,∞)
=
n
1−θ
θ
2p + 2
L2p
0
(∞, 2p
θ )
(M, E),
(M, E),
104
5. INTERSECTIONS OF Lp SPACES
for 1 ≤ p ≤ ∞. Moreover, if 1 ≤ p < ∞ the same result gives
£√ r
¤
√
√
n Lp (M, E), n Lcp (M, E) θ = n L2p2p0 2p0 (M, E).
( 1−θ ,
θ
)
In the extremal case we claim that we have a contractive inclusion
£ r
¤
L∞ (M, E), Lc∞ (M, E) θ ⊂ L∞
( 2 , 2 ) (M, E).
1−θ θ
Indeed, let us consider the multi-linear mappings
T1 : (α, x, β) ∈ L2 (M) × Lr∞ (M, E) × L∞ (M)
T2 : (α, x, β) ∈ L∞ (M) × Lc∞ (M, E) × L2 (M)
7→ αxβ ∈ L2 (M),
7→ αxβ ∈ L2 (M).
By the definition of Lr∞ (M, E) and Lc∞ (M, E) it is clear that both T1 and T2 are
contractions. In particular, it is easily checked that our claim follows by multi-linear
interpolation, details are left to the reader. Therefore, according to the observation
above and Corollary 4.7, we obtain the lower estimate with constant 1. In other
words, there exists a contractive inclusion
\
¤
£ n
1−θ
1
θ
n
n u + 2p + v L2p
R2p,1 (M, E), C2p,1
(M, E) θ ⊂
( u , v ) (M, E).
1−θ
u,v∈{2p0 ,∞}
θ
To prove the converse, we consider again the map given in (5.25). Arguing as in
the proof of Theorem 5.16 and according to Lemmas 5.14 and 5.15, we obtain the
following equivalences
°
°
n
kxk[Rn2p,1 (M,E),C2p,1
∼ °u(x)°[Hr (A),Hc (A)]
for p ≤ 2,
(M,E)]θ
θ
2p
°
° 2p
n
kxk[Rn2p,1 (M,E),C2p,1
∼ °u(x)°[Lr MO(A),Lc MO(A)] for p ≥ 2,
(M,E)]θ
2p
θ
2p
with constants independent of p, q, n. On the other hand, Berg’s theorem gives
isometric inclusions
¤θ
¤
£ r
£ r
c
c
(A) ,
(A), H2p
(A) θ ⊂ H2p
H2p (A), H2p
£ r
¤
£
¤θ
L2p MO(A), Lc2p MO(A) θ ⊂ Lr2p MO(A), Lc2p MO(A) .
Now we can use duality an obtain
¤θ
£ r
c
(A)
'
H2p (A), H2p
£ r
¤θ
c
L2p MO(A), L2p MO(A)
'
¤∗
£ r
c
H(2p)0 (A), H(2p)
0 (A)
θ
£ r
¤∗
c
H(2p)0 (A), H(2p)
(A)
0
θ
for 1 ≤ p < ∞,
for 1 < p ≤ ∞,
where the constants in the first isomorphism remain bounded as p → 1 and the
constants in the second one remain bounded as p → ∞. Therefore, recalling the
terminology used above
¤∗
£ r
c
H(2p)0 (A), H(2p)
= Z(2p)0 (A, θ),
0 (A)
θ
we deduce the following equivalence
n
kxk[Rn2p,1 (M,E),C2p,1
(M,E)]θ ∼ ku(x)kZ(2p)0 (A,θ) .
According to Proposition 5.13, the right hand side is controlled by
o
n
c0 (p, θ) max ku(x)kH2p , ku(x)k[hr ,H2p ]θ , ku(x)k[H2p ,hc ]θ , ku(x)k[hr2p ,hc2p ]θ .
2p
2p
2p
2p
The first two terms are estimated as in Theorem 5.16
ku(x)kH2p (A)
2p
=
1
n 2p kxk2p ,
2p
5.4. INTERPOLATION OF 4-TERM INTERSECTIONS
ku(x)k[hr
2p
2p (A),H2p (A)]θ
1
n 2q kxkL2p
≤
(
(M,E)
2pq
,∞)
p−q
105
.
Note that the latter term is the norm of x in
n
1−θ
1
θ
u + 2p + v
L2p
( u
v
1−θ , θ )
(u, v) = (2p0 , ∞).
(M, E) with
Taking adjoints and replacing θ by 1 − θ, we obtain
ku(x)k[H2p (A),hc
2p
2p (A)]θ
1
≤ n 2p
θ
+ 2p
0
kxkL2p
(∞,
(M,E)
2p0
)
θ
.
It remains to estimate the last term in the maximum. As in Theorem 5.16
°
√ °
ku(x)khr2p (A) =
n °E(xx∗ )°2p ,
°
√ °
n °E(x∗ x)°2p .
ku(x)khc2p (A) =
Thus, by complex interpolation we conclude
√
ku(x)k[hr2p ,hc2p ]θ ≤ n kxkL2p
(
(M,E)
2p0 2p0
,
)
1−θ
θ
.
Combining the estimates obtained above, the assertion follows.
¤
Remark 5.19. As we shall see in Chapter 7 below, another useful way to write
the intersection space appearing on the right side of Theorem 5.18 is in the following
form
\
1−η
η
n α + β L2p
(M, E).
4pq
(
, 4pq )
α,β∈{2p,2q}
(1−η)(2p−α) η(2p−β)
Remark 5.20. It might be of independent interest to mention that our methods
n
(M, E) form interpolation
immediately imply that the spaces Rn2p,1 (M, E) and C2p,1
families with respect to the index p. In other words, for any 1 ≤ p ≤ ∞
¤
£ n
R∞,1 (M, E), Rn2,1 (M, E) 1/p ' Rn2p,1 (M, E),
£ n
¤
n
n
C∞,1 (M, E) , C2,1
(M, E) 1/p ' C2p,1
(M, E).
Moreover, using anti-linear duality we may replace intersections by sums and extend
our results to the whole range 1 ≤ 2p ≤ ∞. These generalizations of Theorems 5.16
and 5.18 are out of the scope of this paper.
CHAPTER 6
n
Factorization of Jp,q
(M, E)
Let (X1 , X2 ) be a pair of operator spaces containing a von Neumann algebra M
as a common two-sided ideal. We define the amalgamated Haagerup tensor product
X1 ⊗M,h X2 as the quotient of the Haagerup tensor product X1 ⊗h X2 by the closed
subspace I generated by the differences
x1 γ ⊗ x2 − x1 ⊗ γx2
with γ ∈ M.
We shall be interested only in the Banach space structure of the operator spaces
X1 ⊗M,h X2 . In particular, we shall write X1 ⊗M X2 to denote the underlying
Banach space of X1 ⊗M,h X2 . According to the definition of the Haagerup tensor
product and recalling the isometric embeddings Xj ⊂ B(Hj ), we have
½° X
¾
°¡ X
¢1/2 °
¢1/2 °
°¡
°
°
°
(6.1) kxkX1 ⊗M X2 = inf °
x1k x∗1k
x∗2k x2k
,
°
°
°
k
B(H1 )
k
B(H2 )
where the infimum runs over all possible decompositions of x + I into a finite sum
X
x=
x1k ⊗ x2k + I.
k
Remark 6.1. Our definition (6.1) of the norm in X1 ⊗M X2 uses the operator
space structure of X1 and X2 since the row and column square functions live in
B(Hj ) but not necessarily in Xj . However, in the sequel it will be important to
note that much less structure on (X1 , X2 ) is needed to define the norm in X1 ⊗M X2 .
Indeed, we just need to impose conditions under which the row and column square
functions become closed operations in X1 and X2 respectively. In particular, this
is guaranteed if X1 is a right M-module and X2 is a left M-module. On the other
hand, note that this structure does not provide us with a natural operator space
structure on X1 ⊗M X2 , as we did above with X1 ⊗M,h X2 .
Let us consider any pair of indices 1 ≤ p, q ≤ ∞ satisfying q ≤ p and let us
define 1/r = 1/q − 1/p. Then, given a positive integer n, the rest of this paper will
be devoted to study the following intersection spaces
\
1
1
1
n
Jp,q
(M, E) =
n u + p + v Lp(u,v) (M, E).
u,v∈{2r,∞}
n
The aim of this chapter is the following factorization result for the spaces Jpq
(M, E).
Theorem 6.2. If 1 ≤ q ≤ p ≤ ∞, we have
n
n
Jp,q
(M, E) ' Rn2p,q (M, E) ⊗M C2p,q
(M, E),
isomorphically. Moreover, the constants are independent of p, q, n.
107
n
6. FACTORIZATION OF Jp,q
(M, E)
108
6.1. Amalgamated tensors
Before any other consideration, let us simplify the expression (6.1) for the
amalgamated Haagerup tensor product in Theorem 6.2 above. Since Rn2p,q (M, E)
n
and C2p,q
(M, E) coincide with L2p (M) as a set, the product ab of any two elements
n
(a, b) ∈ Rn2p,q (M, E) × C2p,q
(M, E) is well-defined and the amalgamation over M
allows us to identify finite sums
X
X
ak bk '
ak ⊗ bk + I.
k
k
Moreover, it is easily seen that
a1 , a2 , . . . , an ∈ L2p
(u,∞) (M, E)
⇒
b1 , b2 , . . . , bn ∈ L2p
(∞,v) (M, E)
⇒
¡X
¡X
k
k
ak a∗k
b∗k bk
¢ 12
¢ 12
∈ L2p
(u,∞) (M, E),
∈ L2p
(∞,v) (M, E).
In particular, it turns out that (6.1) simplifies in this case as follows
½° X
°¡ X
¢ °
¢1/2 °
°
°¡
°
∗ 1/2 °
∗
n
kxkRn2p,q (M,E)⊗M C2p,q
=
inf
a
a
b
b
°
°
°
°
k k
(M,E)
k k
n
k
k
R2p,q
n
C2p,q
¾
,
where the infimum runs over all possible decompositions
X
x=
ak bk .
k
Of course, this argument holds in a more general context. Indeed, arguing as
in Proposition 4.5, we deduce that the conditional Lp space Lp(u,v) (M, E) embeds
contractively into Ls (M) with
1/s = 1/u + 1/p + 1/v.
Thus, the same arguments lead to the same simplification for the spaces
A =
L2p (M) ⊗M L2p (M),
B =
L2p (M) ⊗M L2p
(M, E),
(∞, 2pq )
C =
L2p
(M, E)
2pq
( p−q
,∞)
D =
L2p
(M, E) ⊗M L2p
(M, E).
( 2pq ,∞)
(∞, 2pq )
p−q
⊗M L2p (M),
p−q
p−q
Our first step in the proof of Theorem 6.2 is the following.
Lemma 6.3. We have
1
1
1
1
1
1
n
Rn2p,q (M, E) ⊗M C2p,q
(M, E) ' n p A ∩ n 2p + 2q B ∩ n 2p + 2q C ∩ n q D,
where the relevant constants in the isomorphism above are independent of p, q, n.
Proof. It is clear that
1
1
1
1
1
1
n
Rn2p,q (M, E) ⊗M C2p,q
(M, E) ⊂ n p A ∩ n 2p + 2q B ∩ n 2p + 2q C ∩ n q D
contractively. To prove the reverse inclusion it suffices to see that
1
1
⊂ Rn2p,q (M, E) ⊗M n 2p L2p (M) = X1 ,
1
⊂ Rn2p,q (M, E) ⊗M n 2q L2p
(M, E) = X2 ,
(∞, 2pq )
n p A ∩ n 2p + 2q C
(6.3)
n 2p + 2q B ∩ n q D
1
1
1
1
(6.2)
1
p−q
6.1. AMALGAMATED TENSORS
109
with constants independent of p, q, n and also
n
X1 ∩ X2 ⊂ Rn2p,q (M, E) ⊗M C2p,q
(M, E)
(6.4)
with absolute constants. In fact, the three inclusions can be proved using the same
principle, which obviously works in a much more general setting. Indeed, let us
1
1
1
p A ∩ n 2p + 2q C and δ > 0, we may
prove (6.2). If x is a norm
P one element in n P
find decompositions x =
k a1k a2k and x =
k c1k c2k satisfying the following
estimates
°
n 1 °¡ X
o
¢1 °
¢1 °
1 °¡ X
°
°
°
max n 2p °
a1k a∗1k 2 ° , n 2p °
a∗2k a2k 2 °
≤ 1 + δ,
k
k
2p
2p
°
°
°
o
n 1 °¡ X
¢1 °
¢1 °
1 °¡ X
°
c1k c∗1k 2 ° 2p
c∗2k c2k 2 °
≤ 1 + δ.
max n 2q °
, n 2p °
k
L
(
k
(M,E)
2pq
,∞)
p−q
2p
Let us consider the following element in L2p (M)
³X
´1/2
X
γ=
a∗2k a2k +
c∗2k c2k + δD1/p
.
k
k
Since γ is invertible as a measurable operator, we may define ξ by x = ξγ. Moreover,
we also define (a02k , c02k ) by a2k = a02k γ and c2k = c02k γ. This gives rise to the
following expressions for x that will be used below
¡X
¢
x = ξγ =
a1k a02k γ,
k
¡X
¢
x = ξγ =
c1k c02k γ.
k
The norm of x in X1 is estimated as follows
kxkX1 ≤ kξkRn2p,q (M,E) kγk
1
n 2p L2p (M)
,
where the norm of ξ in the space Rn2p,q (M, E) is given by
°
°
°
n 1 °X
1 °X
°
°
°
max n 2p °
a1k a02k ° , n 2q °
c1k c02k ° 2p
k
However, since
k
2p
X
k
°
°
1 °X
°
S1 = n 2p °
a1k a02k °
2p
Similarly, we have
(
X
k
°
°
°
¢1 °
1 °¡ X
° °¡ X 0 ∗ 0 ¢ 12 °
≤ n 2p °
a1k a∗1k 2 ° °
a2k a2k °
k
∗
k
c02k c02k = γ −1 (
k
≤
=
n
X
k
L2q (M)
°1/2
n° X
°
°
sup °
αc1k c∗1k α∗ °
k
1
n 2q
∞
≤ 1 + δ.
c∗2k c2k )γ −1 ≤ 1
o
¯
¯ kαkL
≤1
2pq (N )
Lq (M)
n° ¡ X
¢1 °
°
°
sup °α
c1k c∗1k 2 °
k
k
2p
and therefore we deduce
°
n° X
1
°
°
S2 = n 2q sup °α
c1k c02k °
1
2q
(M,E)
2pq
,∞)
p−q
©
ª
= max S1 , S2 .
X
∗
a∗2k a2k )γ −1 ≤ 1,
a02k a02k = γ −1 (
we obtain
k
L
o
p−q
o
¯
¯ kαkL
≤
1
2pq (N )
L2q (M)
p−q
o
¯
¯ kαkL
≤ 1 ≤ 1 + δ.
2pq (N )
Thus, we have proved
kξkRn2p,q (M,E) ≤ 1 + δ.
p−q
n
6. FACTORIZATION OF Jp,q
(M, E)
110
1
It remains to estimate the norm of γ in n 2p L2p (M)
°
°X
°
´ 12
³° X
√
1
1
1
1
°
°
°
°
n 2p kγk2p ≤ n 2p °
a∗2k a2k ° + °
c∗2k c2k ° + δ
≤ 2(1 + δ) + δ 2 n 2p .
k
k
p
p
+
In conclusion, letting δ → 0 we obtain
kxkX1 ≤
√
2.
This proves (6.2) and inclusion (6.3) is proved in a similar way. To prove (6.4) we
just need to observe that the common factor Rn2p,q (M, E) in X1 ∩ X2 is on the left,
in contrast with the previous situations where the common factor was on the right.
The only consequence is that the roles of ξ and γ above must be exchanged.
¤
Lemma 6.4. We have
A =
Lp (M),
B =
Lp(∞, 2pq ) (M, E),
C =
Lp( 2pq ,∞) (M, E),
p−q
D =
Lp( 2pq , 2pq ) (M, E),
p−q
p−q p−q
isometrically whenever the indices p and q satisfy 1 ≤ q ≤ p < ∞ or 1 < q ≤ p ≤ ∞.
Proof. Let us define
1/s = 1/u + 1/p + 1/v = 1/s1 + 1/s2
with the indices s1 and s2 given by
1/s1
1/s2
=
=
1/u + 1/2p,
1/2p + 1/v.
Then we obtain the following estimate
°X
°
°X
°
°
°
°
°
ak bk ° p
= sup °
αak bk β °
°
k
k
L(u,v) (M,E)
s
α,β
° °¡ X
° ¡X
1
¢1 °
¢
° °
°
°
b∗k bk 2 β °
≤ sup °α
ak a∗k 2 ° °
k
k
s1
s2
α,β
°
°¡ X
°
°¡ X
1
1
¢
¢
°
°
°
°
= °
ak a∗k 2 ° 2p
b∗k bk 2 °
°
k
L(u,∞) (M,E)
k
L2p
(M,E)
(∞,v)
,
where the supremum runs over all α in the unit ball in Lu (N ) and all β in the
unit ball of Lv (N ). Therefore, taking infima on the right we obtain a contractive
inclusion
2p
p
L2p
(u,∞) (M, E) ⊗M L(∞,v) (M, E) ⊂ L(u,v) (M, E).
This shows at once the lower estimate for all isometries and for 1 ≤ q ≤ p ≤ ∞.
That is, with no restriction on the indices. In order to show the reverse inequalities,
2pq
or ∞. We shall obtain
we restrict the indices u and v to be either p−q
2p
Lp(u,v) (M, E) ⊂ L2p
(u,∞) (M, E) ⊗M L(∞,v) (M, E)
contractively. To do so we begin with some remarks. First, the isometry A = Lp (M)
is very well-known and there is nothing to prove. Thus, the case q = p is trivial
since the spaces on the left collapse into L2p (M) ⊗M L2p (M) while the spaces
6.1. AMALGAMATED TENSORS
111
on the right coincide with Lp (M). Therefore, we just need to consider the cases
1 ≤ q < p < ∞ and 1 < q < p ≤ ∞. In both cases we have
2pq
2<
< ∞.
p−q
In other words, we may assume that 2 < min(u, v) < ∞. In particular, we are
in position to apply the standard Grothendieck-Pietsch separation argument as in
Theorem 3.16 and Observation 3.17. This will be our main tool in the proof. Let
us consider the following norm on Lp (M)
(
)
°¡ X
°¡ X
¢1 °
¢ 12 °
°
°
°
∗ 2°
∗
|||x||| = inf °
ak ak ° 2p
bk bk ° 2p
°
k
L(u,∞) (M,E)
k
L(∞,v) (M,E)
P
where the infimum runs over all decompositions of x into a finite sum k ak bk with
2p
ak , bk ∈ L2p (M). Since this norm majorizes that of L2p
(u,∞) (M, E)⊗M L(∞,v) (M, E)
in Lp (M), it suffices by density to see that the norm in Lp(u,v) (M, E) controls ||| |||
from above. To that aim, given x ∈ Lp (M), we consider a norm one functional
φ : (Lp (M), ||| |||) → C
satisfying |φ(x)| = |||x|||.
Then, the reverse inequalities will follow from
n
o
¯
(6.5)
|φ(x)| ≤ sup kαxβkLs (M) ¯ α ∈ BLu (N ) , β ∈ BLv (N ) .
Applying the standard Grothendieck-Pietsch separation argument, we may find
positive elements α ∈ BLu (N ) and β ∈ BLv (N ) satisfying
|φ(ab)| ≤ kαakLs1 (M) kbβkLs2 (M) .
Then, taking qα and qβ to be the support projections of α and β respectively, we
obtain after the usual arguments (see e.g. Theorem 3.16) a right M-module map
Ψ : qα Ls1 (M) → qβ Ls02 (M) determined by
³
´
φ(ab) = trM Ψ(αa)bβ .
In particular, there exists mΨ ∈ BLs0 (M) satisfying Ψ(αa) = mΨ αa so that
n
o
¯
¯
¡
¢¯
|φ(x)| = ¯trM mΨ αxβ ¯ ≤ sup kαxβkLs (M) ¯ α ∈ BLu (N ) , β ∈ BLv (N ) .
This completes the proof of inequality (6.5) and thus the proof is concluded.
¤
Let us observe that Lemmas 6.3 and 6.4 give Theorem 6.2 for every pair of
n
(M, E). However, this is for several reasons
indices (p, q) except for the case of J∞,1
one of the most important factorization results that we need. In order to factorize
n
(M, E), we note that Lemma 6.3 is still valid. Moreover, due to the
the space J∞,1
obvious isometries
X1 ⊗M M = X1
and
M ⊗M X2 = X2
for any right (resp. left) M-module, we deduce that the first three isometries in
Lemma 6.4 are trivial in the limit case p = ∞. Hence, we just need to show that
the isometry for D still holds in the case (p, q) = (∞, 1). Again, we recall that
the lower estimate can be proved as in Lemma 6.4 so that it suffices to prove the
upper estimate. Unfortunately, the application of Grothendieck-Pietsch separation
argument in this case is much more delicate and we need some preparation. After
some auxiliary results in the next paragraph, we will go back to this question.
n
6. FACTORIZATION OF Jp,q
(M, E)
112
6.2. Conditional expectations and ultraproducts
We study certain ultraproduct von Neumann algebras and the corresponding
conditional expectations. These auxiliary results will be used to factorize the norm
n
of J∞,1
(M, E) as explained above.
Lemma 6.5. Let F be a finite dimensional subspace of M∗ and let G be a finite
dimensional subspace of M. Then, given any δ > 0 there exists a linear mapping
ω : F → M∗
satisfying the following properties:
i) kωkcb ≤ 1 + δ.
ii) The space ω(F ∩ N ∗ ) is contained in N∗ .
iii) The following estimate holds for any f ∈ F and g ∈ G
¯
¯
¯g(ω(f )) − f (g)¯ ≤ δ kf kF kgkG .
Proof. Let (f1 , f2 , . . . fk ; f1∗ , f2∗ , . . . , fk∗ ) be an Auerbach basis of F ∩ N ∗ . That
is, (f1 , f2 , . . . , fk ) is a basis of F ∩ N ∗ with kfj k = 1 for 1 ≤ j ≤ k and the fj∗ ’s are
functionals on F ∩ N ∗ satisfying fi∗ (fj ) = δij . Let us take Hahn-Banach extensions
f1∗ , . . . , fk∗ : F → C of f1∗ , . . . , fk∗ respectively. Then we may define the projection
P : f ∈ F 7→
k
X
fj∗ (f )fj ∈ F ∩ N ∗ .
j=1
∗
Now, using P we may also consider an Auerbach basis (fk+1 , . . . , fn ; fk+1
, . . . , fn∗ )
of (1F − P)(F). Then, given any k + 1 ≤ j ≤ n we consider the linear functional
fj∗ : F → C defined by the relation
³
´
fj∗ (f ) = fj∗ f − P(f ) .
∗
) of G. This
Finally we consider an Auerbach basis (g1 , g2 , . . . , gm ; g1∗ , g2∗ , . . . , gm
allows us to define the following set for any ε > 0
n
o
¯
¯
¯
C(ε) = conv ω : F → M∗ ¯ ω(F ∩ N ∗ ) ⊂ N∗ , ¯gk (ω(fj )) − fj (gk )¯ ≤ ε .
Let us assume that
(1 + δ) BCB(F,M∗ ) ∩ C(ε) = ∅,
where C(ε) denotes the closure of C(ε) in the σ(CB(F, M∗ ), F ⊗ M) topology. We
will show below that C(ε) is not empty. Thus, by the Hahn-Banach theorem we
may find a linear functional ξ : CB(F, M∗ ) → C such that
¡
¢
¡
¢
Re ξ(ω1 ) ≤ 1 ≤ Re ξ(ω2 )
for all ω1 ∈ (1+ δ) BCB(F,M∗ ) and ω2 ∈ C(ε). This implies kξkCB(F,M∗ )∗ ≤ (1+ δ)−1 .
After identifying the space CB(F, M∗ ) with the minimal tensor product F∗ ⊗min M∗ ,
we consider the associated linear map Tξ : M∗ → F defined by
f ∗ (Tξ (m∗ )) = ξ(f ∗ ⊗ m∗ ).
Then, taking mj = fj∗ ◦ Tξ ∈ M it turns out that
ξ=
n
X
j=1
mj ⊗ fj .
6.2. CONDITIONAL EXPECTATIONS AND ULTRAPRODUCTS
113
We claim that
kξkN(M∗ ,F) = kξkI(M∗ ,F) ≤ (1 + δ)−1 ,
where N(M∗ , F) (resp. I(M∗ , F)) denotes the space of completely nuclear maps
(resp. completely integral maps) from M∗ to F. Indeed, according to the main
result of [10], M∗ is locally reflexive and so the first identity follows from Proposition 4.4 in [10]. The inequality following it holds by Corollary 12.3.4 of [11] and
the fact that kξkCB(F,M∗ )∗ ≤ (1 + δ)−1 . Moreover, since F is finite dimensional
ˆ
N(M∗ , F) = F⊗M
and N(M∗ , F)∗ = CB(F, M∗ ).
This means that given any linear map ω : F → M∗ , we have
n
¯X
¯
¯
¯
|ξ(ω)| = ¯
ω(fj )(mj )¯ ≤ kξkN(M∗ ,F) kωkcb ≤ (1 + δ)−1 kωkcb .
j=1
In particular, the inclusion map j : F → M∗ satisfies
|ξ(j)| ≤ (1 + δ)−1 .
On the other hand we can write
j=
n
X
fj∗ ⊗ fj .
j=1
Given 1 ≤ j ≤ k, let (fj,α )α∈Λ ⊂ N∗ be a net converging to fj in the σ(N ∗ , N )
topology. Thus fj,α ◦ E converges to fj ◦ E in the σ(M∗ , M) topology. When
j = k + 1, k + 2, . . . , n we may also fix a net (fj,α )α∈Λ ⊂ M∗ converging to fj in
the σ(M∗ , M) topology. This implies that the maps ωα : F → M∗ defined by
ωα (f ) =
n
X
fj∗ (f )fj,α
satisfy ωα (F ∩ N ∗ ) ⊂ N∗
j=1
since fj,α ∈ N∗ for j = 1, 2, . . . , k and fj∗ (P(F)) = 0 for j = k + 1, k + 2, . . . , n.
Moreover, for large enough α we clearly find
¯
¯
¯gk (ωα (fj )) − fj (gk )¯ ≤ ε.
This shows that there exists α for which ωα ∈ C(ε). Finally, we have
lim ξ(ωα ) = lim
α
α
n
X
n
X
¡
¢
¡
¢
mj ωα (fj ) = lim
mj fj,α = ξ(j).
j=1
α
j=1
Therefore we may find α such that
ωα ∈ C(ε) and |ξ(ωα )| < 1.
¡
¢
However, any ω ∈ C(ε) satisfies Re ξ(ω) ≥ 1. Therefore, we have a contradiction
so that we can find
ω ∈ (1 + δ) BCB(F,M∗ ) ∩ C(ε).
Such a mapping clearly satisfies
n
m
X
X
¯
¯ ¯X ∗
¡
¢¯¯
¯g(ω(f )) − g(f )¯ = ¯¯
fj (f )gk∗ (g) gk (ω(fj )) − fj (gk ) ¯ ≤ ε
|fj∗ (f )|
|gk∗ (g)|.
j,k
j=1
k=1
Then, recalling the definition of fj∗ and gk∗ we easily obtain
¯
¯
¯g(ω(f )) − g(f )¯ ≤ εmnk1F − Pk kf kF kgkG ≤ 2εmn2 kf kF kgkG
n
6. FACTORIZATION OF Jp,q
(M, E)
114
for all f ∈ F and g ∈ G. Therefore, taking ε < δ/2mn2 the assertion follows.
In the following we use the notation
(xi )• = Equivalence class of (xi ) in
Y
¤
Xi .
i,U
Lemma 6.6. There exist an ultrafilter U on an index set I and a linear map
Y
α : M∗ →
M∗
i,U
satisfying the
i) The
ii) The
iii) The
following properties:
map α is a complete contraction.
Q
space α(N ∗ ) is contained in i,U N∗ .
following identity holds for all ψ ∈ M∗ and m ∈ M
limi,U m(α(ψ)i ) = ψ(m).
Proof. Let I be the set of tuples (F, G) with F a finite dimensional subspace
of M∗ and G a finite dimensional subspace of M. Let U be an ultrafilter containing
all the subsets of I of the form
n
o
¯
IF,G = (F0 , G0 ) ¯ F ⊂ F0 , G ⊂ G0 .
Note that this can be done since these sets have the finite intersection property.
For fixed F, G we choose ωF,G : F → M∗ satisfying the assumptions of Lemma 6.5
for δ = (dim F dim G)−1 . Then we define
¡
α(ψ) = ωF,G (ψ))• .
Note that for (F, G) ∈ Ihψi,h0i this is well-defined. Hence α is well-defined on M∗ .
It is easilyQchecked α is linear and completely contractive. By construction we have
¤
α(N ∗ ) ⊂ i,U N∗ and α(ψ)(m) = limi,U m(α(ψ)i ) for all m ∈ M.
Lemma 6.7. There exists normal conditional expectations
³Y
´∗
³Y
´∗
³Y
´∗
E:
M∗ →
N∗
and E :
M∗ → M∗∗ .
i,U
i,U
i,U
Moreover, they are related to each other by the identity E
∗∗
Proof. Let us consider the map
Y
Y
EU : (xi )• ∈
M 7→ (E(xi ))• ∈
i,U
By strong density of
Q
◦ E = E ◦ E.
i,U
N.
¡Q
¢∗
M in
i,U M∗ , we may define
³Y
´∗
³Y
´∗
E:
M∗ →
N∗
i,U
i,U
i,U
with predual EU
∗ . On the other hand, the map
Y
M∗ 7→ limi,U ϕi (·) ∈ M∗
γ : (ϕi )• ∈
i,U
1Q
clearly satisfies α ◦ γ =
with α being the map constructed in Lemma 6.6.
i,U M∗
Taking adjoints we find that
³Y
´∗
γ ∗ : M∗∗ →
M∗
i,U
n
6.3. FACTORIZATION OF THE SPACE J∞,1
(M, E)
115
is an injective ∗-homomorphism so that the adjoint E of α
³Y
´∗
E:
M∗ → M∗∗
i,U
is a normal conditional expectation. We are interested
Q in proving the relation
E∗∗ ◦ E = E ◦ E. Note that since α(N ∗ ) is contained in i,U N∗ , we have
³Y
´∗
E
N∗ ⊂ N ∗∗ .
i,U
∗∗
In particular, both maps E ◦ E and E ◦ E end in N ∗∗ . However, if we predualize
this identity it turns out that it suffices to see that
α ◦ E∗ = EU
∗ ◦ α|N ∗ .
This is a reformulation of Lemma 6.6 iii). The proof is complete.
¤
n
6.3. Factorization of the space J∞,1
(M, E)
According to [55],
Y
Lp (MU ) =
Lp (M) with
i,U
MU =
³Y
i,U
M∗
´∗
and 1 ≤ p < ∞.
Then, by Lemma 6.7 the inclusion below is a complete contraction
ξp : Lp (M∗∗ ) → Lp (MU ).
Lemma 6.8. The map ξ1 : L1 (M∗∗ ) → L1 (MU ) satisfies
¡
¢
1
1
ξ1 (ax)(yb) = ξ1 (a2 ) 2 xy ξ1 (b2 ) 2
∗∗
for all (x, y) ∈ Lr∞ (M, E) × Lc∞ (M, E) and all positive elements a, b ∈ L+
2 (N ).
Proof. Accordingly to the terminology used in Lemma 6.7, we may identify
M∗∗ with its image γ ∗ (M∗∗ ) in MU . In particular, the map ξp can be regarded as
an M∗∗ bimodule map satisfying
1/p
1/p
ξp (Dψ ) = Dψ◦E
for every functional ψ ∈ M∗ with associated density Dψ ∈ L1 (M∗∗ ) so that ψ ◦ E is
a functional on MU with associated density Dψ◦E ∈ L1 (MU ). Therefore, we have
1/2
1/2
1/2
1/2
ξ1 (Dψ ) = Dψ◦E Dψ◦E = ξ2 (Dψ )ξ2 (Dψ ).
Now, let us consider a, b ∈ L2 (M∗∗ ) and define Dψ = aa∗ + b∗ b with corresponding
1/2
1/2
positive functional ψ. Let m1 , m2 ∈ M∗∗ such that a = Dψ m1 and b = m2 Dψ .
Assuming that m1 and m2 are ψ-analytic and using the bimodule property of ξ1
we obtain
¡ 1/2
1/2 ¢
ξ1 (ab) = ξ1 Dψ m1 m2 Dψ
¡ ψ
¢
ψ
= ξ1 σ−i/2
(m1 )Dψ σi/2
(m2 )
=
1/2
1/2
ψ◦E
ψ◦E
σ−i/2
(m1 )Dψ◦E Dψ◦E σi/2
(m2 )
1/2
1/2
= Dψ◦E m1 m2 Dψ◦E
=
1/2
1/2
ξ2 (Dψ )m1 m2 ξ2 (Dψ ).
n
6. FACTORIZATION OF Jp,q
(M, E)
116
Thus, by approximation with ψ-analytic elements, we conclude ξ1 (ab) = ξ2 (a)ξ2 (b).
Assuming in addition that a, b are positive and x, y ∈ M∗∗
¡
¢
1
1
(6.6)
ξ1 (ax)(yb) = ξ2 (ax)ξ2 (yb) = ξ2 (a)xyξ2 (b) = ξ1 (a2 ) 2 xy ξ1 (b2 ) 2 .
∗∗
This proves the assertion with a, b ∈ L+
2 (M ) and x, y ∈ M. Before considering
r
c
elements (x, y) ∈ L∞ (M, E)×L∞ (M, E), we observe that ξp (Lp (N ∗∗ )) is contained
in the space
³Y
´∗
Y
Lp (NU ) =
Lp (N ) with NU =
N∗ .
i,U
i,U
Indeed, according to Lemma 6.7 we know that E∗∗ ◦ E = E ◦ E so that E(Lp (NU )) is
contained in Lp (N ∗∗ ). According to (6.6) and the density of M in Lr∞ (M, E) and
∗∗
Lc∞ (M, E), to prove the assertion it suffices to see that given a, b ∈ L+
2 (N ) and
x, y ∈ M we have
° ¡
°
°1/2 °
°1/2
¢°
(6.7) °ξ1 (ax)(yb) °L1 (MU ) ≤ kakL2 (N ∗∗ ) °E(xx∗ )°M °E(y ∗ y)°M kbkL2 (N ∗∗ ) .
By (6.6) we have
° ¡
¢°
°ξ1 (ax)(yb) °
L
1 (MU )
³
´1/2
³
´1/2
≤ trMU ξ2 (a)xx∗ ξ2 (a)
trMU ξ2 (b)y ∗ yξ2 (b)
.
Now, using E∗∗ ◦ E = E ◦ E and ξ2 : L2 (N ∗∗ ) ,→ L2 (NU ), we have
³
´1/2
³
¡
¢´1/2
trMU ξ2 (a)xx∗ ξ2 (a)
= trMU E ◦ E ξ2 (a)xx∗ ξ2 (a)
³
´1/2
= trMU ξ2 (a)E(xx∗ )ξ2 (a)
°
°1/2
≤ kξ2 (a)kL2 (NU ) °E(xx∗ )°M .
Similarly, we have
³
´1/2 °
°1/2
trMU ξ2 (b)y ∗ yξ2 (b)
≤ °E(y ∗ y)°M kξ2 (b)kL2 (NU ) .
Therefore, (6.7) follows since ξ2 is a contraction. This concludes the proof.
¤
Proposition 6.9. Any x ∈ M satisfies
kxkL∞
(M,E)⊗M L∞
(M,E) = kxkL∞
(M,E) .
(2,∞)
(∞,2)
(2,2)
Proof. The lower estimate can be proved as in Lemma 6.4. To prove the
upper estimate, we consider the following norm on M
½° X
¾
°¡ X
¢ °
¢1/2 °
°¡
°
°
∗ 1/2 °
∗
|||x||| = inf °
E(ak ak )
E(bk bk )
°
°
°
k
L∞ (N )
k
L∞ (N )
P
where the infimum runs over all decompositions of x into a finite
¡ sum ¢k ak bk
with ak , bk ∈ M. Let x ∈ M and take a norm one functional φ : M, ||| ||| → C
satisfying |||x||| = |φ(x)|. Then it is clear that it suffices to see that
o
n
¯
(6.8)
|φ(x)| ≤ sup kαxβkL1 (M) ¯ α, β ∈ BL2 (N ) .
Applying the Grothendieck-Pietsch separation argument, we may find states ψ1 and
ψ2 in N ∗ with associated densities D1 and D2 in L1 (N ∗∗ ) satisfying the following
inequality
1
1
|φ(ab)| ≤ ψ1 (aa∗ ) 2 ψ2 (b∗ b) 2 .
n
6.3. FACTORIZATION OF THE SPACE J∞,1
(M, E)
1/2
117
1/2
By Kaplansky’s density theorem, Dj M is norm dense in Dj M∗∗ for j = 1, 2.
Therefore, taking ej to be the support of Dj for j = 1, 2, we may consider the map
Ψ : e1 L2 (M∗∗ ) → e2 L2 (M∗∗ )
determined by the relation
³
´
1
1
1
­ 1
®
φ(ab) = D22 b∗ , Ψ(D12 a) = trM∗∗ Ψ(D12 a)bD22 .
1
1
From this it easily follows that Ψ(D12 am) = Ψ(D12 a)m for all m ∈ M. Hence, by
density we deduce that Ψ commutes on L2 (M∗∗ ) with the right action on M∗∗ . In
particular, we may find a contraction mΨ ∈ M∗∗ such that
1
1
Ψ(D12 a) = mΨ D12 a.
This gives
³
´
1
1
φ(ab) = trM∗∗ mΨ D12 abD22 .
On the other hand, by Goldstein’s theorem there exists a net (mλ ) in the unit
ball of M which converges to mΨ in the σ(M∗∗ , M∗ ) topology. Therefore, since
1/2
1/2
D1 abD2 ∈ L1 (M∗∗ ) we deduce that
³
´
1
1
φ(ab) = lim trM∗∗ mλ D12 abD22 .
λ
By Lemma 6.6 we have
³
´
1
1
trM∗∗ mλ D12 abD22 = ψ
1
D12
1
abD22
³
¡
(mλ ) = trMU mλ α ψ
¢´
1
D12
1
abD22
.
Observing that ξ1 = α we may apply Lemma 6.8 to obtain
³
´
³
1
1
¢´
¡
1
= limi,U trM mλ ξ2 (D12 )i ab ξ2 (D22 )i .
trMU mλ α ψ 12
2
D1 abD2
Therefore we conclude
³
´
1
1
φ(x) = lim limi,U trM mλ ξ2 (D12 )i x ξ2 (D22 )i .
λ
Moreover, since there exist nets (αi ) and (βi ) in L+
2 (N ) such that
1
(αi )• = ξ2 (D12 )
and
1
(βi )• = ξ2 (b 2 ),
we obtain the following expression for φ(x)
¡
¢
φ(x) = lim limi,U trM mλ αi x βi .
λ
Recalling that limi,U kαi k2 ≤ 1 and limi,U kβi k2 ≤ 1, we obtain
n
o
¯
|φ(x)| ≤ sup kαxβkL1 (M) ¯ α, β ∈ BL2 (N ) .
This proves (6.8) and implies the assertion. The proof is completed.
¤
CHAPTER 7
Mixed-norm inequalities
In this chapter we construct an isomorphic embedding
n
(M, E) ,→ Lp (A; `nq ),
Jp,q
with A being a sufficiently large von Neumann algebra. After that, we conclude
by studying the right analogue of inequality (Σpq ) (see the Introduction) in the
noncommutative and operator space levels. As we shall see, this appears as a
n
particular case of our embedding of Jp,q
(M, E) into Lp (A; `nq ) when considering the
so-called asymmetric Lp spaces, a particular case of conditional noncommutative
Lp spaces. These results complete the mixed-norm inequalities explored in this
paper, which will be instrumental in the last chapter when dealing with operator
space Lp embeddings.
n
(M, E) into Lp (A; `nq )
7.1. Embedding of Jp,q
Now we use the factorization results proved in the previous chapter to identify
n
(M, E) as an interpolation scale in q. This will give rise to an
the spaces Jp,q
n
isomorphic embedding of Jp,q
(M, E) into the space Lp (A; `nq ), with A determined
by the map (5.25)
n
X
u : x ∈ M 7→
xk ⊗ δk ∈ A⊕n .
k=1
Here xk = πk (x, −x) in the terminology of Chapter 5. That is, A⊕n denotes the
direct sum A ⊕ A ⊕ . . . ⊕ A (with n terms) of the N -amalgamated free product
∗N Ak with Ak = M ⊕ M for 1 ≤ k ≤ n and πk denotes the natural embedding of
Ak into A. The following lemma generalizes the main result in [22].
Lemma 7.1. If 1 ≤ p ≤ ∞, the map
n
u : x ∈ Jp,1
(M, E) 7→
n
X
xk ⊗ δk ∈ Lp (A; `n1 )
k=1
is an isomorphism with complemented image and constants independent of p, n.
Proof. Since the result is clear for p = 1, we shall assume in what follows
that 1 < p ≤ ∞. According to Theorem 6.2, we identify the intersection space
n
n
Jp,1
(M, E) with the amalgamated tensor Rn2p,1 (M, E) ⊗M C2p,1
(M, E). Now, using
the characterization of Lp (A; `n1 ) given in [16], we have


°¡ X
°
°¡ X
°

°
°
¢
¢
1/2 °
°
°
∗ 1/2 °
∗
°u(x)°
=
inf
a
a
b
b
,
°
°
°
°
kj kj
kj kj
Lp (A;`n
1)

L2p (A)
L2p (A) 
j,k
j,k
119
120
7. MIXED-NORM INEQUALITIES
with the infimum running over all possible decompositions
X
xk =
akj bkj
j
and where sum above is required to converge in the norm topology for 1 ≤ p < ∞
and in the weak operator
of x
P topology otherwise. Then, given any decomposition
n
into a finite sum x = j αj βj with αj ∈ Rn2p,1 (M, E) and βj ∈ C2p,1
(M, E), we
have
X
xk =
akj bkj with akj = πk (αj , αj ) and bkj = πk (βj , −βj ).
j
This observation provides the following estimate
°X
°
°
°
°
°
°u(x)°
(7.1)
≤
Λ
(α)
⊗
e
°
°
r,k
1k
n
Lp (A;`1 )
n )
k
L2p (A;R2p
°X
°
°
°
× °
Λc,k (β) ⊗ ek1 °
n
k
L2p (A;C2p )
P
for any possible decomposition x = j αj βj and where
h¡ X
¢1/2 ¡ X
¢1/2 i
,
Λr,k (α) = πk
αj αj∗
,−
αj αj∗
j
j
i
h¡ X
X
¢
¢
¡
1/2
1/2
.
,−
βj∗ βj
Λc,k (β) = πk
βj∗ βj
j
j
We want to reformulate (7.1) in terms of the square functions
¡X
¢1/2
¡ X ∗ ¢1/2
Sr (α) =
αj αj∗
and Sc (β) =
βj βj
.
j
j
By the definition of the map u, we find
n° ¡
°
°
¢°
°u(x)°
°u Sr (α) °
≤
inf
Lp (A;`n )
L
o
° ¡
¢°
°
°
u
S
(β)
,
c
n
n )
L2p (A;C2p
2p (A;R2p )
1
P
where the infimum runs over decompositions x =
k αj βj . However, applying
Corollary 5.3 as we did in the proof of Lemma 5.14, the spaces Rn2p,1 (M, E) and
n
n
n
)
) and L2p (A; C2p
(M, E) are isomorphic to their images via u in L2p (A; R2p
C2p,1
respectively. In particular, we obtain the following inequality up to a constant
independent of p, n
n°
o
°
°
°
°
°
°u(x)°
°Sr (α)° n
°Sc (β)° n
.
inf
.
n
(M,E)
L (A;` )
R
C
(M,E)
p
1
2p,1
2p,1
n
⊗M C2p,1
(M, E). Thus, we
The right hand side is the norm of x in
have proved that u is a bounded map. Now we prove the reverse estimate arguing
by duality. First we note that it easily follows from Theorem 4.2 and Remark 4.3
that
n
Jp,1
(M, E) = Kpn0 ,∞ (M, E)∗ for 1 < p ≤ ∞
Rn2p,1 (M, E)
where, following the notation of [22], the latter space is given by
X
1
1
1
Kpn0 ,∞ (M, E) =
n−( u + p + v ) Lu (N )Lρuv (M)Lv (N ),
u,v∈{2p0 ,∞}
with ρuv determined by 1/u + 1/ρuv + 1/v = 1/p0 . We claim that
n
(7.2)
w : x ∈ Kpn0 ,∞ (M, E) 7→
1X
xk ⊗ δk ∈ Lp0 (A; `n∞ )
n
k=1
n
7.1. EMBEDDING OF Jp,q
(M, E) INTO Lp (A; `n
q)
121
is a contraction. It is not difficult to see that the result follows from our claim.
Indeed, using the anti-linear duality bracket we find that
n
hu(x), w(y)i =
n
¡
¢
1X
1X
trA πk (x, −x)∗ πk (y, −y) =
trM (x∗ y) = hx, yi.
n
n
k=1
k=1
∗
n
In particular, it turns out that w u is the identity map on Jp,1
(M, E) and uw∗ is a
n
n
bounded projection from Lp (A; `1 ) onto u(Jp,1 (M, E)). Thus, u is an isomorphism
and its image is complemented with constants independent of p, n. Therefore, it
remains to prove our claim. Since the initial space of w is a sum of four Banach
spaces indexed by the pairs (u, v) with u, v ∈ {2p0 , ∞}, it suffices to see that the
restriction of w to each of these spaces is a contraction. Let us start with the case
(u, v) = (∞, ∞). In this case the associated space is simply n−1/p Lp0 (M) and we
have by complex interpolation in 1 ≤ p0 ≤ ∞ that
n
°
1
1°
°X
°
≤ n− p kxkLp0 (M) .
x k ⊗ δk °
°
n
)
Lp0 (A;`n
∞
k=1
Indeed, when p = 1 we have an identity while for p = ∞ the assertion follows by the
triangle inequality. Therefore, recalling from [29] that the spaces Lp0 (M, `n∞ ) form
an interpolation scale in p0 , our claim follows. The space associated to u = 2p0 = v
is
1
L2p0 (N )L∞ (M)L2p0 (N ).
n
Here we use the characterization of the norm of Lp0 (A; `n∞ ) given in [16]
¾
½
n
°X
°
³
´
°
°
xk ⊗ δk °
= inf
kakL2p0 (A) sup kdk kL∞ (A) kbkL2p0 (A) .
°
Lp0 (A;`n
∞)
k=1
xk =adk b
1≤k≤n
Then we consider the decompositions of xk that come from decompositions of x.
In other words, any decomposition x = αyβ with α, β ∈ L2p0 (N ) and y ∈ L∞ (M)
gives rise to the decompositions xk = απk (y, −y)β. Hence, since
kπk (y, −y)kL∞ (A) = kykL∞ (M)
for
1 ≤ k ≤ n,
we obtain the desired estimate
n
°X
°
n
o
°
°
xk ⊗ δk °
≤ inf
kαkL2p0 (N ) kykL∞ (M) kβkL2p0 (N ) = kxk2p0 ·∞·2p0 .
°
k=1
Lp0 (A;`n
∞)
x=αyβ
Finally, it remains to consider the cross terms associated to (u, v) = (2p0 , ∞) and
(u, v) = (∞, 2p0 ). Since both can be treated using the same arguments, we only
consider the case (u, v) = (2p0 , ∞) whose associated space is
1
L2p0 (N )L2p0 (M)L∞ (N ).
n
Here we observe that given any α ∈ L2p0 (N ), the left multiplication mapping
1
1
2 + 2p
Lα :
n
X
k=1
ak ⊗ δk ∈ L2p0 (A; `n2p0 ) 7→ α
n
X
ak ⊗ δk ∈ Lp0 (A; `n∞ )
k=1
is a bounded map with norm ≤ kαkL2p0 (A) . Indeed, by complex interpolation we
only study the extremal cases. Noting that the result is trivial for p0 = ∞, it suffices
122
7. MIXED-NORM INEQUALITIES
to see it for p0 = 1. Let us assume that
n
X
kak k2L2 (A) ≤ 1.
k=1
¡P ∗
¢1/2
Then we define the invertible operator β =
(Dφ being the
k ak ak + δDφ
density associated to the state φ of A) so that ak = bk β for 1 ≤ k ≤ n where the
operators b1 , b2 , . . . , bn satisfy
n
X
b∗k bk ≤ 1.
k=1
This provides a factorization αak = αbk β from which we deduce
n
° X
°
³
´
°
°
ak ⊗δk °
≤
kαk
sup
kb
°α
k kL∞ (A) kβkL2 (A) ≤ (1+δ) kαkL2 (A) .
L
(A)
2
n
L1 (A;`∞ )
k=1
1≤k≤n
Thus, we conclude letting δ → 0. Now assume that x is a norm 1 element of
1
0
0
1
1 L2p (N )L2p (M)L∞ (N ),
2
n + 2p
so that for any δ > 0 we can find a factorization x = αy such that
kαkL2p0 (N ) ≤ 1 and
1
1
kykL2p0 (M) ≤ (1 + δ) n 2 + 2p .
Then, since α ∈ L2p0 (N ) we have
n
X
k=1
xk ⊗ δk = α
n
X
πk (y, −y) ⊗ δk = Lα
k=1
In particular,
n
°
1°
°X
°
x k ⊗ δk °
°
n
Lp0 (A;`n
∞)
n
³X
´
πk (y, −y) ⊗ δk .
k=1
≤
n
°X
°
1
°
°
kαkL2p0 (N ) °
πk (y, −y) ⊗ δk °
n
)
L2p0 (A;`n
2p0
≤
n
´1/2p0
0
1 ³X
kπk (y, −y)k2p
≤ 1 + δ.
L2p0 (A)
n
k=1
k=1
k=1
Letting δ → 0, we conclude that the mapping defined in (7.2) is a contraction. ¤
Theorem 7.2. If 1 ≤ p ≤ ∞, then
£ n
¤
n
n
Jp,1 (M, E), Jp,p
(M, E) θ ' Jp,q
(M, E)
with 1/q = 1 − θ + θ/p and with relevant constants independent of n.
Proof. By Theorem 4.6 and Observation 4.8, we have
£ n
¤
n
n
(M, E)
Jp,1 (M, E), Jp,p
(M, E) θ ⊂ Jp,q
contractively. To prove the reverse inclusion we consider the map u : M → A⊕n .
1
n
Step 1. Since Jp,p
(M, E) = n p Lp (M), we clearly have
n (M,E) .
ku(x)kLp (A;`np ) = kxkJp,p
n
Moreover, u(Jp,p
(M, E)) is clearly contractively complemented in Lp (A; `np ). On
n
the other hand, according to Lemma 7.1, Jp,1
(M, E) is isomorphic to its image
n
via u, which is complemented in Lp (A; `1 ) with constants not depending on p, n.
n
7.1. EMBEDDING OF Jp,q
(M, E) INTO Lp (A; `n
q)
123
n
n
Therefore, the norm of x in the space [Jp,1
(M, E), Jp,p
(M, E)]θ turns out to be
equivalent to the norm of
n
X
xk ⊗ δk
in Lp (A; `nq ).
k=1
Thus, it remains to see that
n
°X
°
°
°
xk ⊗ δk °
(7.3)
°
Lp (A:`n
q)
k=1
n (M,E) .
. kxkJp,q
Step 2. We need an auxiliary result. Let us consider the bilinear map
n
Λ : (a, b) ∈ Rn2p,q (M, E) × C2p,q
(M, E) 7→
n
X
πk (ab, −ab) ⊗ δk ∈ Lp (A; `nq ).
k=1
We claim that Λ is bounded by c(p, q) ∼ (p − q)/(pq + q − p). By Theorem 5.16,
we may use bilinear interpolation and it suffices to show our claim for the extremal
values of q. Let us note that, since we are applying Theorem 5.16 for Rn2p,q (M, E)
n
(M, E), the operator norm of Λ is controlled by the square of the relevant
and C2p,q
constant in Theorem 5.16, as we have claimed. The boundedness of Λ for q = 1
follows from Theorem 6.2 and Lemma 7.1. The estimate for q = p is much easier
n
³X
°
°
°
°p ´ p1
1
°Λ(a, b)°
°
°
= n p kabkp ≤ kakR2p,p kbkC2p,p .
=
π
(ab,
−ab)
k
L (A;`n )
p
p
p
k=1
Step 3. Now it is easy to deduce inequality (7.3) from the boundedness of the
mapping Λ. Indeed, according to Theorem 6.2 we know that (7.3) is equivalent to
n
°X
°
°³ X
°³ X
´ 12 °
´ 12 °
°
°
°
°
°
°
(7.4)
x k ⊗ δk °
.°
aj a∗j ° n °
b∗j bj ° n ,
°
k=1
j
Lp (A:`n
q)
j
R2p,q
C2p,q
P
for any decomposition of x into a finite sum j aj bj . Let us assume that the index
j runs from 1 to m for some finite m. Then we consider the matrix amplifications
N̂ = Mm ⊗ N and M̂ = Mm ⊗ M. Similarly, we consider  = Mm ⊗ A. According
to (5.3), we have
¡
¢ ∗n
 = Mm ⊗ A1 ∗N̂ Mm ⊗ A2 ∗N̂ · · · ∗N̂ Mm ⊗ An = M̂ ⊕ M̂ N̂ .
By Step 2, we deduce
n
°X
°
°
°
(7.5)
xk ⊗ δk °
°
k=1
Lp (Â;`n
q)
. kakR̂n kbkĈ n
2p,q
2p,q
where the elements xk are given by xk = πk (ab, −ab) and
¢
¡
R̂n2p,q = Rn2p,q M̂, 1Mm ⊗ E ,
¢
¡
n
n
Cˆ2p,q
= C2p,q
M̂, 1M ⊗ E .
m
n
To prove the P
remaining estimate (7.4) we fix x in Jp,q
(M, E) and decompose it into
a finite sum j aj bj with m terms. Then, we define the row and column matrices
a=
m
X
j=1
aj ⊗ e1j
and b =
m
X
j=1
bj ⊗ ej1 .
124
7. MIXED-NORM INEQUALITIES
According to (7.5), it just remains to show that
n
°
°X
°
°
xk ⊗ δk °
°
k=1
n
°
°X
°
°
xk ⊗ δk °
°
=
Lp (Â;`n
q)
k=1
Lp (A;`n
q)
kakR̂n
=
°³ X
´ 12 °
°
°
aj a∗j °
°
kbkĈ n
=
°³ X
´ 12 °
°
°
b∗j bj °
°
2p,q
2p,q
j
j
,
Rn
2p,q
n
C2p,q
,
.
The first identity follows easily from
xk
=
=
πk (ab, −ab)
X
¡
¢
πk aj bj ⊗ e11 , −aj bj ⊗ e11
j
³X
´
X
πk
aj bj , −
aj bj ⊗ e11
=
xk ⊗ e11 ,
=
j
j
where the third identity P
holds because πk is an N̂ -bimodule map. The relation
xk = xk ⊗ e11 shows that k xk ⊗ δk lives in fact in the (1, 1) corner, which by [30]
is isometrically isomorphic to Lp (A; `nq ). The identities for a and b are proved in
the same way, so that we only consider the first of them. We have
n 1
o
1
kakR̂n = max n 2p kakL2p (M̂) , n 2q kakL2p
(M̂,1M ⊗E) .
2p,q
(
It is clear that
°³ X
´ 12 °
°
°
kakL2p (M̂) = °
aj a∗j °
j
m
2pq
,∞)
p−q
L2p (M)
.
Therefore, we just need to show that
kakL2p
(
(M̂,1Mm ⊗E)
2pq
,∞)
p−q
°³ X
´ 12 °
°
°
=°
aj a∗j °
L2p2pq
j
(
p−q
,∞)
(M,E)
.
By definition, we have
kakL2p
(
2pq
,∞)
p−q
(M̂,1Mm ⊗E)
°
°1
= sup °αaa∗ α∗ °q2
α
° h³ X
´
i ° 12
°
°
= sup °α
aj a∗j ⊗ e11 α∗ °
α
j
α
j
q
° h³ X
´ 12
i°
°
°
= sup °α
aj a∗j
⊗ e11 ° ,
where the supremum runs over all α such that
kαkL 2pq (N̂ ) ≤ 1.
p−q
If (αij ) are the m × m entries of α, we clearly have
α
h³ X
j
aj a∗j
´ 12
m
i X
³X
´ 12
⊗ e11 =
αs1
⊗ es1 .
aj a∗j
s=1
j
2q
n
7.1. EMBEDDING OF Jp,q
(M, E) INTO Lp (A; `n
q)
125
Therefore, we obtain the following estimate
° h³ X
i°
´ 12
°
°
⊗ e11 °
aj a∗j
°α
j
L2q (M̂)
=
°³ X h X
i 12
hX
i 12 ´ 12 °
°
°
∗
aj a∗j αs1
αs1
aj a∗j
°
°
=
°³ X
´ 21 °
´ 12 ³ X
°
°
∗
aj a∗j °
αs1
αs1
°
≤
°X
°
°
°
αs1 ⊗ es1 °
°
s
j
j
s
s
j
L
2pq
p−q
L2q (M)
L2q (M)
(N̂ )
°³ X
´ 12 °
°
°
aj a∗j °
°
j
L2p2pq
(
p−q
,∞)
(M,E)
.
Hence, the result follows since the column projection is contractive on L 2pq (N̂ ). ¤
p−q
The following is a major result in this paper.
Theorem 7.3. If 1 ≤ q ≤ p ≤ ∞, the map
n
X
n
xk ⊗ δk ∈ Lp (A; `nq )
u : x ∈ Jp,q (M, E) 7→
k=1
is an isomorphism with complemented image and constants independent of n.
Proof. It is an immediate consequence of Lemma 7.1 and Theorem 7.2
¤
Remark 7.4. If 1 ≤ q ≤ p ≤ ∞, let us define
X
1
n−1+ ρuv Lu (N )Lρuv (M)Lv (N )
Kpn0 ,q0 (M, E) =
u,v∈{2r,∞}
where 1/r = 1/q − 1/p and ρuv is determined by 1/u + 1/ρuv + 1/v = 1/p0 . Note
that this definition is consistent with the space Kpn0 ,∞ (M, E) introduced in the proof
of Lemma 7.1. Arguing by duality as in [22], we easily conclude from Theorem 7.3
that we have an isomorphism with complemented image
n
1X
(7.6)
ξf ree : x ∈ Kpn0 ,q0 (M, E) 7→
xk ⊗ δk ∈ Lp0 (Af ree ; `nq0 )
n
k=1
where Af ree denotes the usual free product algebra A from Theorem 7.3. Moreover,
it is important to note that replacing free products by tensors and freeness by
noncommutative independence as in [22], Theorem 7.3 and (7.6) hold in the range
1 < p0 ≤ q 0 ≤ ∞. In other words, we replace Af ree by the tensor product algebra
Aind = M ⊗ M ⊗ · · · ⊗ M
with n terms and xk is now given by
xk = 1 ⊗ · · · 1 ⊗ x ⊗ 1 ⊗ · · · ⊗ 1
where x is placed in the k-th position. The question now is whether or not (7.6)
holds in the non-free setting for p0 = 1. Using recent results from [18], we shall see
in a forthcoming paper that we have an isomorphism with complemented image
n
X
n
ξind : x ∈ K1,2 (M, E) 7→
xk ⊗ δk ∈ Lp0 (Aind ; OHn ).
k=1
n
0
Let us note that the same question for K1,q
0 (M, E) with q 6= 2 is still open.
126
7. MIXED-NORM INEQUALITIES
7.2. Asymmetric Lp spaces and noncommutative (Σpq )
Let M be a von Neumann algebra equipped with a n.f. state ϕ. Now let
2 ≤ u, v ≤ ∞ be such that 1/p = 1/u + 1/v for some 1 ≤ p ≤ ∞. Then, we
define the asymmetric Lp space associated to the pair (u, v) as the M-amalgamated
Haagerup tensor product
(7.7)
L(u,v) (M) = Lru (M) ⊗M,h Lcv (M),
where we recall that Lrq (M) and Lcq (M) were defined in (1.3) for 2 ≤ q ≤ ∞. That
is, we consider the quotient of Lru (M)⊗h Lcv (M) by the closed subspace I generated
by the differences x1 γ ⊗ x2 − x1 ⊗ γx2 with γ ∈ M. Recall that we are using the
notation ⊗M,h instead of ⊗M because, in contrast with the previous chapters, we
shall be interested here in the operator space structure rather than in the Banach
space one. By a well known factorization argument (see e.g. Lemma 3.5 in [46]),
the norm of an element x in L(u,v) (M) is given by
kxk(u,v) = inf kαkLu (M) kβkLv (M) .
x=αβ
According to this observation, it turns out that asymmetric Lp spaces arise as a
particular case of the amalgamated noncommutative Lp spaces defined in Chapter 2
when we take q = ∞ and N to be M itself. Asymmetric Lp spaces were introduced
in [22] for finite matrix algebras. There the amalgamated Haagerup tensor product
used in (7.7) was not needed in [22] to define the asymmetric Schatten classes. In
fact, if M is the algebra Mm of m × m matrices and X is an operator space, we can
define the vector-valued asymmetric Schatten class
m
m
m
S(u,v)
(X) = Cu/2
⊗h X ⊗h Rv/2
.
Note that this definition is consistent with (7.7). Indeed, recalling that
m
Lru (Mm ) = Cu/2
⊗h Rm
and
m
Lcv (Mm ) = Cm ⊗h Rv/2
,
it can be easily checked from the definition of the Haagerup tensor product that
m
(C) isometrically. Moreover, according to
we have Lru (Mm ) ⊗M,h Lcv (Mm ) = S(u,v)
[22] any linear map u : X1 → X2 satisfies
°
°
°
°
n
n
(7.8)
kukcb = sup °1Mn ⊗ u : S(u,v)
(X1 ) → S(u,v)
(X2 )°.
n≥1
In particular, since it is clear that
¢
¡ r
n
mn
(C)
S(u,v)
Lu (Mm ) ⊗M,h Lcv (Mm ) = S(u,v)
for all
m ≥ 1,
we have the following completely isometric isomorphism
m
L(u,v) (Mm ) = S(u,v)
(C).
Remark 7.5. Lp (M) is completely isometric to Lr2p (M) ⊗M,h Lc2p (M).
We conclude by generalizing the inequalities (Σpq ) stated in the Introduction to
the noncommutative setting. Moreover, we shall seek for a completely isomorphic
embedding rather than a Banach space one. As we shall see, this appears as a
particular case of Theorem 7.3 module the corresponding identifications. Indeed,
let M be a von Neumann algebra equipped with a n.f. state ϕ and let us consider
the particular situation in which (M, N , E) above are replaced by
³
´
(7.9)
(Mm , Nm , Em ) = Mm ⊗ M, Mm , 1Mm ⊗ ϕ .
7.2. ASYMMETRIC Lp SPACES AND NONCOMMUTATIVE (Σpq )
If Dϕ is the associated density, we consider the
¡ ¢
¡ r
¢
m
ρr : D1/2
xij ∈ S∞
L2 (M) 7→
ϕ
¡ ¢
¡ c
¢
m
ρc : xij D1/2
7→
ϕ ∈ S∞ L2 (M)
127
densely defined maps
¡ ¢
xij ∈ Lr∞ (Mm , Em ),
¡ ¢
xij ∈ Lc∞ (Mm , Em ).
Lemma 7.6. The maps ρr and ρc extend to isometric isomorphisms.
Proof. By [11, p.56] we have
m
°³ X
° 1/2 ¡ ¢°
¢´°
¡
°1/2
°
∗
1/2
°Dϕ zij ° m r
=
trM D1/2
°
°
ϕ zik zjk Dϕ
S (L (M))
∞
2
k=1
m
°³ X
°
= °
Mm
´°1/2
°
∗
ϕ(zik zjk
) °
Mm
k=1
°
h¡ ¢¡ ¢ i°1/2
°¡ ¢°
∗ °
°
= ° zij °Lr
= °1Mm ⊗ ϕ zij zij °
Mm
∞ (Mm ,Em )
The proof of the second isometric isomorphism follows from [11, p.54] instead.
.
¤
Now, assuming that M, N and E are given as in (7.9), our aim is to identify
n
the intersection space Jp,q
(Mm , Em ) in terms of asymmetric Lp spaces. More
concretely, let us define the following intersection of asymmetric spaces
\
1
1
n
Jp,q
(M) =
n u + v L(u,v) (M).
u,v∈{2p,2q}
Lemma 7.7. If m ≥ 1, we have an isometry
¡ n
¢
n
Spm Jp,q
(M) = Jp,q
(Mm , Em ).
Proof. By definition we have
n
Jp,q
(Mm , Em ) =
\
1
1
1
n u + p + v Lp(u,v) (Mm , Em ).
2pq
u,v∈{ p−q
,∞}
Since the powers of n fit, it clearly suffices to show that
¡
¢
Spm L(2p,2p) (M) = Lp(∞,∞) (Mm , Em ),
¡
¢
Spm L(2p,2q) (M) = Lp(∞, 2pq ) (Mm , Em ),
Spm
¡
L(2q,2p) (M)
¢
p−q
=
¡
¢
Spm L(2q,2q) (M) =
Lp( 2pq ,∞) (Mm , Em ),
p−q
Lp( 2pq , 2pq ) (Mm , Em ).
p−q p−q
The first isometry follows from Remark 7.5, we have Lp (Mm ) at both sides. The
last one follows from Example 4.1 (b), which uses one of Pisier’s identities stated
in Chapter 1. It remains to see the second and third isometries. By complex
interpolation on q, it suffices to assume q = 1 since the case q = p has been already
considered. In that case, we have to show that
¡
¢
Spm L(2p,2) (M) = Lp(∞,2p0 ) (Mm , Em ),
¡
¢
Spm L(2,2p) (M) = Lp(2p0 ,∞) (Mm , Em ).
128
7. MIXED-NORM INEQUALITIES
When p = 1, the isometry follows again from Remark 7.5. Therefore, by complex
interpolation one more time, it suffices to assume that (p, q) = (∞, 1). In that case,
we note that
¡
¢
¡ c
¢
m
m
L(∞,2) (M) = S∞
L2 (M) = Lc∞ (Mm , Em ) = L∞
S∞
(∞,2) (Mm , Em ),
¡
¢
¡
¢
m
m
S∞
L(2,∞) (M) = S∞
Lr2 (M) = Lr∞ (Mm , Em ) = L∞
(2,∞) (Mm , Em ).
We have used Lemma 7.6 in the second identities. This completes the proof.
¤
Let Am stand for the usual amalgamated free product von Neumann algebra
constructed out of Mm = Mm ⊗ M with Nm = Mm . According to (5.3),
¡ we have¢
Am = Mm ⊗ A with A = (M ⊕ M)∗n . In particular, Lp (Am ; `nq ) = Spm Lp (A; `nq )
and we deduce the result below from Theorem 7.3 and Lemma 7.7.
Theorem 7.8. If 1 ≤ q ≤ p ≤ ∞, the map
n
X
n
u : x ∈ Jp,q
(M) 7→
xk ⊗ δk ∈ Lp (A; `nq )
k=1
is a cb-isomorphism with cb-complemented image and constants independent of n.
Remark 7.9. The o.s.s. of Lp (A; `nq ) was introduced in Chapter 1.
Remark 7.10. Let M be the matrix algebra Mm equipped with its normalized
trace τ and let us consider the direct sum M⊕n = M ⊕ M ⊕ · · · ⊕ M with n terms.
We equip this algebra with the (non-normalized) trace τn = τ ⊕ τ ⊕ · · · ⊕ τ . In this
case, given an operator space X, we could also define the space
\
Jp,q (M⊕n ; X) =
Lru (M⊕n ) ⊗M⊕n ,h L∞ (M⊕n ; X) ⊗M⊕n ,h Lcv (M⊕n ).
u,v∈{2p,2q}
The analogue of Theorem 7.8 with
i) (p, q) being (p, 1),
ii) freeness replaced by noncommutative independence,
iii) vector-values in some operator space X as explained above,
was the main result in [22]. In our situation, a vector-valued analogue of Theorem
7.8 also holds in the context of finite matrix algebras. However, note that the use
of free probability requires to define vector-valued Lp spaces for free product von
Neumann algebras [19], which is beyond the scope of this paper.
Remark 7.11. The constants in Theorems 7.2, 7.3 and 7.8 are also independent
of p, q unless p ∼ ∞ is large and q ∼ 1 is small simultaneously. In that case the
argument in Theorem 7.2 produces the singularity (p − q)/(pq + q − p) which affects
Theorems 7.3 and 7.8. Moreover, it arises as a byproduct of the use of Burkholder
inequality in Proposition 5.10, which is used in the proof of Theorem 5.16, a key
tool in Theorems 7.2. Of course, this seems to be a removable singularity. It is also
worthy of mention that this will not cause any singularity in our main application,
the construction of operator space Lp embeddings in the next chapter. Indeed, in
that case we will work all the time with q = 2 and we have
(p − q)/(pq + q − p) = (p − 2)/(p + 2) < 1.
CHAPTER 8
Operator space Lp embeddings
Given 1 ≤ p < q ≤ 2 and a von Neumann algebra M, we conclude this
paper by constructing a completely isomorphic embedding of Lq (M) into Lp (A)
for some sufficiently large von Neumann algebra A. In the first section we embed the
Schatten class Sq into Lp (A) for some QWEP von Neumann algebra A. Roughly
speaking, the proof is almost identical to our argument in [24] after replacing
Corollary 1.9 there by Theorem D in the Introduction. Thus, we shall omit some
details in our construction. The second paragraph is devoted to the stability of
hyperfiniteness and there we will present the transference argument mentioned in
the Introduction. Finally, the last paragraph contains our construction for general
von Neumann algebras.
8.1. Embedding Schatten classes
We shall prove an Lp version of [24, Theorem D], from which the embedding of
the Schatten class Sq into Lp (A) will follow for 1 ≤ p < q ≤ 2 and A a sufficiently
large QWEP von Neumann algebra. The main embedding result in Xu’s paper [71]
claims that any quotient of a subspace of Cp ⊕p Rp cb-embeds in Lp (A) for some
sufficiently large von Neumann algebra A whenever 1 ≤ p < 2. In particular, if
1 ≤ p < q ≤ 2, both Rq and Cq embed completely isomorphically in Lp (A) since
both are in QS(Cp ⊕p Rp ). The last assertion follows as in [24, Lemma 2.1]. More
precisely, Xu’s construction holds either with A being the Araki-Woods quasi-free
CAR factor and also with Shlyakhtenko’s generalization of it in the free setting
[59]. In any case, A can be chosen to be a QWEP type IIIλ factor, 0 < λ ≤ 1.
Our first embedding result generalizes Xu’s embedding.
Theorem 8.1. If for some 1 ≤ p ≤ 2
(X1 , X2 ) ∈ QS(Cp ⊕2 OH) × QS(Rp ⊕2 OH),
there exist a cb-embedding X1 ⊗h X2 → Lp (A), for some QWEP algebra A.
Sketch of the proof. The argument is very close to that of [24]. By
injectivity of the Haagerup tensor product, we may assume that X1 is a quotient
of Cp ⊕2 OH and X2 is a quotient of Rp ⊕2 OH. Therefore, the cb-isomorphisms
below follow from [24, Lemma 2.5] by duality
³¡
´
¢±
X1 'cb H11,cp ⊕2 H12,oh ⊕2 K11,cp ⊕2 K12,oh graph(Λ1 )⊥ ,
´
³¡
¢±
X2 'cb H21,rp ⊕2 H22,oh ⊕2 K21,rp ⊕2 K22,oh graph(Λ2 )⊥ ,
129
130
8. OPERATOR SPACE Lp EMBEDDINGS
for certain closed subspaces Hij , Kij (1 ≤ i, j ≤ 2) of `2 and
Λ1 : K11,cp0
Λ2 : K21,rp0
→
→
K12,oh ,
K22,oh ,
injective closed densely-defined operators with dense range. Let us set
¡
¢±
Z1 = K11,cp ⊕2 K12,oh graph(Λ1 )⊥ ,
¡
¢±
Z2 = K21,rp ⊕2 K22,oh graph(Λ2 )⊥ .
Then, we have the following cb-isometric inclusion
(8.1)
X1 ⊗h X2
⊂
⊕2
⊕2
⊕2
⊕2
⊕2
Z 1 ⊗ h Z2
H11,cp ⊗h X2
X1 ⊗h H21,rp
H12,oh ⊗h Z2
Z1 ⊗h H22,oh
H12,oh ⊗h H22,oh .
According to [71], we know that OH ∈ QS(Cp ⊕p Rp ) and that any element in
QS(Cp ⊕p Rp ) completely embeds in Lp (A) for some QWEP type III factor A. This
eliminates the last term in (8.1). The second and third terms embed into Sp (X2 )
and Sp (X1 ) completely isometrically. On the other hand, since OH ∈ QS(Cp ⊕p Rp )
and we have by hypothesis
X1 ∈ QS(Cp ⊕2 OH) and X2 ∈ QS(Rp ⊕2 OH),
both X1 and X2 are cb-isomorphic to an element in QS(Cp ⊕p Rp ). Applying Xu’s
theorem [71] one more time, we may eliminate these terms. Finally, for the fourth
and fifth terms on the right of (8.1), we apply [24, Lemma 2.6] and the self-duality
of OH to rewrite them as particular cases of the first term Z1 ⊗h Z2 . This reduces
the proof to construct a complete embedding of Z1 ⊗h Z2 into Lp (A).
By [24, Lemma 2.8] we may assume that the graphs appearing in the terms
Z1 and Z2 are graphs of diagonal operators dλ1 and dλ2 . Moreover, exactly as in
the proof of [24, Theorem D] we may use polar decomposition, perturbation and
complementation and assume that Z1 and Z2 are given by
¡
¢±¡
¢⊥
,
Cp ⊕2 OH Cp0 ∩ `oh
Z1 = Cp + `oh
2 (λ)
2 (λ) =
¢
¡
¢±¡
⊥
Rp ⊕2 OH Rp0 ∩ `oh
,
Z2 = Rp + `oh
2 (λ)
2 (λ) =
with λ1 , λ2 , . . . ∈ R+ strictly positive and
n
o
0 ⊕2 OH ,
Cp0 ∩ `oh
(λ)
=
span
(δ
,
λ
δ
)
∈
C
k
k
k
p
2
n
o
oh
Rp0 ∩ `2 (λ) = span (δk , λk δk ) ∈ Rp0 ⊕2 OH .
Before going on we adapt some terminology from [24].P Given a sequence
γ1 , γ2 , . . . of strictly positive numbers, the diagonal map dγ = k γk ekk is regarded
as the density of a n.s.s.f. weight ψ on B(`2 ). We also keep from [24] the same
terminology for its restriction ψn to the subalgebra qn B(`2 )qn (where qn denotes the
8.1. EMBEDDING SCHATTEN CLASSES
131
Pn
projection k=1 eP
kk ) and for the state ϕn determined by the relation ψn = kn ϕn
n
with kn given by k=1 γk . If 1 ≤ p ≤ 2, we define
o
n¡ 1
1
1
1
1
1
1
1 ¢¯
0
0
0
0
Jp0 ,2 (ψn ) = dψ2pn zdψ2pn , dψ2pn zdψ4 n , dψ4 n zdψ2pn , dψ4 n zdψ4 n ¯ z ∈ qn B(`2 )qn
as a subspace of the direct sum
¡
¢
¡
¢
¡
¢
¡
¢
Lnp0 = Cpn0 ⊗h Rpn0 ⊕2 Cpn0 ⊗h OHn ⊕2 OHn ⊗h Rpn0 ⊕2 OHn ⊗h OHn .
In other words, we may regard Jp0 ,2 (ψn ) as an intersection of some weighted forms
of the asymmetric Schatten classes (see Section 7.2) considered above. On the
other P
hand, by a simple perturbation argument, we may assume as in [24] that
n
kn = k=1 γk is an integer. Then, if we set Mn = qn B(`2 )qn , we easily find the
complete isometry
Jp0 ,2 (ψn ) = Jpk0n,2 (Mn ),
see Section 7.2 again. Hence, according to Theorem 7.8, we obtain a complete
embedding of Jp0 ,2 (ψn ) into a cb-complemented subspace of Lp0 (An ; OHkn ), where
An stands for the kn -fold reduced free product of qn B(`2 )qn ⊕ qn B(`2 )qn . Let us
now consider the dual space
Kp,2 (ψn ) = Jp0 ,2 (ψn )∗ .
A simple duality argument provides a cb-embedding
(8.2)
ω : x ∈ Kp,2 (ψn ) 7→
kn
1 X
xj ⊗ δj ∈ Lp (An ; OHkn ),
kn j=1
with cb-complemented image and constants independent of n.
Let us now go back to our study of the Haagerup tensor product Z1 ⊗h Z2 .
Let us recall that the finite weights ψn are restrictions of the n.s.s.f. weight ψ to
the subalgebras qn B(`2 )qn , which are directed by inclusion. In particular, we may
consider the direct limit
[
Kp,2 (ψ) =
Kp,2 (ψn ).
n≥1
A fairly simple adaptation of [24, Lemma 2.4] to the Lp case gives
Y
¡
¢
¡
¢
oh
(8.3) Z1 ⊗h Z2 = Cp + `oh
2 (λ) ⊗h Rp + `2 (λ) = Kp,2 (ψ) →
n,U
Kp,2 (ψn ).
The last step being the natural embedding of a direct limit into the corresponding
ultraproduct. According to [55], this reduces the problem to the finite-dimensional
case, which follows from Xu’s cb-embedding [71] of OH into Lp (B) since
¯ = Lp (A0n ),
Kp,2 (ψn ) → Lp (An ; OHkn ) → Lp (An ⊗B)
see [24] for a rigorous explanation. We have therefore constructed a cb-embedding
³Y
´∗
Z1 ⊗h Z2 → Lp (A) with A =
A0n ∗ .
n,U
The fact that A is QWEP is justified as in the proof of [24, Theorem D].
¤
Corollary 8.2. If 1 ≤ p < q ≤ 2, the Schatten class Sq cb-embeds into Lp (A).
Proof. Since Sq = Cq ⊗h Rq , combine [24, Lemma 2.1] and Theorem 8.1.
¤
132
8. OPERATOR SPACE Lp EMBEDDINGS
8.2. Embedding into the hyperfinite factor
Now we want to show that the cb-embedding Sq → Lp (A) can be constructed
with A being a hyperfinite type III factor. Moreover, we shall prove some more
general results to be used in the next paragraph, where the general cb-embedding
will be constructed. We first set a transference argument, based on a Rosenthal
type inequality for noncommuting identically distributed random variables in L1
from [18], which enables us to replace freeness by some sort of independence.
Let N be a σ-finite von Neumann subalgebra of some algebra A and let us
consider a family A1 , A2 , . . . of von Neumann algebras with N ⊂ Ak ⊂ A. As usual
we require the existence of a n.f. conditional expectation EN : A → N . We recall
that (Ak )k≥1 is a system of indiscernible independent copies over N (i.i.c. in short)
when
­
®
i) If a ∈ A1 , A2 , . . . , Ak−1 and b ∈ Ak , we have
EN (ab) = EN (a)EN (b).
ii) There exist a von Neumann algebra A containing N , a normal faithful
conditional expectation E0 : A → N and homomorphisms πk : A → Ak
such that
EN ◦ πk = E0
and the following holds for every strictly increasing function α : N → N
¡
¢
¡
¢
EN πj1 (a1 ) · · · πjm (am ) = EN πα(j1 ) (a1 ) · · · πα(jm ) (am ) .
iii) There exist n.f. conditional expectations Ek : A → Ak such that
EN = E0 πk−1 Ek
for all
k ≥ 1.
Further, when the first condition above also holds for a in the algebra generated
by A1 , . . . , Ak−1 , Ak+1 , . . . and the second condition holds for any permutation α
of the integers, we shall say that (Ak )k≥1 are symmetrically independent copies
(s.i.c. in short) over N . In what follows, given a probability space (Ω, µ), we shall
write ε1 , ε2 , . . . to denote an independent family of Bernoulli random variables on
Ω equidistributed on ±1. We now present the key inequality in [18].
Lemma 8.3. The following inequalities hold for x ∈ L1 (A)
a) If (Ak )k≥1 are i.i.c. over N , we have
Z °X
n
°
°
°
εk πk (x)°
dµ
°
Ω
∼
L1 (A)
k=1
inf
x=a+b+c
nkakL1 (A) +
√ °
√ °
1°
1°
n°E0 (bb∗ ) 2 °L1 (N ) + n°E0 (c∗ c) 2 °L1 (N ) .
b) If moreover, E0 (x) = 0 and (Ak )k≥1 are s.i.c. over N , then
n
°X
°
°
°
πk (x)°
°
k=1
∼
inf
x=a+b+c
L1 (A)
nkakL1 (A) +
√ °
1°
n°E0 (bb∗ ) 2 °L
1 (N )
+
√ °
1°
n°E0 (c∗ c) 2 °L
1 (N )
Proof. We claim that
n
n
n
°
°X
°
°X
°
1°
°X
°
°
°
°
°
πk (x)°
≤°
εk πk (x)°
≤ 2°
πk (x)°
°
2
L1 (A)
L1 (A)
L1 (A)
k=1
k=1
k=1
.
8.2. EMBEDDING INTO THE HYPERFINITE FACTOR
133
for any choice of signs ε1 , ε2 , . . . , εn whenever A1 , A2 , . . . are symmetric independent
copies of A over N and E0 (x) = 0. This establishes (a) ⇒ (b) and so, since the first
assertion is proved in [18], it suffices to prove our claim. Such result will follow
from the more general statement
n
n
°X
°
°X
°
°
°
°
°
εk πk (xk )°
≤ 2°
πk (xk )°
,
°
k=1
L1 (A)
k=1
L1 (A)
for any family x1 , x2 , . . . , xn in L1 (A) with E0 (xk ) = 0 for 1 ≤ k ≤ n. Indeed,
take xk = x for the upper estimate and xk = εk x for the lower estimate. Since we
assume that N is σ-finite we may fix a n.f. state ϕ. We define φ = ϕ ◦ EN and
φ0 = ϕ ◦ E0 . According to [6] we have
σtϕ ◦ E0 = E0 ◦ σtφ0
and
σtϕ ◦ EN = EN ◦ σtφ .
Moreover, since EN = EN ◦ Ek we find φ = φ ◦ Ek which implies
σtφ ◦ Ek = Ek ◦ σtφ .
In particular, σtφ (Ak ) ⊂ Ak for k ≥ 1. Therefore, given any subset S of {1, 2, . . . , n}
we find a φ-invariant conditional expectation ES : A → AS where the von Neumann
algebra AS = hAk | k ∈ Si. We claim that
ES (πj (a)) = 0 whenever E0 (a) = 0
and
j∈
/ S.
Indeed, let b ∈ AS and a as above. Then we deduce from symmetric independence
¡
¢
¡
¢
¡
¢
¡
¢
φ ES (πj (a))b = φ πj (a)b = φ EN (πj (a)b) = φ E0 (a)EN (b) = 0.
Thus we may apply Doob’s trick
n
°X
°
° ³X
´°
°
°
°
°
πk (xk )°
= °ES
πk (xk ) °
°
k∈S
L1 (A)
k=1
L1 (A)
n
°X
°
°
°
≤°
πk (xk )°
k=1
L1 (A)
.
Then, the claim follows taking {1, 2, . . . , n} = S1 ∪ S−1 with Sα = {k : εk = α}.
¤
Remark 8.4. The inequalities above generalize the noncommutative Rosenthal
inequality [30] to the case p = 1 for identically distributed variables and under such
notions of noncommutative independence. Of course, in the case 1 < p < 2 we have
much stronger results from [28, 30] and there is no need of proving any preliminary
result for our aims in this case.
Let us now generalize our previous definition of the space Kp,2 (ψ) to general
von Neumann algebras. Let M be a given von Neumann algebra, which we assume
σ-finite for the sake of clarity. Let M be equipped with a n.s.s.f. weight ψ.
In other words, ψ is given by an increasing sequence (a net in the general case)
of pairs (ψn , qn ) such that the qn ’s are increasing finite projections in M with
limn qn = 1 in the strong operator topology and σtψ (qn ) = qn . Moreover, the
ψn ’s are normal positive functionals on M with support qn and satisfying the
compatibility condition ψn+1 (qn xqn ) = ψn (x). As above, we shall write kn for the
number ψn (qn ) ∈ (0, ∞) and (again as above) we may and will assume that the
kn ’s are nondecreasing positive integers. In what follows we shall write dψn for the
density on qn Mqn associated to the n.f. finite weight ψn . If 1 ≤ p ≤ 2, we define
the space Jp0 ,2 (ψn ) as the closure of
n¡ 1
o
1
1
1
1
1
1
1 ¢¯
0
0
0
0
dψ2pn zdψ2pn , dψ2pn zdψ4 n , dψ4 n zdψ2pn , dψ4 n zdψ4 n ¯ z ∈ qn Mqn
134
8. OPERATOR SPACE Lp EMBEDDINGS
in the direct sum
Lnp0 = Lp0 (qn Mqn ) ⊕2 L(2p0 ,4) (qn Mqn ) ⊕2 L(4,2p0 ) (qn Mqn ) ⊕2 L2 (qn Mqn ).
In other words, after considering the n.f. state ϕn on qn Mqn determined by the
n
relation ψn = kn ϕn and recalling the definition of the spaces Jp,q
(M) from Section
7.2, we may regard Jp0 ,2 (ψn ) as the 4-term intersection space
\
1
1
Jp0 ,2 (ψn ) =
kn u + v L(u,v) (qn Mqn ) = Jpk0n,2 (qn Mqn ).
u,v∈{2p0 ,4}
Now we take direct limits and define
[
Jp0 ,2 (ψ) =
Jp0 ,2 (ψn ),
n≥1
where the closure is taken with respect to the norm of the space
Lp0 = Lp0 (M) ⊕2 L(2p0 ,4) (M) ⊕2 L(4,2p0 ) (M) ⊕2 L2 (M).
To define the space Kp,2 (ψ) we also proceed as above and consider
Ψn : Lnp → L1 (qn Mqn )
given by
1
0
1
1
0
1
0
1
1
0
1
1
Ψn (x1 , x2 , x3 , x4 ) = dψ2pn x1 dψ2pn + dψ2pn x2 dψ4 n + dψ4 n x3 dψ2pn + dψ4 n x4 dψ4 n .
This gives ker Ψn = Jp0 ,2 (ψn )⊥ and we define
Kp,2 (ψn ) = Lnp / ker Ψn
and
Kp,2 (ψ) =
[
Kp,2 (ψn ),
n≥1
where the latter is understood as a quotient of Lp . In other words, we may regard
the space Kp,2 (ψn ) as the sum of the corresponding dual weighted asymmetric Lp
spaces considered in the definition of Jp0 ,2 (ψn )
X
1
1
(8.4)
Kp,2 (ψn ) =
kn u + v L(u,v) (qn Mqn ).
u,v∈{2p,4}
Thus, using ψn = kn ϕn backwards and taking direct limits
Kp,2 (ψ) = Lp (M) + L(2p,4) (M) + L(4,2p) (M) + L2 (M),
where the sum is taken in Lp (M) and the embeddings are given by
jc (x) : x ∈ L(2p,4) (M) 7→
xdβψ ∈ Lp (M),
jr (x) : x ∈ L(4,2p) (M) 7→
dβψ x ∈ Lp (M),
with β = 1/2p − 1/4, while the embedding of L2 (M) into Lp (M) is given by
j2 (x) = dβψ xdβψ .
Remark 8.5. It will be important below to observe that our definition of
Kp,2 (ψn ) is slightly different to the one given in the previous paragraph. Indeed,
according to the usual duality bracket hx, yi = tr(x∗ y), we should have defined
X
Kp,2 (ψn ) =
kn −γ(u,v) L(u,v) (qn Mqn )
u,v∈{2p,4}
8.2. EMBEDDING INTO THE HYPERFINITE FACTOR
with
γ(u, v) =
135
1
1
+
.
2(u/2)0
2(v/2)0
This would give Kp,2 (ψn ) = Jp0 ,2 (ψn )∗ and
X
1
1
1
Kp,2 (ψn ) =
kn u + v L(u,v) (qn Mqn ).
kn
u,v∈{2p,4}
However, we prefer to use (8.4) in what follows for notational convenience.
Now we set some notation to distinguish between independent and free random
variables. If we fix a positive integer n, the von Neumann algebra Anind will denote
the kn -fold tensor product of qn Mqn while Anfree will be the kn -fold free product
of qn Mqn ⊕ qn Mqn . In other words, if we set
e n,j = qn Mqn and An,j = qn Mqn ⊕ qn Mqn
A
for 1 ≤ j ≤ kn , we define the following von Neumann algebras
e n,j ,
An
= ⊗j A
ind
Anfree
=
We also consider the natural embeddings
e n,j → An
πj : A
and
ind
ind
∗j An,j .
πfj ree : An,j → Anfree
into the j-th components of the algebras Anind and Anfree respectively.
Proposition 8.6. If 1 ≤ p ≤ 2, the map
n
ξind
: x ∈ Kp,2 (ψn ) 7→
kn
X
¡
¢
j
πind
(x) ⊗ δj ∈ Lp Anind ; OHkn
j=1
is a completely isomorphic embedding with relevant constants independent of n.
Before proceeding with the proof of Proposition 8.6, we need a more in depth
discussion on the cb-embedding of OH. Given 1 < p < 2, Xu constructed in [71]
a complete embedding of OH into Lp (A) with A hyperfinite, while for p = 1 the
corresponding cb-embedding was recently constructed in [18]. The argument can
be sketched with the following chain
¡
¢
OH ,→ Cp ⊕p Rp /graph(dλ )⊥ 'cb Cp + Rp (λ) ,→ Lp (A).
Indeed, arguing as in [24, Lemma 2.1/Remark 2.2] and applying [24, Lemma 2.8],
we see how to regard OH as a subspace of a quotient of Cp ⊕p Rp by the annihilator
of some diagonal map dλ : Cp0 → Rp0 . By the action of dλ , the annihilator of
its graph is the span of elements of the form (δk , −δk /λk ). This suggest to regard
the quotient above as the sum of Cp with a weighted form of Rp . This establishes
the cb-isomorphism in the middle. Then, it is natural to guess that the complete
embedding into Lp (A) should follow from a weighted form of the noncommutative
Khintchine inequality. The first inequality of this kind was given by Pisier and
Shlyakhtenko in [50] for generalized circular variables and further investigated in
[26, 70]. However, if we want to end up with a hyperfinite von Neumann algebra A,
we must replace generalized circulars by their Fermionic analogues. More precisely,
given a complex Hilbert space H, we consider its antisymmetric Fock space F−1 (H).
Let c(e) and a(e) denote the creation and annihilation operators associated with
136
8. OPERATOR SPACE Lp EMBEDDINGS
a vector e ∈ H. Given an orthonormal basis (e±k )k≥1 of H and a family (µk )k≥1
of positive numbers, we set fk = c(ek ) + µk a(e−k ). The sequence (fk )k≥1 satisfies
the canonical anticommutation relations and we take A to be the von Neumann
algebra generated by the fk ’s. Taking suitable µk ’s depending only on p and the
eigenvalues of dλ , the Khintchine inequality associated to the system of fk ’s provides
the desired cb-embedding. Namely, let φ be the quasi-free state on A determined
by the vacuum and let dφ be the associated density. Then, if (δk )k≥1 denotes the
unit vector basis of OH, the cb-embedding has the form
1
1
w(δk ) = ξk dφ2p fk dφ2p = ξk fp,k
for some scaling factors (ξk )k≥1 . The necessary Khintchine type inequalities for
1 < p < 2 follow from the noncommutative Burkholder inequality [28]. In the L1
case, the key inequalities follow from Lemma 8.3, see [18] for details. With this
construction, the von Neumann algebra A turns out to be the Araki-Woods factor
arising from the GNS construction applied to the CAR algebra with respect to the
quasi-free state φ. In fact, using a conditional expectation, we can replace the µk ’s
by a sequence (µ0k )k≥1 such that for every rational 0 < λ < 1 there are infinitely
many µ0k ’s with µ0k = λ/(1 + λ). According to the results in [1], we then obtain the
hyperfinite type III1 factor R.
On the other hand, there exists a slight modification of this construction which
will be used below. Indeed, using the terminology introduced above and following
[18] there exists a mean-zero γp ∈ Lp (R) given by a linear combination of the fp,k ’s
such that
j
w(δj ) = πind
(γp )
defines a completely isomorphic embedding
w : OHkn → Lp (R⊗kn )
with constants independent of n. Here R⊗kn denotes the kn -fold tensor product
of R. Moreover, given any von Neumann algebra A, idLp (A) ⊗ w also defines an
isomorphism
¡
¢
¯ ⊗kn ).
idLp (A) ⊗ w : Lp A; OHkn → Lp (A⊗R
We refer to [18, 71] for a more detailed exposition on the cb-embedding of OH.
Proof of Proposition 8.6. By Theorem 7.8, this is true for
ξfnree : x ∈ Kp,2 (ψn ) 7→
kn
X
¢
¡
πfj ree (x, −x) ⊗ δj ∈ Lp Anfree ; OHkn .
j=1
Indeed, it follows from a simple duality argument (see e.g. Remark 7.4) taking
Remark 8.5 into account. According to the preceding discussion on OH, we deduce
that
n
X
¡
¢
j
¯ ⊗kn
w ◦ ξfnree : x ∈ Kp,2 (ψn ) 7→
πfj ree (x, −x) ⊗ πind
(γp ) ∈ Lp Anfree ⊗R
j=1
also provides a cb-isomorphism. Now, we consider
e n,j
B
j
e n,j ) ⊗ π j (R),
= πind
(A
ind
Bn,j
j
= πfj ree (An,j ) ⊗ πind
(R),
8.2. EMBEDDING INTO THE HYPERFINITE FACTOR
137
for 1 ≤ j ≤ kn . It is clear from the construction that both families of von Neumann
algebras are s.i.c. over the complex field. Therefore, Lemma 8.3/Remark 8.4 apply
e n,j ’s holds due to the
in both cases (note that the mean-zero condition for the B
fact that γp is mean-zero) and hence
°
°
°
°
n
°idS m ⊗ w ◦ ξfnree (x)° ∼c °idS m ⊗ w ◦ ξind
(x)°
p
m
Sp (Kp,2 (ψn )).
p
p
p
holds for every element x ∈
In particular, we obtain
°
°
°
°
°
°
n
n
n
kxkSpm (Kp,2 (ψn )) ∼ °idSpm ⊗w◦ξf ree (x)°p ∼ °idSpm ⊗w◦ξind
(x)°p ∼ °idSpm ⊗ξind
(x)°p
since w and ξfnree are cb-isomorphic embeddings. This completes the proof.
¤
Remark 8.7. Proposition 8.6 extends (8.2) to general von Neumann algebras,
where freeness is replaced by noncommutative independence. The only difference
between both results is the factor 1/kn , which follows from Remark 8.5.
Remark 8.8. The transference argument applied in the proof of Proposition
8.6 gives a result which might be of independent interest. Given a von Neumann
algebra A, let us construct the tensor product Aind of infinitely many copies of
A ⊕ A. Similarly, the free product Af ree of infinitely many copies of A ⊕ A will be
considered. Following our terminology, we have maps
j
πind
: A → Aind
If 1 < p < q < 2, we claim that
°X
°
°
°
j
(8.5) °
πind
(x, −x) ⊗ δj °
j
πfj ree : A → Af ree .
and
Lp (Aind ;`q )
°X
°
°
°
∼cb °
πfj ree (x, −x) ⊗ δj °
j
Lp (Af ree ;`q )
,
where the symbol ∼cb is used to mean that the equivalence also holds (with absolute
constants) when taking the matrix norms arising from the natural operator space
structures of the spaces considered. The case q = 2 follows by using exactly the same
argument as in Proposition 8.6. In fact, the same idea works for general indices.
Indeed, we just need to embed `q into Lp completely isometrically and then use
the noncommutative Rosenthal inequality [30]. Recall that the cb-embedding of `q
into Lp is already known at this stage of the paper as a consequence of Corollary
8.2. At the time of this writing, it is still open whether or not (8.5) is still valid for
other values of (p, q).
Our main goal in this paragraph is to generalize the complete embedding in
Proposition 8.6 to the direct limit Kp,2 (ψ). Of course, this is possible using an
ultraproduct procedure. However, this would not preserve hyperfiniteness. We will
now explain how the proof of Proposition 8.6 allows to factorize the cb-embedding
¯ ⊗kn ) via a three term K-functional. We will combine this
Kp,2 (ψn ) → Lp (Anind ⊗R
with the concept of noncommutative Poisson random measure from [20] to produce
a complete embedding which preserves the direct limit mentioned above.
Let us consider the operator space
1
1
1
r
c
p
Krc
(ψn ) = knp Lp (qn Mqn ) + kn2 L2p (qn Mqn ) + kn2 L2p (qn Mqn ),
p
where the norms in the Lp spaces considered above are calculated with respect to
the state ϕn arising from the relation ψn = kn ϕn . More precisely, the operator
space structure is determined by
n 1
o
1
1
kxkSpm (Krpcp (ψn )) = inf knp kx1 kSpm (Lp ) + kn2 kx2 kS m (Lrp ) + kn2 kx3 kS m (Lcp ) ,
p
2
p
2
138
8. OPERATOR SPACE Lp EMBEDDINGS
where the infimum runs over all possible decompositions
α
x = x1 + dα
ϕn x2 + x3 dϕn ,
with dϕn standing for the density associated to ϕn and α = 1/p − 1/2. Note that
p
Krc
(ψn ) coincides algebraically with Lp (qn Mqn ). There exists a close relation
p
p
between Krc
(ψn ) and conditional Lp spaces. Indeed, let us consider the conditional
p
expectation Eϕn : Mm (qn Mqn ) → Mm given by
¡
¢ ¡
¢ ³ ψn (xij ) ´
Eϕn (xij ) = ϕn (xij ) =
.
kn
Lemma 8.9. We have isometries
¡ r
¢
Spm L2p (qn Mqn ) =
¡ c
¢
Spm L2p (qn Mqn ) =
¡
¢
1
m p Lrp Mm (qn Mqn ), Eϕn ,
¡
¢
1
m p Lcp Mm (qn Mqn ), Eϕn .
Moreover, these isometries are densely determined by
1
1 °
1 °
1
° 12 − 2p
1°
°dϕn adϕ2pn ° m rp
p °d 2p ad 2p °
=
m
,
ϕ
ϕ
n
n Lr (M (q Mq ),E
S (L (qn Mqn ))
m n
n
ϕ )
p
2
1
1
1 °
° 2p
°dϕn adϕ2 n− 2p ° m cp
S (L (qn Mqn ))
p
2
1
p
n
1 °
° 1
= m °dϕ2pn adϕ2pn °Lc (M
1
p
p
m (qn Mqn ),Eϕn )
.
1
In particular, using the relation dϕp n = dϕ2 n dα
ϕn , we conclude
kxkSpm (Kprcp (ψn ))
=
inf
x=xp +xr +xc
n 1
o
1°
1°
1°
1°
knp kxp kp + kn2 °Eϕn (xr x∗r ) 2 °S m + kn2 °Eϕn (x∗c xc ) 2 °S m .
p
Proof. We have
1 °
1
° 12 − 2p
°dϕn adϕ2pn ° m rp
S (L (q
p
2
n Mqn ))
p
°
°
1
1
¡ 1− 1
− 1 ¢1 °
°
= °trM dϕ2 n 2p adϕp n a∗ dϕ2 n 2p 2 °
Spm
°¡
1
1 ¢1 °
1
¢¡
−
−
°
°
= ° idMm ⊗ ϕn dϕn2p adϕp n a∗ dϕn2p 2 °
Spm
.
Then, normalizing the trace on Mm and recalling that
°
°¡
°
1
1
1 ¢1 °
¢¡ − 1
¡ 1
1°
− 1 ¢1 °
°
°
° idMm ⊗ϕn dϕn2p adϕp n a∗ dϕn2p 2 ° m = m p °Eϕn dϕ2pn adϕp n a∗ dϕ2pn 2 °
Sp
Lp (Mm (qn Mqn ))
when regarding the conditional expectation as a mapping
¡
¢
Eϕn : Lp Mm (qn Mqn ) → Lp (Mm ),
we deduce the assertion. The column case is proved in the same way.
¤
Proposition 8.10. Let R be the hyperfinite III1 factor and φ the quasi-free
p
state on R considered above. Let us consider the space Krc
(φ ⊗ ψn ), defined as we
p
did above. Then, there exists a completely isomorphic embedding
p
ρn : Kp,2 (ψn ) → Krc
(φ ⊗ ψn ).
p
Moreover, the relevant constants in ρn are independent of n.
8.2. EMBEDDING INTO THE HYPERFINITE FACTOR
139
Proof. We have
1
1
1
r
c
p
¯ n Mqn ) + kn2 L2p (R⊗q
¯ n Mqn ) + kn2 L2p (R⊗q
¯ n Mqn ).
Krc
(φ ⊗ ψn ) = knp Lp (R⊗q
p
The embedding is given by ρn (x) = γp ⊗x, with γp the element of Lp (R) introduced
¯ n Mqn ) → Mm and letting
after Proposition 8.6. Indeed, taking Eφ⊗ϕn : Mm (R⊗q
m
x ∈ Sp (Kp,2 (ψn )), we may apply Lemma 8.9 and obtain
kγp ⊗ xkSpm (Krpcp (φ⊗ψn ))
o
n 1
1°
1°
1°
1°
=
inf
knp kxp kp + kn2 °Eφ⊗ϕn (xr x∗r ) 2 °S m + kn2 °Eφ⊗ϕn (x∗c xc ) 2 °S m .
γp ⊗x=xp +xr +xc
p
p
Therefore, Lemma 8.3 and Remark 8.4 give
kn
kn
°X
°
°X
°
°
°
°
°
j
j
kγp ⊗ xkSpm (Krpcp (φ⊗ψn )) ∼ °
πind
(γp ⊗ x)° ∼ °
πind
(γp ) ⊗ πfj ree (x, −x)° .
p
k=1
j=1
Hence, the assertion follows as in Proposition 8.6. The proof is complete.
p
¤
Let M be a von Neumann algebra equipped with a n.s.s.f. weight ψ and let
us write (ψn , qn )n≥1 for the associated sequence of qn -supported weights. Then we
define the following direct limit
[ p
p
Krc
(ψ) =
Krcp (ψn ).
p
n≥1
p
We are interested in a cb-embedding Krc
(ψ) → Lp (A) preserving hyperfiniteness.
p
In the construction, we shall use a noncommutative Poisson random measure. Let
us briefly review the main properties of this notion from [20] before stating our
result. Let Mfsa stand for the subspace of self-adjoint elements in M which are
ψ-finitely supported. Let Mπ denote the projection lattice of M. We write e ⊥ f
for orthogonal projections. A noncommutative Poisson random measure is a map
λ : (M, ψ) → L1 (A, Φψ ), where (A, Φψ ) is a noncommutative probability space
and the following conditions hold
i) λ : Mfsa → L1 (A) is linear.
ii) Φψ (eiλ(x) ) = exp(ψ(eix − 1)) for x ∈ Mfsa .
iii) If e, f ∈ Mπ and e ⊥ f , λ(eMe)00 and λ(f Mf )00 are strongly independent.
These properties are not yet enough to characterize λ, see below. Let us recall
that two von Neumann subalgebras A1 , A2 of A are called strongly independent if
a1 a2 = a2 a1 and Φψ (a1 a2 ) = Φψ (a1 )Φψ (a2 ) for any pair (a1 , a2 ) in A1 × A2 . The
construction of λ follows by a direct limit argument. Indeed, let us show how to
produce λ : (qn Mqn , ψn ) → L1 (An , Φψn ). We define
An = Msym (qn Mqn ) =
∞
Y
⊗ksym qn Mqn ,
k=0
where ⊗ksym qn Mqn denotes the subspace of symmetric tensors in the k-fold tensor
product (qn Mqn )⊗k . In other words, if Sk is the symmetric group of permutations
of k elements, the space ⊗ksym qn Mqn is the range of the conditional expectation
1 X
xπ(1) ⊗ xπ(2) ⊗ · · · ⊗ xπ(k) .
Ek (x1 ⊗ x2 ⊗ · · · ⊗ xk ) =
k!
π∈Sk
140
8. OPERATOR SPACE Lp EMBEDDINGS
Then we set
λ(x) = (λk (x))k≥0 ∈ Msym (qn Mqn )
with λk (x) =
k
X
j
πind
(x),
j=1
and properties (i), (ii) and (iii) hold when working with the state
∞
¡
¢ X
exp(−ψn (1))
Φψn (zk )k≥1 =
ψn ⊗ · · · ⊗ ψn (zk ).
|
{z
}
k!
k=0
k times
In the following, it will be important to know the moments with respect to this
state. Given m ≥ 1, Π(m) will be the set of partitions of {1, 2, . . . , m}. On the
other hand, given an ordered family (xα )α∈Λ in M, we shall write
→
Y
xα
α∈Λ
for the directed product of the xα ’s. Then, the moments are given by the formula
r
→
³ Y
´
X
Y
¡
¢
iv) Φψn λ(x1 )λ(x2 ) · · · λ(xm ) =
ψn
xj .
k=1
σ∈Π(m)
σ={σ1 ,...,σr }
j∈σk
Now we can say that properties (i)-(iv) determine the Poisson random measure
λ for any given n.s.s.f. weight ψ in M. According to a uniqueness result from
[18] which provides a noncommutative form of the Hamburger moment problem, it
turns out that there exists a state preserving embedding
Msym (qn1 Mqn1 ) → Msym (qn2 Mqn2 )
for n1 ≤ n2 and such that the map λ = λn1 constructed for qn1 may be obtained
as a restriction of λn2 . This allows to take direct limits. More precisely, let us
define the algebra Msym (M) as the ultra-weak closure of the direct limit of the
Msym (qn Mqn )’s. Then, there exists a n.f. state Φψ on Msym (M) and a map λ
which assigns to every self-adjoint operator x (with supp(x) ≤ e for some ψ-finite
projection e in M) a self-adjoint unbounded operator λ(x) affiliated to Msym (M)
and such that
Φψ (eiλ(x) ) = exp(ψ(eix − 1)).
Theorem 8.11. Let 1 ≤ p ≤ 2 and ψ be a n.s.s.f. weight on M. Then there
exists a von Neumann algebra A, which is hyperfinite when M is hyperfinite, and
a completely isomorphic embedding
p
Krc
(ψ) → Lp (A).
p
Proof. Let us set the s-fold tensor product
³
£
¤´
¯ qn Mqn ⊕ qn Mqn
Bn,s = L∞ [0, 1]⊗
⊗s
.
p
Given s ≥ kn , we define the mapping Λn,s : Krc
(ψn ) → Lp (Bn,s ) by
p
1 ¢
¡ 1
Λn,s dϕ2pn xdϕ2pn
=
s
X
³
1
1 ´
j
πind
1[0,kn /s] ⊗ dϕ2pn (x, −x) dϕ2pn
j=1
=
s
X
j=1
1
1
¢ 2p
j ¡
2p
dn,s
πind
1[0,kn /s] ⊗ (x, −x) dn,s
,
8.2. EMBEDDING INTO THE HYPERFINITE FACTOR
141
where dn,s is the density associated to the s-fold tensor product state
hZ 1
i
1
φn,s =
· dt ⊗ (ϕn ⊕ ϕn )
= φn ⊗ φn ⊗ · · · ⊗ φn .
|
{z
}
2
⊗s
0
s times
If we tensor Λn,s with the identity map on Mm , the resulting mapping gives a sum
of symmetrically independent mean-zero random variables over Mm . Therefore,
p
taking x ∈ Spm (Krc
(ψn )) and applying Lemma 8.3/Remark 8.4
p
n
1
1
°
°
°
°
° o
¡
¢°
°Λn,s dϕ2pn xdϕ2pn ° ∼ inf s p1 kakp + s 12 °Eφ (bb∗ ) 12 ° m + s 12 °Eφ (c∗ c) 12 ° m ,
n
n
p
S
S
p
p
where Eφn denotes the conditional expectation
³
£
¤´
¯ qn Mqn ⊕ qn Mqn → Mm
Eφn : Mm L∞ [0, 1]⊗
and the infimum runs over all possible decompositions
1
(8.6)
1
1[0,kn /s] ⊗ dϕ2pn (x, −x) dϕ2pn = a + b + c.
Multiplying at both sizes of (8.6) by 1[0,kn /s] ⊗ 1, we obtain a new decomposition
which vanishes over (kn /s, 1]. Thus, since this clearly improves the infimum above,
we may assume this property in all decompositions considered. Moreover, we claim
that we can also restrict the infimum above to those decompositions a + b + c which
are constant on [0, kn /s]. Indeed, given any decomposition of the form (8.6) we
take averages at both sizes and produce another decomposition a0 + b0 + c0 given
by the relations
Z ksn
¡
¢
¡
¢
s
a0 , b0 , c0 = 1[0,kn /s] ⊗
a(t), b(t), c(t) dt.
kn 0
Then, our claim is a consequence of the inequalities
ka0 kp
°
°
∗
°Eφ (b0 b0 ) 12 ° m
n
Sp
°
1°
°Eφn (c∗0 c0 ) 2 ° m
S
p
≤ kakp ,
°
1°
≤ °Eφ (bb∗ ) 2 °
Spm
n
,
°
1°
≤ °Eφn (c∗ c) 2 °S m .
p
The first one is justified by means of the inequality
° Z λ
°
1°1
°
λp °
a(t) dt°
≤ kakLp ([0,λ]⊗M)
,
¯
λ 0
Lp (M)
which follows easily by complex interpolation. The two other inequalities arise as a
consequence of Kadison’s inequality E(x)E(x∗ ) ≤ E(xx∗ ) applied to the conditional
expectation
Z
1 λ
E=
· dt.
λ 0
Our considerations allow us to assume
¡
¢
¡
¢
a, b, c = 1[0,kn /s] ⊗ xp , xr , xc
for some xp , xr , xc ∈ Spm (Lp (qn Mqn )). This gives rise to
n 1
1 ¢°
1°
1°
°
°
° o
¡ 1
°Λn,s dϕ2pn xdϕ2pn ° ∼ inf knp kxp kp + kn2 °Eϕn (xr x∗r ) 12 ° m + kn2 °Eϕn (x∗c xc ) 12 ° m ,
p
S
S
p
p
142
8. OPERATOR SPACE Lp EMBEDDINGS
where the infimum runs over all possible decompositions
1
1
dϕ2pn xdϕ2pn = xp + xr + xc .
p
This shows that Λn,s : Krc
(ψn ) → Lp (Bn,s ) is a completely isomorphic embedding
p
with constants independent of n or s. We are not ready yet to take direct limits.
Before that, we use the algebraic central limit theorem to identify the moments in
the limit as s → ∞. To calculate the joint moments we set
1
ζn = (ϕn ⊕ ϕn ) and ζn,s = ζn ⊗ ζn ⊗ · · · ⊗ ζn
{z
}
|
2
s times
and recall that the map Λn,s corresponds to
s
³
´
X
j
un,s (x) =
πind
1[0,kn /s] ⊗ (x, −x) .
j=1
Then, the joint moments are given by
¡
¢
φn,s un,s (x1 ) · · · un,s (xm )
Z
m
s
→
³ Y
´
Y
X
ji
ji
=
πind
(1[0,kn /s] )(t) dt ζn,s
πind
(xi , −xi )
j1 ,j2 ,...,jm =1
X
=
[0,1]s i=1
X
σ∈Π(m)
(j1 ,...,jm )∼σ
σ={σ1 ,...,σr }
1≤i≤m
r
³ k ´r Y
n
s
ζn
→
h³ Y
xi , (−1)|σk |
i∈σk
k=1
→
Y
xi
´i
,
i∈σk
where |σk | denotes the cardinality of σk and we write (j1 , . . . , jm ) ∼ σ when ja = jb
if and only if there exits 1 ≤ k ≤ r such that a, b ∈ σk . Therefore, recalling that
ζn = 12 (ϕn ⊕ ϕn ), the only partitions which contribute to the sum above are the
even partitions satisfying |σk | ∈ 2N for 1 ≤ k ≤ r. Let us write Πe (m) for the set
of even partitions. Then, using ψn = kn ϕn we deduce
r
→
X
¡
¢
|{(j1 , . . . , jm ) ∼ σ}| Y ³ Y ´
φn,s un,s (x1 ) · · · un,s (xm ) =
ψ
xi
n
sr
i∈σ
k=1
σ∈Πe (m)
σ={σ1 ,...,σr }
=
X
σ∈Πe (m)
σ={σ1 ,...,σr }
k
r
→
³Y
´
Y
s!
ψ
x
.
n
i
sr (s − r)!
i∈σ
k=1
k
Therefore, taking limits
¡
¢
lim φn,s un,s (x1 ) · · · un,s (xm ) =
s→∞
X
r
Y
ψn
→
³Y
σ∈Πe (m) k=1
σ={σ1 ,...,σr }
´
xi .
i∈σk
These moments coincide with the moments of the Poisson random process
λ : (qn Mqn , ψn ) → L1 (Msym (qn Mqn ), Φψn ).
Hence, the noncommutative version of the Hamburger moment problem from [18]
provides a state preserving homomorphism between the von Neumann algebra
which generate the operators
n
o
¯
eiun,s (x) ¯ x ∈ qn Mqn , s ≥ 1
8.2. EMBEDDING INTO THE HYPERFINITE FACTOR
143
and the von Neumann subalgebra of Msym (qn Mqn ) generated by
o
n
¯
eiλ(x) ¯ x ∈ (qn Mqn )fsa .
In particular, taking An = Msym (qn Mqn )
1
1 °
° 2p
°d xd 2p ° m p
ψn
ψn S (K (ψn ))
p
∼
rcp
1 ¢°
1
°
° 1
°
¡ 1
lim °Λn,s dψ2pn xdψ2pn °S m (Lp (Bn,s )) = °dΦ2pψ λ(x)dΦ2pψ °S m (Lp (An )) .
n
n
s→∞
p
p
Now, we use from [20] that
[¡
¢
¡
¢
Msym (qn Mqn ), Φψn
Msym (M), Φψ =
n≥1
exists. Therefore, the map
1 ¢
1
1
¡ 1
Λ dψ2p xdψ2p = dΦ2pψ λ(x)dΦ2pψ
extends to a complete embedding
p
p
Krc
(ψ) = limn Krc
(ψn ) → Lp (Msym (M)).
p
p
Moreover, if M is hyperfinite so is Msym (qn Mqn ) and hence Msym (M).
¤
Corollary 8.12. Let 1 ≤ p ≤ 2 and ψ be a n.s.s.f. weight on M. Then there
exists a von Neumann algebra A, which is hyperfinite when M is hyperfinite, and
a completely isomorphic embedding
Kp,2 (ψ) → Lp (A).
Proof. Let us set
³
£
¤´
¯ (R⊗q
¯ n Mqn ) ⊕ (R⊗q
¯ n Mqn )
RBn,s = L∞ [0, 1]⊗
⊗s
.
By Proposition 8.10 and Theorem 8.11, the map
p
Λn,s ◦ ρn : Kp,2 (ψn ) → Krc
(φ ⊗ ψn ) → Lp (RBn,s )
p
provides a complete isomorphism with constant independent of n and s. Using
the algebraic central limit theorem to take limits in s and the noncommutative
version of the Hamburger moment problem one more time, we obtain a complete
embedding
¡
¢
p
¯ n Mqn ) .
Λn ◦ ρn : Kp,2 (ψn ) → Krc
(φ ⊗ ψn ) → Lp Msym (R⊗q
p
Taking direct limits we obtain a cb-embedding which preserves hyperfiniteness.
¤
Corollary 8.13. If 1 ≤ p < q ≤ 2, Sq cb-embeds in Lp (A) with A hyperfinite.
Proof. By the complete isometry Sq = Cq ⊗h Rq and [24, Remark 2.2], it
suffices to embed the first term Z1 ⊗ Z2 on the right of (8.1) into Lp (A) for some
hyperfinite von Neumann algebra A. However, following the proof of Theorem
8.1, we know that Z1 ⊗ Z2 embeds completely isomorphically into Kp,2 (ψ), where
ψ denotes some n.s.s.f. weight on B(`2 ). Therefore, the assertion follows from
Corollary 8.12.
¤
144
8. OPERATOR SPACE Lp EMBEDDINGS
Remark 8.14. Theorem 8.1 easily generalizes to the context of Corollary 8.13.
More precisely, given operator spaces X1 ∈ QS(Cp ⊕2 OH) and X2 ∈ QS(Rp ⊕2 OH)
and combining the techniques applied so far, it is rather easy to find a hyperfinite
type III factor A and a completely isomorphic embedding
X1 ⊗h X2 → Lp (A).
Remark 8.15. In contrast with Corollary 8.2, where free products are used,
the complete embedding of Sq into Lp given in Corollary 8.13 provides estimates
on the dimension of A in the cb-embedding
Sqm → Lp (A).
Indeed, a quick look at our construction shows that
¡
¢
¡
¢
n
Sqm = Cqm ⊗h Rqm → Kp,2 (ψn ) → Lp Anind ; OHkn = Lp M⊗k
; OHkn ,
n
with n ∼ m log m, see [18] for this last assertion. This chain essentially follows
from [24, Remark 2.2], the complete isometry (8.3) and Proposition 8.6. On the
other hand, given any parameter γ > 1/2kn and according once more to [18], we
n
with
know that OHkn embeds completely isomorphically into Spwn for wn = kγk
n
αp
constants depending only on γ and that kn ∼ n . Combining the embeddings
mentioned so far, we have found a complete embedding Sqm → SpM with
αp
M ∼ mβp m .
8.3. Embedding for general von Neumann algebras
Let 1 ≤ p < q ≤ 2 as in the statement of our main result in the Introduction.
We will encode complex interpolation in a suitable graph. This follows Pisier’s
approach [49] to the main result in [17]. Indeed, given 0 < θ < 1, let µθ be
the harmonic measure of the point z = θ. This is a probability measure on the
boundary ∂S (with density given by the Poisson kernel in the strip) that can be
written as µθ = (1 − θ)µ0 + θµ1 , with µj being probability measures supported by
∂j and such that
Z
f (θ) =
f dµθ
∂S
for any bounded analytic f extended non-tangentially to ∂S. Let
n¡
o
¢¯
H2 = f|∂0 , f|∂1 ¯ f : S → C analytic ⊂ L2 (∂0 ) ⊕ L2 (∂1 ).
We need operator-valued versions of this space given by subspaces
³ c
´ ³
´
0
r
r
r
r
H2p
L2p (∂0 ) ⊗h Lr2p0 (M) ⊕ Loh
0 ,2 (M, θ) ⊂
2 (∂1 ) ⊗h L4 (M) = Op0 ,0 ⊕ Op0 ,1 ,
³
´ ³
´
r 0
c
H2p
Lc2p0 (M) ⊗h L2p (∂0 ) ⊕ Lc4 (M) ⊗h Loh
(∂
)
= Opc0 ,0 ⊕ Opc0 ,1 .
0 ,2 (M, θ) ⊂
1
2
More precisely, if M comes equipped with a n.s.s.f. weight ψ and dψ denotes the
r
associated density, H2p
0 ,2 (M, θ) is the subspace of all pairs (f0 , f1 ) of functions in
r
r
Op0 ,0 ⊕Op0 ,1 such that for every scalar-valued analytic function g : S → C (extended
non-tangentially to the boundary) with g(θ) = 0, we have
Z
Z
1
1
0
4−
(1 − θ)
g(z) dψ 2p f0 (z) dµ0 (z) + θ
g(z)f1 (z) dµ1 (z) = 0.
∂0
∂1
8.3. EMBEDDING FOR GENERAL VON NEUMANN ALGEBRAS
c
Similarly, the condition on H2p
0 ,2 (M, θ) is
Z
Z
1
1
4 − 2p0
(1 − θ)
g(z)f0 (z) dψ
dµ0 (z) + θ
∂0
145
g(z)f1 (z) dµ1 (z) = 0.
∂1
We shall also need to consider the subspaces
Z
Z
n
o
1
1
¯
4 − 2p0
r
¯
f0 dµ0 + θ
Hr,0 =
(f0 , f1 ) ∈ H2p0 ,2 (M, θ) (1 − θ)
dψ
f1 dµ1 = 0 ,
∂
Z∂1
Z 0
o
n
1
1
¯
−
0
4
c
¯ (1 − θ)
f1 dµ1 = 0 .
Hc,0 =
(f0 , f1 ) ∈ H2p
f0 dψ 2p dµ0 + θ
0 ,2 (M, θ)
∂1
∂0
Remark 8.16. In order to make all the forthcoming duality arguments work,
we need to introduce a slight modification of these spaces for p = 1. Indeed, in that
case the spaces defined above must be regarded as subspaces of
³
´ ³
´
r
r
¯
H∞,2
(M, θ) ⊂
Lc2 (∂0 )⊗M
⊕ Loh
2 (∂1 ) ⊗h L4 (M) ,
³
´ ³
´
c
¯ r2 (∂0 ) ⊕ Lc4 (M) ⊗h Loh
H∞,2
(M, θ) ⊂
M⊗L
(∂
)
.
1
2
The von Neumann algebra tensor product used above is the weak closure of the
minimal tensor product, which in this particular case coincides with the Haagerup
tensor product since we have either a column space on the left or a row space on
the right. In particular, the only difference is that we are taking the closure in the
weak operator topology.
Lemma 8.17. Let M be a finite von Neumann algebra equipped with a n.f. state
θ
ϕ and let dϕ be the associated density. If 2 ≤ q 0 < p0 and 2q1 0 = 1−θ
2p0 + 4 , we have
complete contractions
³
´
1
1
1
0
0
r
ur : dϕ2q x ∈ Lr2q0 (M) 7→ 1 ⊗ dϕ2p x, 1 ⊗ dϕ4 x + Hr,0 ∈ H2p
0 ,2 (M, θ)/Hr,0 ,
³
´
1
1
1
0
0
c
uc : xdϕ2q ∈ Lc2q0 (M) 7→ xdϕ2p ⊗ 1, xdϕ4 ⊗ 1 + Hc,0 ∈ H2p
0 ,2 (M, θ)/Hc,0 .
Proof. By symmetry, it suffices to consider the column case. Let x be an
element in Mm (Lc2q0 (M)) of norm less than 1. According to our choice of 0 < θ < 1,
we find that
£
¤
Mm (Lc2q0 (M)) = Mm (Lc2p0 (M)), Mm (Lc4 (M)) θ .
Thus, there exists f : S → Mm (M) analytic such that f (θ) = x and
n
o
1 °
1°
°
°
0
max sup °f (z)dϕ2p °Mm (Lc (M)) , sup °f (z)dϕ4 °Mm (Lc (M)) ≤ 1.
2p0
z∈∂0
4
z∈∂1
If 1 ≤ s ≤ ∞ and j ∈ {0, 1}, we claim that
1 °
1 °
°
°
°f| dϕ2s °
(8.7)
≤ sup °f (z)dϕ2s °Mm (Lc
∂j
Mm (Lc (M)⊗ Lrs (∂j ))
2s
h
2
2s (M))
z∈∂j
.
Before proving our claim, let us finish the proof. Taking fj = f|∂j , we have
³
´
1
1
1¢
1
¡
0
0
f0 dϕ2p , f1 dϕ4 − xdϕ2p ⊗ 1, xdϕ4 ⊗ 1 ∈ Hc,0 .
Indeed by analyticity, we have
Z
Z
1
1
1
0
0
4−
(1 − θ)
f0 dϕ2p dϕ 2p dµ0 + θ
∂0
1
Z
1
f1 dϕ4 dµ1 =
∂1
1
1
f dϕ4 dµθ = f (θ)dϕ4 = xdϕ4 .
∂S
146
8. OPERATOR SPACE Lp EMBEDDINGS
This implies from (8.7) applied to (s, j) = (p0 , 0) and (s, j) = (2, 1) that
° ¡
°
°¡
1 ¢°
1
1¢
0 °
°
°
2p0
4 °
≤
f
d
,
f
d
≤ 1.
°uc xdϕ2q °
°
0 ϕ
1 ϕ °
c
c
Mm (H2p0 ,2 /Hc,0 )
Mm (H2p0 ,2 )
Hence,
¡ it remains
¢ to prove our claim (8.7). We must show that the identity map
L∞ ∂j ; Lc2s (M) → Lc2s (M) ⊗h Lr2s (∂j ) is a complete contraction. By complex
interpolation, we have
¢
£
¡
¢
¡
¢¤
¡
L∞ ∂j ; Lc2s (M) = L∞ ∂j ; M , L∞ ∂j ; Lc2 (M) 1 ,
s
£
¤
Lc2s (M) ⊗h Lr2s (∂j ) = M ⊗h Lr2 (∂j ), Lc2 (M) ⊗h Lc2 (∂j ) 1
s
£
¤
= M ⊗min Lr2 (∂j ), Lc2 (M) ⊗min Lc2 (∂j ) 1 .
s
In other words, we must study the identity mappings
M ⊗min L∞ (∂j )
⊗min L∞ (∂j )
Lc2 (M)
→ M ⊗min Lr2 (∂j ),
→ Lc2 (M) ⊗min Lc2 (∂j ).
However, this automatically reduces to see that we have complete contractions
L∞ (∂j )
L∞ (∂j )
→
→
Therefore it suffices to observe that
°Z
°
°
°
kf k2Mm (Lr2 (∂j )) = °
f f ∗ dµj °
≤
Mm
∂j
°Z
°
°
°
kf k2Mm (Lc2 (∂j )) = °
f ∗ f dµj °
≤
Mm
∂j
L2r (∂j ),
L2c (∂j ).
µj (∂j ) sup kf (z)k2Mm = kf k2Mm (L∞ (∂j )) ,
z∈∂j
µj (∂j ) sup kf (z)k2Mm = kf k2Mm (L∞ (∂j )) .
z∈∂j
This completes the proof for 1 < p ≤ 2. In the case p = 1, we have overlooked the
c
(M, θ) (see Remark 8.16 above) is slightly different.
fact that the definition of H∞,2
The only consequence of this point is that we also need the inequality
°
°
°f| °
≤ sup kf (z)kM (M) .
r
∂0
¯ 2 (∂0 ))
Mm (M⊗L
z∈∂0
m
However, this is proved once more as above. The proof is complete.
¤
To continue with the argument we need to introduce some predual spaces. This
requires to extend our definition (1.3) to the case 1 ≤ q ≤ 2. This is easily done as
follows
£
¤
Lrq (M) = L1 (M), Lr2 (M) 2 ,
£
¤ q0
Lcq (M) = L1 (M), Lc2 (M) 2 .
q0
Lemma 8.17 is closely related to [24, Lemma 2.1] and a similar result holds on the
preduals. More precisely, we begin by defining the operator-valued Hardy spaces
which arise as subspaces
´
³
´ ³
c
c
c
H2p,2
(M, θ) ⊂
L2p (∂0 ) ⊗h Lc2p (M) ⊕ Loh
2 (∂1 ) ⊗h L 43 (M) ,
p+1
´
³
´ ³
r
r
r
(∂
)
,
H2p,2 (M, θ) ⊂
L 2p (M) ⊗h L2p (∂0 ) ⊕ Lr4 (M) ⊗h Loh
1
2
3
p+1
formed by pairs (f0 , f1 ) respectively satisfying
Z
Z
p+1
−3
(1 − θ)
g(z)f0 (z) dµ0 (z) + θ
g(z)dϕ2p 4 f1 (z) dµ1 (z)
∂0
∂1
=
0,
8.3. EMBEDDING FOR GENERAL VON NEUMANN ALGEBRAS
Z
(1 − θ)
Z
p+1
g(z)f1 (z)dϕ2p
g(z)f0 (z) dµ0 (z) + θ
∂0
− 34
dµ1 (z)
=
147
0,
∂1
for all scalar-valued analytic function g : S → C (extended non-tangentially to the
0
0
boundary) with g(θ) = 0. The subspaces Hr,0
and Hc,0
are defined accordingly. In
other words, we have
Z
Z
n
o
p+1
¯
−3
0
c
¯
Hc,0 =
(f0 , f1 ) ∈ H2p,2 (M, θ) (1 − θ)
f0 dµ0 + θ
dϕ2p 4 f1 dµ1 = 0 ,
Z∂0
Z∂1
n
o
p+1
¯
−3
0
r
Hr,0
=
(f0 , f1 ) ∈ H2p,2
(M, θ) ¯ (1 − θ)
f0 dµ0 + θ
f1 dϕ2p 4 dµ1 = 0 .
∂0
∂1
Lemma 8.18. Let M be a finite von Neumann algebra equipped with a n.f. state
ϕ and let dϕ be the associated density. Taking the same values for p, q and θ as
above, we have complete contractions
³
´
q+1
p+1
3
0
c
0
wr : dϕ2q x ∈ Lc2q (M) 7→ 1 ⊗ dϕ2p x, 1 ⊗ dϕ4 x + Hc,0
∈ H2p,2
(M, θ)/Hc,0
,
q+1
q+1
2q
wc : xdϕ
³ p+1
´
3
0
r
0
∈ Lr2q (M) 7→ xdϕ2p ⊗ 1, xdϕ4 ⊗ 1 + Hr,0
∈ H2p,2
(M, θ)/Hr,0
.
q+1
Proof. If 1 ≤ s ≤ ∞ and Mm = Mm (M), we have
³£
¢
¤ ´
¡
(8.8)
S1m Lr2s (M) = S1m L1 (M), Lr2 (M) 1
s+1
s0
h ¡
¢ m¡
¢ i
m
= S1 L1 (M) , S2 L2 (M) S2m 1
s0
¢ m
¡
m
m
2s (Mm )S
2s (M) S
= S 2s L s+1
2s0 = L s+1
2s0 .
s+1
Indeed, the second isometry follows by dualizing the second isometry below
m
° 1³X
´°
° 1 ¡ ¢°
°
° 2
°dϕ2 xij °
α
x
=
sup
,
d
°
ϕ
ik kj °
Mm (Lr (M))
2
(8.9)
°¡ ¢ 1 °
° xij dϕ2 °
Mm (Lc2 (M))
kαkS m ≤1
2
=
sup
kβkS m ≤1
2
k=1
L2 (Mm )
m
°³ X
´ 1°
°
°
xik βkj dϕ2 °
°
k=1
L2 (Mm )
.
The first one is needed for the analysis of the mapping wr . The isometries (8.9) are
well-known in operator space theory, see e.g. p.56 in [11]. Alternatively, one may
argue directly as we did in Lemma 8.9. The third isometry in (8.8) follows from
Theorem 3.2. Now, let us consider an element of norm less than 1
q+1
¢
¡
xdϕ2q ∈ S1m Lr2q (M) .
q+1
The isometry above provides a factorization
q+1
m m
xdϕ2q = αβγδ ∈ L 34 (Mm )Lρ1 (Mm )S2p
0 Sρ
2
with α, β, γ, δ in the unit balls of their respective spaces with
q+1 3
1
=
−
ρ1
2q
4
and
1
1
1
= 0 − 0.
ρ2
2q
2p
Moreover, by polar decomposition and approximation, we may assume that β and δ
are strictly positive elements. In particular, motivated by the complex interpolation
148
8. OPERATOR SPACE Lp EMBEDDINGS
isometry
L
2q
q+1
h
m
(Mm )S2q
L
0 =
we take (βθ , δθ ) = (β
1/(1−θ)
,δ
2p
p+1
1/θ
m
m
(Mm )S2p
0 , L 4 (Mm )S4
3
i
θ
¤
£
= X0 , X1 θ ,
) and define
f : z ∈ S 7→ αβθ1−z γδθz ∈ X0 + X1 .
q+1
Since f is analytic and f (θ) = xdϕ2q , we conclude
3
¡
¢
¢ ¡ p+1
0
.
f|∂0 , f|∂1 ∈ xdϕ2p ⊗ 1, xdϕ4 ⊗ 1 + Hr,0
Therefore, taking f|∂j = fj we obtain the following estimate
° ¡ q+1 ¢°
°
°
°wc xdϕ2q ° m r
S1 (H2p,2 /H0r,0 )
n° °
° °
rp
≤ max °f0 ° m r
, °f1 ° m r
S1 (L
2p
p+1
n°
°
= max °αβθ γ °S m (Lr
2p
p+1
1
o
S1 (L 4 (M)⊗h Loh
2 (∂1 ))
(M)⊗h L2 (∂0 ))
3
o
°
°
, °αγδθ °S m (Lr (M)) ≤ 1,
(M))
1
4
3
where the last inequality follows from (8.8) and the fact that (αβθ , γδθ ) are in the
unit balls of L2p/p+1 (Mm ) and S4m respectively. The assertion for the mapping wr
is proved similarly. The proof is complete.
¤
Proposition 8.19. Let M be a finite von Neumann algebra equipped with a
θ
n.f. state ϕ and let dϕ be the associated density. If 2 ≤ q 0 < p0 and 2q1 0 = 1−θ
2p0 + 4 ,
we have complete isomorphisms
³
´
1
1
1
0
0
r
ur : dϕ2q x ∈ Lr2q0 (M) 7→ 1 ⊗ dϕ2p x, 1 ⊗ dϕ4 x + Hr,0 ∈ H2p
0 ,2 (M, θ)/Hr,0 ,
³
´
1
1
1
0
0
c
uc : xdϕ2q ∈ Lc2q0 (M) 7→ xdϕ2p ⊗ 1, xdϕ4 ⊗ 1 + Hc,0 ∈ H2p
0 ,2 (M, θ)/Hc,0 .
Proof. This follows easily from the identity
Z
¡
¢
trM (x∗ y) =
trM f (z)∗ g(z) dµθ (z),
∂S
valid for any pair of analytic functions f and g such that (f (θ), g(θ)) = (x, y).
Indeed, according to the definition of the mappings ur , uc , wr , wc , this means that
we have
­
®
ur (x1 ), wr (y1 ) = hx1 , y1 i
­
®
uc (x2 ), wc (y2 ) = hx2 , y2 i
for any
(x1 , x2 , y1 , y2 ) ∈ Lr2q0 (M) × Lc2q0 (M) × Lc2q (M) × Lr2q (M).
q+1
q+1
In particular, we deduce
wr∗ ur = idLr
2q 0
(M)
and wc∗ uc = idLc
2q 0
(M) .
Therefore, the assertion follows combining Lemma 8.17 and Lemma 8.18.
¤
8.3. EMBEDDING FOR GENERAL VON NEUMANN ALGEBRAS
149
Let M be a von Neumann algebra equipped with a n.f. state ϕ and let Mm
be the tensor product Mm ⊗ M. Then, if Em = idMm ⊗ ϕ : Mm → Mm denotes
the associated conditional expectation, the following generalizes Lemma 7.7.
Lemma 8.20. If 1/r = 1/2 − 1/p0 , we have isometries
0
= Cpm0 ⊗h Lr4 (M) ⊗h Rm ,
0
= Cm ⊗h Lc4 (M) ⊗h Rpm0 .
L2p
(2r,∞) (Mm , Em )
2p
L(∞,2r)
(Mm , Em )
Proof. By Kouba’s theorem, it is clear that the spaces on the right form
complex interpolation families with respect to 2 ≤ p0 ≤ ∞. Let us see that the same
happens for the conditional Lp spaces on the left. Indeed, according to Theorem
4.6 we have isometries
h
i
0
∞
4
L2p
(M
,
E
),
L
(M
,
E
)
,
(M
,
E
)
=
L
m
m
m
m
m
m
(4,∞)
(∞,∞)
(2r,∞)
2/p0
h
i
0
4
L2p
L∞
.
(∞,4) (Mm , Em ), L(∞,∞) (Mm , Em )
(∞,2r) (Mm , Em ) =
0
2/p
0
0
0
This reduces the problem to the cases p = 2 and p = ∞. If p = 2, we have r = ∞
and both spaces on the left coincide with L4(∞,∞) (Mm , Em ) = L4 (Mm ). On the
other hand, the spaces on the right are respectively given by
¤
£
C2m ⊗h Lr4 (M) ⊗h Rm = Cm ⊗h M ⊗h Rm , Rm ⊗h Lr2 (M) ⊗h Rm θ ,
¤
£
Cm ⊗h Lc4 (M) ⊗h Rpm0 = Cm ⊗h M ⊗h Rm , Cm ⊗h Lc2 (M) ⊗h Cm θ .
In other words, we find the spaces Lr4 (Mm ) and Lc4 (Mm ) which coincide with
L4 (Mm ) at the Banach space level. It remains to check the validity of the assertion
for p0 = ∞, but this is exactly the content of [24, Lemma 1.8].
¤
Let M be a von Neumann algebra equipped with a n.f. state ϕ and let N
be a von Neumann subalgebra of M. Let E : M → N denote the corresponding
conditional expectation. In Chapter 5, we defined the spaces
1
1
0
1
1
0
Rn2p0 ,2 (M, E)
= n 2p0 Lr2p0 (M) ∩ n 4 L2p
(2r,∞) (M, E),
n
C2p
0 ,2 (M, E)
= n 2p0 Lc2p0 (M) ∩ n 4 L2p
(∞,2r) (M, E),
with 1/r = 1/2 − 1/p0 and in Chapter 6 we proved the isomorphism
(8.10)
n
Jpn0 ,2 (M, E) ' Rn2p0 ,2 (M, E) ⊗M C2p
0 ,2 (M, E).
In fact, to be completely fair we should say that we have slightly modified the
n
definition of Rn2p0 ,2 (M, E) and C2p
0 ,2 (M, E) by considering the row/column o.s.s. of
L2p0 (M). However, the new definition coincides with the former one at the Banach
space level. Hence, since we do not even have an operator space structure for
these spaces, our modification is only motivated by notational convenience below.
Namely, inspired by Lemma 8.20, we introduce the operator spaces
1
1
1
1
Rn2p0 ,2 (M) =
n 2p0 Lr2p0 (M) ∩ n 4 Lr4 (M),
n
C2p
=
0 ,2 (M)
n 2p0 Lc2p0 (M) ∩ n 4 Lc4 (M).
These spaces give rise to the complete isomorphism
(8.11)
n
Jpn0 ,2 (M) 'cb Rn2p0 ,2 (M) ⊗M,h C2p
0 ,2 (M).
150
8. OPERATOR SPACE Lp EMBEDDINGS
Indeed, taking (M, N , E) = (Mm , Mm , Em ) in (8.10) we have
¡
¢
Spm0 Jpn0 ,2 (M) = Jpn0 ,2 (Mm , Em )
n
Rn2p0 ,2 (Mm , Em ) ⊗Mm C2p
0 ,2 (Mm , Em )
¡
¢
¡ n
¢
m
n
m
= S(2p0 ,∞) R2p0 ,2 (M) ⊗Mm S(∞,2p
0 ) C2p0 ,2 (M)
³
´
n
= Cpm0 ⊗h Rn2p0 ,2 (M) ⊗M,h C2p
⊗h Rpm0
0 ,2 (M)
¡
¢
n
= Spm0 Rn2p0 ,2 (M) ⊗M,h C2p
0 ,2 (M) ,
'
where the third identity follows from Lemma 8.20 after taking in consideration
n
our new definition of Rn2p0 ,2 (M, E) and C2p
0 ,2 (M, E). In other words, (8.10) and
(8.11) are the amalgamated and operator space versions of the same factorization
isomorphism. Now assume that M is equipped with a n.s.s.f. weight ψ, given
by the increasing sequence (ψn , qn )n≥1 . Then, we may generalize the factorization
result above in the usual way. Namely, assuming by approximation that kn = ψn (1)
are positive integers, we define
1
1
0
R2p0 ,2 (ψn )
=
kn2p Lr2p0 (qn Mqn ) ∩ kn4 Lr4 (qn Mqn ),
C2p0 ,2 (ψn )
=
kn2p Lc2p0 (qn Mqn ) ∩ kn4 Lc4 (qn Mqn ).
1
1
0
This gives the complete isomorphism
\
Jp0 ,2 (ψn ) 'cb R2p0 ,2 (ψn ) ⊗M,h C2p0 ,2 (ψn ) =
1
knu
+ v1
L(u,v) (qn Mqn ).
u,v∈{2p0 ,4}
Then, taking direct limits we obtain the space
\
Jp0 ,2 (ψ) = R2p0 ,2 (ψ) ⊗M,h C2p0 ,2 (ψ) =
L(u,v) (M).
u,v∈{2p0 ,4}
Lemma 8.21. Let M be a von Neumann algebra equipped with a n.s.s.f. weight
ψ. Then, there exists a n.s.s.f. weight ξ on B(`2 ) such that the following complete
isomorphisms hold
r
H2p
0 ,2 (M, θ) ⊗h R
'cb
R2p0 ,2 (ψ ⊗ ξ),
c
⊗h H2p
0 ,2 (M, θ)
'cb
C2p0 ,2 (ψ ⊗ ξ).
C
Proof. By symmetry, we only consider the column case. Let us first observe
that H2 is indeed the graph of an injective closed densely-defined (unbounded)
operator with dense range. This is quite similar to [24, Remark 2.2]. It follows
from the three lines lemma that for z = a + ib
a
|f (z)| ≤ kf|∂0 k1−a
L2 (∂0 ,µa ) kf|∂1 kL2 (∂1 ,µa ) .
Since µa and µθ have the same null sets, we deduce that
πj (f ) = f|∂j ∈ L2 (∂j , µθ )
are injective for j = 0, 1 when restricted to analytic functions. Thus, the mapping
Λ(π0 (f )) = π1 (f ) is an injective closed densely-defined operator with dense range
and H2 is its graph. Let Λ = u|Λ| be the polar decomposition. Since Mu = 1 ⊗ u∗
defines a complete isometry (recall that Λ has dense range)
c
oh
Lc4 (M) ⊗h Loh
2 (∂1 ) → L4 (M) ⊗h L2 (∂0 ),
8.3. EMBEDDING FOR GENERAL VON NEUMANN ALGEBRAS
151
c
we may replace Λ by |Λ| in the definition of H2p
0 ,2 (M, θ). Using [24, Lemma 2.8],
we may also replace L2 (∂0 ) by `2 and the operator |Λ| by a diagonal operator dλ .
These considerations provide a cb-isomorphism
³
´ ³
´
c
C ⊗h H2p
C ⊗h Lc2p0 (M) ⊗h Rp0 ∩ C ⊗h Lc4 (M) ⊗h `oh
(λ)
,
0 ,2 (M, θ) 'cb
2
where `oh
2 (λ) is the weighted form of OH which arises from the action of dλ . The
assertion follows by a direct limit argument. Indeed, the n.s.s.f. weight ψ on M
is given by the sequence (ψn , qn )n≥1 . On the other hand, we may consider the
n.s.s.f. weight ξ on B(`2 ) determined by the sequence (ξn , πn )n≥1 , where πn is the
projection onto the first n coordinates and ξn is the finite weight on πn B(`2 )πn
given by
n
³ ¡X
1
1
¢ ´ X
0
4−
ξn πn
xij eij πn =
γk xkk with γk 2p = λk .
ij
k=1
k0n
Let us define the parameters
= ξn (1) and wn = kn k0n . Then, arguing as we did
in [24, Lemma 2.4], it turns out that the intersection space above is the direct limit
of the following sequence of spaces
³ ¡
1 ³
1
¡
¢´
¢´
0
¯ n B(`2 )πn ∩ wn4 Lc4 qn Mqn ⊗π
¯ n B(`2 )πn .
wn2p Lc2p0 qn Mqn ⊗π
However, the latter space is C2p0 ,2 (ψn ⊗ ξn ). This completes the proof.
¤
Proposition 8.22. The predual space of
r
c
H2p
0 ,2 (M, θ) ⊗M,h H2p0 ,2 (M, θ)
p
embeds completely isomorphically into Krc
(φ ⊗ ψ ⊗ ξ) for some n.s.s.f. weight ξ
p
on B(`2 ) and where φ denotes the quasi-free state over the hyperfinite III1 factor R
considered in Proposition 8.10.
Proof. According to Lemma 8.21
r
c
H2p
0 ,2 (M, θ) ⊗M,h H2p0 ,2 (M, θ)
³
´
³
´
r
c
=
H2p
⊗M⊗B(`
C ⊗h H2p
0 ,2 (M, θ) ⊗h R
0 ,2 (M, θ)
¯
2 ),h
'cb
R2p0 ,2 (ψ ⊗ ξ) ⊗M⊗B(`
C2p0 ,2 (ψ ⊗ ξ) 'cb Jp0 ,2 (ψ ⊗ ξ).
¯
2 ),h
However, Jp0 ,2 (ψ ⊗ ξ) is a direct limit of spaces
³
´
n
¯ n B(`2 )πn .
Jp0 ,2 (ψn ⊗ ξn ) = Jpw0 ,2
qn Mqn ⊗π
According to Proposition 8.10, the direct limit
n
Kp,2 (ψ ⊗ ξ) = limn Kp,2
(ψn ⊗ ξn )
p
of the corresponding predual spaces cb-embeds into Krc
(φ ⊗ ψ ⊗ ξ).
p
¤
Now we are ready to prove our main result. In the proof we shall need to
r
c
work with certain quotient of H2p
0 ,2 (M, θ) ⊗M,h H2p0 ,2 (M, θ). Namely, recalling
the subspaces Hr,0 and Hc,0 , we set
¡ r
¢
¡ c
¢
Q2p0 ,2 (M, θ) = H2p
0 ,2 (M, θ)/Hr,0 ⊗M,h H2p0 ,2 (M, θ)/Hc,0 .
r
c
We claim that Q2p0 ,2 (M, θ) is a quotient of H2p
0 ,2 (M, θ)⊗M,h H2p0 ,2 (M, θ). Indeed,
according to the definition of the M-amalgamated Haagerup tensor product of two
152
8. OPERATOR SPACE Lp EMBEDDINGS
operator spaces (see Chapter 6), we may write Q2p0 ,2 (M, θ) as a quotient of the
Haagerup tensor product
¢
¡ c
¢
¡ r
Λ2p0 ,2 (M, θ) = H2p
0 ,2 (M, θ)/Hr,0 ⊗h H2p0 ,2 (M, θ)/Hc,0
by the closed subspace spanned by the differences x1 γ ⊗ x2 − x1 ⊗ γx2 , with
γ ∈ M. Therefore, it suffices to see that the space Λ2p0 ,2 (M, θ) is a quotient
r
c
of H2p
0 ,2 (M, θ) ⊗h H2p0 ,2 (M, θ). However, this follows from the projectivity of the
Haagerup tensor product and our claim follows.
Let us prove our main embedding result.
Theorem 8.23. Let 1 ≤ p < q ≤ 2 and let M be a von Neumann algebra.
Then, there exists a sufficiently large von Neumann algebra A and a completely
isomorphic embedding of Lq (M) into Lp (A), where both spaces are equipped with
their respective natural operator space structures. Moreover, we have
i) If M is QWEP, we can choose A to be QWEP.
ii) If M is hyperfinite, we can choose A to be hyperfinite.
Proof. Let us first assume that M is finite. According to Theorem 8.11 and
Proposition 8.22, it suffices to prove that the operator space Lq0 (M) is completely
isomorphic to a quotient of
r
c
H2p
0 ,2 (M, θ) ⊗M H2p0 ,2 (M, θ).
This follows from Proposition 8.19 since
Lq0 (M) 'cb Lr2q0 (M) ⊗M,h Lc2q0 (M) 'cb Q2p0 ,2 (M, θ).
The construction of the cb-embedding for a general von Neumann algebra M can
be obtained by using Haagerup’s approximation theorem [12] and the fact that
direct limits are stable in our construction. Indeed, Haagerup theorem shows that
for every σ-finite von Neumann algebra M, the space Lq (M) is complemented in
a direct limit of Lq spaces over finite von Neumann algebras. Finally, if M is any
von Neumann algebra, we observe that Lq (M) can always be written as a direct
limit of Lq spaces associated to σ-finite von Neumann algebras. On the other hand,
the stability of hyperfiniteness follows directly from our construction. Indeed, our
construction goes as follows
³
´
p
r
c
Lq (M) → H2p
→ Krc
(φ ⊗ ψ ⊗ ξ) → Lp (A)
0 ,2 (M, θ) ⊗M,h H2p0 ,2 (M, θ)
p
∗
where the first embedding follows as above, the second from Proposition 8.22 and
the last one from Theorem 8.11. In particular, it turns out that the von Neumann
algebra A is of the form
¡
¢
¯ ⊗B(`
¯
A = Msym R⊗M
2) ,
which is hyperfinite when M is hyperfinite and is a factor when M is a factor.
Finally, it remains to justify that the QWEP is preserved. If M is QWEP, there
exists a completely isometric embedding of Lq (M) into Lq (MU ) with MU of the
form
³Y
´∗
MU =
S1n .
n,U
Since we know from Corollary 8.2 that the Schatten class Sqn embeds completely
isomorphically into Lp (An ) with relevant constants independent of n and An being
8.3. EMBEDDING FOR GENERAL VON NEUMANN ALGEBRAS
QWEP, we find a completely isomorphic embedding
³Y
Lq (M) → Lp (AU ) with AU =
n,U
An ∗
´∗
153
.
This proves the assertion since AU is QWEP. The proof is complete.
¤
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