GAFA, Geom. funct. anal.
Vol. 10 (2000) 389 – 406
1016-443X/00/020389-18 $ 1.50+0.20/0
c Birkhäuser Verlag, Basel 2000
GAFA
Geometric And Functional Analysis
EMBEDDINGS OF NON-COMMUTATIVE Lp -SPACES
INTO NON-COMMUTATIVE L1 -SPACES, 1 < p < 2
Marius Junge
Abstract
It will be shown that for 1 < p < 2 the Schatten p-class is isometrically
isomorphic to a subspace of the predual of a von Neumann algebra.
Similar results hold for non-commutative Lp (N, τ )-spaces defined by
a finite trace on a finite von Neumann algebra. The embeddings rely
on a suitable notion of p-stable processes in the non-commutative
setting.
Introduction and Notation
The theory of p-stable processes is a classical tool in probability and analysis and in particular in the theory of Banach spaces. The intention of this
paper is to present non-commutative versions of a p-stable process. It is
a general phenomenon in non-commutative functional analysis that point
sets disappear after quantization. We proceed in a similar way by constructing linear isomorphic or isometric embeddings of a non-commutative
Lp -space in the predual of a suitable von Neumann algebra which looks
like the integral against a p-stable process when restricted to an arbitrary
commutative subalgebra.
The theory of embeddings of classical Lp -spaces started with the work
of Bretagnolle, Dacuhna-Castelle and Krivine [BrDK] and later Bretagnolle
and Dacuhna-Castelle [BrD] between 1966 and 1969. In particular, they
found embeddings of Lq -spaces and also Orlicz spaces in Lp -spaces based
on the Lévy-Khintchine representation of infinite divisible random variables. Finite dimensional results were obtained with combinatorical tools
by Kwapien and Schütt and extended by Schütt and Raynaud [KS1,2], [RS].
All these results motivated the general problem of determining the set of
those p’s such that the spaces `np are uniformly represented in a given Banach space. Indeed, due to the fundamental work of Maurey-Pisier [MaP],
This research is partially supported by procope 1997/1998.
390
M. JUNGE
GAFA
see also [Pi1,2], the set of those p’s is closely related to the geometry of
the underlying Banach space. Johnson and Schechtman [JS] studied the
concrete problem of embedding `nq into `m
p nearly isometrically and started
the investigation of the minimal number m(ε, q, p, n) for (1 + ε)-isomorphic
embeddings. Bourgain, Lindenstrauss and Milman [BoLM] invented the iteration method and they were able to extend the results to arbitrary finite
dimensional subspaces of Lp . Moreover, they clarified the behaviour of the
function m(ε, q, p, n) for all values 0 < p < q < 2. Talagrand [LT], [Ta], improved the estimates for embeddings of arbitrary n-dimensional subspaces
of Lp into `m
p .
In this paper, we are interested in two kinds of results: finite dimensional uniformly isomorphic results and infinite dimensional isometric results. In the first part of the paper, we will construct embeddings of the
m(n,q)
Schatten class Sqn in Sp
(for precise definitions of the Schatten classes
see below). In the second part of the paper, we use probabilistic tools to
obtain infinite dimensional isometric embeddings for finite von Neumann
algebras. In the previous version of this paper, we intended to include the
general semi-finite case using ultra product techniques. However, in the
mean time we developed a different (and we think better) approach based
on Parthasarathy’s [P] concept. These results are beyond the scope of this
paper.
Before stating our results precisely, let us recall the definition of noncommutative Lp -spaces. We assume that τ is a semi-finite, normal, faithful
trace on a von Neumann algebra N . A trace function τ : N+ → [0, ∞] is
defined on the positive elements N+ of N and satisfies
τ (λx) = λτ (x) ,
τ (yy ∗ ) = τ (y ∗ y) ,
τ (x + y) = τ (x) + τ (y)
for all positive λ ∈ R, all positive x ∈ N+ and all y ∈ N . τ is called
semi-finite, if for all x > 0 there is 0 < y ≤ x such that τ (y) is finite. Then,
the definition ideal
X
n
n
n
X
X
∗
∗
m(τ ) :=
xk yk n ∈ N ,
τ (xk xk ) < ∞ ,
τ (yk yk ) < ∞
k=1
k=1
k=1
is dense in N with respect to the weak operator topology. τ is called normal,
if
τ sup xi = sup τ (xi )
i
i
for every bounded increasing net with supremum in N . τ is called faithful,
if τ (x) = 0 implies x = 0 for all positive x. The non-commutative Lp space Lp (N, τ ) is defined to be the closure of m(τ ) with respect to the
Vol. 10, 2000
EMBEDDINGS OF NON-COMMUTATIVE Lp -SPACES
391
(quasi-)norm
1/p
kxkp := τ (x∗ x)p/2
.
We refer to [T] for basic information of traces and we refer to [N], [Te]
for the realization of Lp (N, τ ) as unbounded operators affiliated to N and
to [FK], [Y] for more information about non-commutative Lp -spaces. Let
us note that the usual duality between Lp (N, τ ) and Lp0 (N, τ ) is valid.
Moreover, for 0 < p ≤ 1 the space Lp (N, τ ) is p-normed [FK, Theorem 4.9],
i.e.
kx + ykpp ≤ kxkpp + kykpp .
We will use B(H) for the space of bounded, linear operators on a Hilbert
space H. In the case N = B(`2 ) or N = Mm = B(`m
2 ), we consider the
usual trace tr and obtain the classical Schatten p-classes Sp = Lp (B(`2 ), tr),
Spm = Lp (Mm , tr), respectively. For notational reasons it is also useful to
consider the normalized τm = tr/m on Mm . Clearly Spm is isometrically
isomorphic to Lp (Mm , τm ). We will use the notation R for the hyperfinite
II1 factor, i.e. closure with respect to the weak operator topology of the
image of the infinite tensor product ⊗n∈N M2 in the GNS-construction with
respect to tracial state σ = ⊗n∈N τ2 . Our finite dimensional result reads as
follows:
Theorem 0.1. Let 1 < p ≤ 2, then there exists a constant C(p) and for
m(n)
all n ∈ N there is a linear isomorphism u : Spn → S1
such that for all
n × n-matrices x
kxkp ≤ ku(x)kp ≤ 32 kxkp ,
n+1
where m(n) ≤ C(p)n .
Let us note that according to Arazy’s results [Ar] there is no embedding
of Sp into Sq except for the trivial cases p = 2 or p = q. This corresponds
to similar results in the commutative setting [LiT]. Apart from rather
classical tools in probability the main new idea in the proof of the theorem above is the consequent use of the notion of independent copies of a
non-commutative random variable. (The notion of freeness introduced by
Voiculescu seems to be less appropriate in this context although it turned
out to be essentially equivalent.) More precisely, if x ∈ Mn is a n × n
matrix, we consider the elements
xj = 1 ⊗ 1 ⊗ · · · 1 ⊗
x
|{z}
1 ⊗ · · · 1 ∈ Mnn .
j-th position
Then x1 , . . . , xn are independent copies of x in Lp (Mnn , τnn ). An isomorphic embedding of Spn into a non-commutative L1 -space can be realized as
392
M. JUNGE
GAFA
follows. If (θj )nj=1 are symmetric p-stable random variables on (Ω, P ), we
have
Z X
n
1
θ
(ω)x
dP (ω) ≤ 4 kθk1 kxkp .
kθk1 kxkp ≤
j
j
8
Ω
L1 (Mnn ,τnn )
j=1
This formula yields some information on how to ‘compute’ the p-norm of
the eigenvalues without explicit reference to the polar composition of x.
Unfortunately, this embedding is not isometric. However, it is closely related to the p-stable process we will construct here. Indeed, our attempt
to the ‘right’ notion of a p-stable process is based on a suitable summation
formula for p-stable processes in the commutative setting and the aforementioned notion of a sequence of independent copies. This summation
formula has been discovered by Marcus and Pisier [MP].
Theorem 0.2. Let N be a von Neumann algebra with a normal, faithful,
tracial state τ and let 1 < p < 2, then there exists a finite von Neumann
algebra M with a normal, faithful, tracial state σ and an isometric linear
embedding u : Lp (N, τ ) → L1 (M ⊗ B(`2 ), σ ⊗ tr), i.e.
ku(x)kL1 (M⊗B(`2 ),σ⊗tr) = kxkp .
Moreover, Lp (N, τ ) is isomorphic to a subspace of L1 (M, σ). In particular,
the non-commutative Lp -space Lp (R, τ ) over the hyperfinite II1 -factor R is
isometrically isomorphic to a subspace of the predual L1 (B(`2 ) ⊗ R, tr ⊗ τ )
of a hyperfinite, semifinite von Neumann algebra and Lp (R, τ ) is isomorphic
to a subspace of L1 (R, τ ) = R∗ .
In fact the method is general enough to obtain embeddings of Lp (N, τ )
in Lq (M, σ) whenever 0 < q < p < 2. Using different tools we will prove in
a subsequent paper
Theorem 0.3. Let N be a semi-finite von Neumann algebra with normal,
semi-finite, faithful trace τ . Then there exists a finite von Neumann algebra
M with a normal, tracial, faithful state σ such that for all 0 < q < p < 2
there is an isometric embedding of Lp (N, τ ) into Lq (M, σ). In particular,
for 0 < q < p < 2, Lp (R ⊗ B(`2 ), τ ⊗ tr) is isometrically isomorphic to a
subspace of Lq (R, σ).
I am grateful to G. Pisier for the joint effort in proving Theorem 0.2
and for further stimulating discussions. I want to thank the referee for his
patient proof-reading.
Vol. 10, 2000
EMBEDDINGS OF NON-COMMUTATIVE Lp -SPACES
1
393
The Finite Dimensional Case
In this section, we will describe how to modify classical embeddings of `np
in Lq such that they can be ‘translated’ into the non-commutative setting.
Let us recall the definition of the p-norm for a vector x = (x1 , . . . , xn )
1/p
X
n
|xi |p
kxkp =
i=1
with the obvious modification for p = ∞. We will make much use of the
following notation: For n ∈ N and j = 1, . . . , n, we consider the map
n
Fj : `n∞ → `n∞
Fj (x)(i1 , . . . , in ) = xij .
Obviously, each Fj is an algebra homomorphism. From the probabilistic
view point, we note that (Fj (x))nj=1 is a (finite) sequence of independent
copies of x. Indeed, let Ω0 = {1, . . . , n} equipped with the normalized
counting measure µ0 ({i}) = 1/n and define x : Ω0 → C by x(i) = xi . Then,
(Fj (x))nj=1 is the standard sequence of independent copies of x defined on
n
the product space Ω = Ωn0 with respect to the product measure µ = µ⊗
0 .
n
Let us denote by ej the standard unit vector basis in C . We will need the
following proposition.
Proposition 1.1. Let 1 ≤ p ≤ ∞, then for all x ∈ Cn
1/p
Z X
n
1
p
|Fj (x)|
dµ ≤ kxkp .
kxkp ≤
2
Ω
j=1
P
In particular, the map w : `np → L1 (Ω, µ; `np ) w(x) = nj=1 Fj (x) ⊗ ej is an
isomorphism up to a constant 2.
The proof relies on the investigation of a certain K-functional. Very recently (September ’99) an elementary proof of Proposition 1.1 with the constant 1 − 1e has been found by Aimo Hinkkanen. Given a random variable h
on a probability space (Ω, µ) the K-functional of the pair (L∞ (Γ,ν),L1 (Γ,ν))
at the value t is given by
K(h, t) := inf h1 ∞ + t h2 1 .
h=h1 +h2
We need the following lemma due to S. Geiss [G, Theorem 3.4].
Lemma 1.2. Let f1 , . . . , fn be random variables on (Γ, ν) and h a random
variable in L1 (Ω0 × Γ, µ0 ⊗ ν) defined by
h(j, ω) := fj (ω) ,
then
Z
1
2 K(h, n)
≤
sup |fj (ωj )|dν(ω1 ) · · · dν(ωn ) ≤ K(h, n) .
Γn j≤n
(1)
394
M. JUNGE
GAFA
If all the fj ’s coincide, i.e. fj = f , then K(h, n) = K(f, n) and
Z
1
K(f,
n)
≤
sup |f (ωj )|dν(ω1 ) · · · dν(ωn ) ≤ K(f, n) .
2
(2)
Γn j≤n
For 1 < p < ∞, we will further need the classical random variable
v = v(p) : (Γ, ν) → R with the tail behavior
ν(v > t) = 1 − exp(−t−p ) .
This variable is well-known in probability and Banach space theory, see
[To]. We assume that (vj )nj=1 are independent copies of v defined on Γn .
The following lemma can be found in [To]. We give a short proof for the
convenience of the reader.
Let 1 < p < ∞, then there is a constant c(p) such that for
Lemma 1.3.
all x ∈ Cn
Z
kxkp = c(p)
sup |v(ωj )xj |dν(ω1 ) · · · dν(ωn ) .
Γn j=1,...,n
Indeed, we can assume kxkp = 1 and xj 6= 0. By independence, we
obtain for the random variables vj (ω1 , . . . , ωn ) = v(ωj )
Y
Prob |vj |≤ |xtj |
Prob sup |vj xj |>t = 1 − Prob sup |vj xj |≤t = 1 −
j
j
=1−
Y
j
p X
|x |
= 1 − exp − t−p
exp − tj
|xj |p
j
j
−p
= 1 − exp(−t
Using 1
− exp(−t−p )
−1
c(p)
≤
t−p ,
= kvk1 =
Z
1
we see that
∞
Z
∞
Prob(v > t)dt ≤ 1 +
0
is finite and
Z
sup |vj xj | =
j
).
t−p dt < ∞
1
∞
Prob sup |xj vj | > t dt = c(p)−1 .
j
0
Proof of Proposition 1.1. Let x =
be a vector and recall x : Ω0 → C
defined by x(i) = xi . The upper estimate is trivial. Indeed,
1
Z X
1 X
1
Z X
n
n
n Z
p
p
p
p
p
p
|Fj (x)|
dµ ≤
|Fj (x)| dµ
=
|x(i)| dµ0 (i)
(xi )ni=1
Ω
Ω j=1
j=1
=
X
n
j=1
1X
|xi |p
n
n
i=1
1/p
j=1
= kxkp .
Ω0
EMBEDDINGS OF NON-COMMUTATIVE Lp -SPACES
Vol. 10, 2000
395
For the lower estimate, we can assume p > 1. Let us consider the random
variable h : Ω0 × Γ → R defined by
h(j, ω) = xj v(ω) .
According to (1) and Lemma 1.3, we get
Z
1
|xj v(ωj )|dν(ω1 ) · · · dν(ωn ) ≥ c(p)
kxkp .
K(h, n) ≥
Γn
Now, let us use the upper estimate of K(h, n) in formula (2) and combine
it again with Lemma 1.3
Z
1
sup h(ij , ωj )d(µ0 ⊗ ν)(i1 , ω1 ) · · · d(µ0 ⊗ ν)(in , ωn )
2 K(h, n) ≤
(Ω0 ×Γ)n j≤n
Z Z
=
sup |xij v(ωj )|dν(ω1 ) · · · dν(ωn ) dµ0 (i1 ) · · · dµ0 (in )
Ωn
0
Γn j≤n
−1
X
n
Z
= c(p)
Ωn
0
−1
Z
1/p
|xij |
dµ0 (i1 ) · · · dµ0 (in )
p
j=1
X
n
1/p
|Fj (x)|
= c(p)
Ω
p
dµ .
j=1
Hence, we get
1
kxkp ≤
2
Z X
n
Ω
1/p
|Fj (x)|
p
dµ ≤ kxkp
j=1
and the proof of Proposition 1.1 is complete.
Corollary 1.4. Let 0 < q ≤ p ≤ ∞, then for all x ∈ Cn
q/p 1/q
Z X
n
1
p
kxk
≤
|F
(x)|
dµ
≤ kxkp .
j
p
21/q
Ω
j=1
P
Hence w(x) = nj=1 Fj (x) ⊗ ej yields an isomorphism of `np into Lq (`np ).
Proof. Apply Proposition 1.1 to p̃ = p/q and x̃j = |xj |q .
Let us recall some standard information about the real and imaginary
part.
Lemma 1.5.
Let 0 < p ≤ ∞, N be a von Neumann algebra with a
∗
x−x∗
semifinite trace τ , x ∈ N , and <(x) = x+x
the real and
2 , =(x) =
2i
imaginary part of x, then
1
k<(x)kp , k=(x)kp ≤ kxkp
1 − 1 max
max{1,2 p 2 }
≤ max{2, 21/p } max k<(x)kp , k=(x)kp .
396
M. JUNGE
GAFA
Proof. If p ≥ 1, the triangle inequality implies the upper estimate. For
0 < p < 1, we use the fact that Lp (N, τ ) is p-normed [FK] to obtain the
upper estimate. For the lower estimate, let us note that
2 <(x)2 + =(x)2 = xx∗ + x∗ x .
√
Moreover, by polar decomposition x = u x∗ x and xx∗ = ux∗ xu∗ , we have
kx∗ kpp = τ u(x∗ x)p/2 u∗ = kxkpp .
This implies for p ≥ 2
k<(x)k2p
= k<(x) kp/2
2
∗
xx + x∗ x ≤
2
≤ kxk2p .
p/2
For p ≤ 2, we use again the fact that Lp/2 (N, τ ) is p/2-normed. The proof
for the imaginary part is identical.
In order to use the probabilistic approach, we extend the definition of
the algebra homomorphisms Fj : Mn → Mnn in the following way:
Fj (x) := 1 ⊗ 1 ⊗ · · · 1 ⊗
x
|{z}
1 ⊗ · · · 1 ∈ Mnn .
j-th position
If we consider the natural embedding of D : `n∞ → Mn given by D(x) = Dx
where Dx (ei ) = xi ei is the corresponding diagonal operator, it is obvious that Fj (D(x)) = D(Fj (x)). Therefore, it is justified to use the same
symbol Fj . Let us recall the normalized trace τm = tr/m. Obviously,
the map x 7→ m1/p x is an isometry between the Schatten class Spm and
Lp (Mm , τm ). In the following, we will use this observation for m = nn and
n
a diagonal matrix Dτ with τ ∈ Rn . Then, we have
kτ kLq (Ω,ν) = kDτ kLq (Mnn ,tr/nn ) .
Now, we will present a rather general result which transforms any real
valued embedding of `np in Lq (Γ, ν) into an embedding of Spn into
Lq (Γ, ν; Lq (Mnn , τnn )).
Proposition 1.6.
Let 0 < q < p ≤ ∞ and u : `np → Lq (Γ, ν) be
an isomorphism such that for all i = 1, . . . , n the functions u(ei ) are real
valued and for all x ∈ Rn
c1 kxkp ≤ ku(x)kq ≤ C1 kxkp .
Then ũ :
Spn
→ Lq (Γ, ν; Lq (Mnn , τnn )) defined by
ũ(x) =
n
X
j=1
u(ej ) ⊗ Fj (x)
EMBEDDINGS OF NON-COMMUTATIVE Lp -SPACES
Vol. 10, 2000
satisfies for all x ∈ Mn
1
1
i 1q − 12
1
2 q max{2, 2 p } max{1, 2
}
397
c1 kxkp ≤ kũ(x)kq
≤ max{2, 2 q } max{1, 2 q − 2 }C1 kxkp .
1
1
1
Proof. Let us consider a selfadjoint matrix x ∈ Mn . By polar decomposition, x = oDσ o∗ , where Dσ (ei ) = σi ei is a diagonal matrix and o is unitary.
Then, we get
Fj (x) = 1 ⊗ 1 ⊗ · · · 1 ⊗
∗
∗
1 ⊗ ··· ⊗ 1
x
|{z}
j-th position
∗
= oo ⊗ oo ⊗ · · · ⊗ oo ⊗ oDσ o∗ oo∗ ⊗ · · · ⊗ oo∗
| {z }
j-th position
= (o ⊗ · · · ⊗ o)Fj (Dσ )(o∗ ⊗ · · · ⊗ o∗ ) .
Note that o ⊗ · · · ⊗ o is again a unitary and therefore the assumptions apply
together with Corollary 1.4
Z X
q
q
kũ(x)kq =
u(ej )(γ)(o⊗···⊗o)Fj (Dσ )(o∗ ⊗···⊗o∗ )
dν(γ)
n
Γ
j
Γ
j
Lq (Mnn ,tr/n )
Z X
q
=
u(ej )(γ)Fj (Dσ )
=
Z X
q
u(e
)(γ)F
(σ)
j
j
Γ
1
= n
n
j
n
X
Lq (Mnn ,tr/nn )
Lq (Ω,µ)
dν(γ)
dν(γ)
Z X
q
u(ej )(γ)σij dν(γ)
(3)
i1 ,...,in =1 Γ
j
n
n
X
X
r/p
1
p
|σ
|
ij
nn
i1 ,...,in =1
j=1
X
r/p
n
q
p
≤ C1
|σj |
= C1q kxkqp .
≤ C1q
j=1
Starting from (3), we can also apply the lower estimates in the assumption
and in Corollary 1.4 to obtain
kũ(x)kqq ≥ cq1 2−1 kxkqp .
Hence, we get
1
c kxkp
21/q 1
≤ kũ(x)kq ≤ C1 kxkp
398
M. JUNGE
GAFA
for all selfadjoint elements. Since ũ preserves the real and imaginary part,
we deduce from Lemma 1.5
1
1
1
1
1 c1 kxkp ≤ kũ(x)kq
2 q max{2, 2 p } max{1, 2 q − 2 }
1
1
≤ max{2, 2 q } max{1, 2 p
− 12
}C1 kxkp .
Proposition 1.6 turns out to be very useful if we combine it with nice
embeddings of `m
p into Lq (Γ, ν). As mentioned in the introduction, good
embeddings have been intensively studied in the literature. For example, we
can consider the classical embedding u(ej ) = θj , where (θj )nj=1 is a sequence
of independent p-stable random variable (i.e. satisfying E(exp(itθj )) =
exp(−cp |t|p )). Then, we obtain the following inequalities.
Corollary 1.7. For 0 < q < p < 2, let (θj )nj=1 be a sequence of symmetric p-stable variables on a probability space (Γ, ν). Then for all matrices
x ∈ Mn
Z X
1
q
q
1
1
1
kθk
kxk
≤
θ
(ω)x
dν(ω)
≤ 2 p + q kθkq kxkp .
j
j
2
1
q
p
+
Lq (Mnn ,τnn )
Γ
2q p
j
The proof of Theorem 0.1 follows from the existence of (1+ε)-isomorphic
embeddings of `np in `m
q . Fortunately, Johnson and Schechtman and later
Bourgain, Lindenstrauss and Milman proved excellent estimates for the
function m = m(n, p, q, ε). In [JS], [BoLM, Corollary 7.6] they showed that
for 0 < q < p < 2 there is an embedding u : `nq → `m
p satisfying
(1 − ε) kxkp ≤ ku(x)kq ≤ (1 + ε) kxkp
where m ≤ C(q, p, ε)n. In combination with Proposition 1.6 this yields.
Theorem 1.8. Let 0 < q < p < 2 and ε > 0. Then there is a constant C(q, p, ε) such that for all n ∈ N there is a natural number m ≤
C(q, p, ε)nn+1 and an isomorphism u : Spn → Sqm such that for all matrices
x ∈ Mn
1 1
(1 − ε)
+q
p
kxkp .
2
1 kxkp ≤ ku(x)kq ≤ (1 + ε)2
+
q
p
2
Remark 1.9. One can slightly improve on the constants using
n
m X
X
w(x) =
uij ei ⊗ ej1 ⊗ xj ,
i=1 j=1
where u = (uij )j=1,...,n,i=1,...,m : Rn → Rm is the map constructed by
Johnson and Schechtman and ej1 are the standard basis of the column
EMBEDDINGS OF NON-COMMUTATIVE Lp -SPACES
Vol. 10, 2000
399
(1+ε)kgk
space (see the next paragraph). For 0 < q < p < 2 this yields a 2 (1−ε)kgkp q
C(q,p,ε)nn+2
. Here kgkp means the Lp isomorphic embedding of Spn into Sq
norm of a normalized, centered gaussian random variable. However, due to
the factor 1/2 in Lemma 1.3, this approach never yields a nearly isometric
embedding.
2
Finite von Neumann Algebras
In this section, we will assume that N is a finite von Neumann algebra, i.e.
there is a semi-finite, normal, faithful trace τ : N → C such that τ (1) = 1.
In this case m(τ ) coincides with N and Lp (N, τ ) is the closure of N with
respect to the p-norm. If L ⊂ N is a subalgebra in N and closed with
respect to the weak operator topology, then the restriction of τ to L is
again a finite, normal, faithful trace. Clearly, the inclusion L ⊂ N induces
an isometric embedding of Lp (L, τ ) into Lp (N, τ ).
Now, we will discuss the infinite tensor product of N . Let M0 =
limn N ⊗n be the direct limit of the tensor products N ⊗n using the canonical
embedding jnm : N ⊗n → N ⊗m
jnm (x) = x ⊗ 1| ⊗ 1{z
· · · ⊗ 1} .
n−m times
This embedding also yields a densely defined linear map
jnm : Lp (N ⊗n , τ ⊗n ) → Lp (N ⊗m , τ ⊗n ) .
Q
Here, τ n (x1 ⊗ · · · ⊗ xn ) = nk=1 τ (xk ) is the natural n-fold tensor product
of τ . In particular, we obtain the Hilbert space H = limn L2 (N ⊗n , τ ⊗n )
on which M0 acts by the left regular representation π(xy) = xy. Let
M ⊂ B(H) be the closure of π(M0 ) with respect to the weak operator
topology. Then σ(x) = (1, π(x)1) yields a continuous extension of the
canonical traces τ ⊗n . By continuity and density, σ extends to a normal
trace on M . By definition, the representation π : M → B(H) is faithful.
Hence, for every strictly positive element m ∈ M , there exists x ∈ N ⊗n
such that
0 < x, π(m)(x) = σ(x∗ mx) = σ(m1/2 m1/2 xx∗ ) ≤ σ(m)σ(x∗ xmxx∗ ) .
This shows that S
σ is faithful on M . It turns out that Lp (M, σ) is indeed the
norm closure of n N ⊗n with respect to the p-norm. Moreover, the predual
M∗ = L1 (M, σ) is the infinite tensor product of L1 (N, τ ) with respect to the
operator space projective tensor norm, see e.g. [BP]. The most
S important
example of this construction is the hyperfinite factor R = cl( n M2⊗n ) with
400
M. JUNGE
GAFA
respect to the normalized trace on R given by τR = ⊗n∈N τ2 and the closure
is taken with respect to the weak operator topology.
R
In the commutative case N = L∞ (Ω, µ) with τ (x) = Ω xdµ, we see
that
n
n
H = lim L2 (N ⊗ , τ ⊗ ) = L2 (ΩN , µ⊗N ) .
N ⊗n
n
(ΩN , µ⊗N )
⊂ L∞
corresponds to the bounded functions depending only
on the first n coordinates. In particular, we have M∗ = L1 (ΩN , µ⊗N ). Let
us note that the hyperfinite factor R is a non-commutative analogue of
L∞ [0, 1] = L∞ ({0, 1}N ) equipped with the normalized Haar measure on the
group {0, 1}N . If M is another finite von Neumann algebra with normalized,
faithful, normal trace σ, then L∞ ([0, 1]; M ) is isomorphic to the subalgebra
¯ ⊂ R⊗M
¯ .
L∞ ({0, 1}N )⊗M
Let λ denote the Lebesgue measure on [0, 1]. Then, Lp ([0, 1], λ; Lp (M, σ)) is
isometrically isomorphic to a subspace of Lp (R ⊗ M, τR ⊗ σ) for 0 < p ≤ ∞.
This space is isometrically complemented for 1 ≤ p ≤ ∞.
As before, we use the following notation of a sequence of independent
copies. Given an element x ∈ N the sequence (xj )j ⊂ M is defined by
xj = 1 ⊗ · · · ⊗ 1 ⊗
x
|{z}
⊗1 ⊗ · · · ∈ M .
j-th position
Indeed, xj corresponds to Fj (x) in the previous section. For our purpose
the following lemma from [MP], plays a central role. (See [MP] for a simple
proof due to J. Zinn.)
Lemma 2.1 ([MP, Lemma 1.4]). Let 0 < p < 2 and v ∈ Lp (0, 1) be a real
valued random variable, positive for 0 < p < 1 and symmetric if 1 ≤ p < 2.
For a sequence of independent copies (vj )j of v, the random variable
X −1/p
X=
Γj vj
j
converges almost surely to a p-stable random variable. In particular for all
0 < q < p there exists a constant c(p, q) such that
X
1/p
−1/p q 1/q
E
Γj vj = c(q, p) E|v|p
.
Pj
j
Here Γj =
k=1 Xk is the sum of j independent identically distributed
variables (Xk )k with Poisson distribution Prob(Xk > t) = exp(−t).
In the following, we will assume that the sequence (Γj )j is defined on
EMBEDDINGS OF NON-COMMUTATIVE Lp -SPACES
Vol. 10, 2000
[0, 1]. We will need the standard basis


0 0 0 ···
 0 0 0 · · ·



· · ·

ej1 = 
 1 0 0 · · ·



· · ·
401
← j-th row
···
of the column space Cq ⊂ Sq = Lq (B(`2 ), tr). Now, we can prove the main
theorem.
Theorem 2.2. Let 0 < q < p < 2, N a von Neumann algebra with a
faithful, normal, tracial state τ and x ∈ N , then
X −1/p
u(x) =
Γj xj ⊗ ej1 ∈ Lq [0, 1], λ; Lq (M ⊗ B(`2 ), σ ⊗ tr)
j
is convergent and there is a constant c(q, p) such that
ku(x)kq = c(q, p) kxkp .
In particular, Lp (N, τ ) isometrically embeds into Lq (R ⊗ M ⊗ B(`2 ), τR ⊗
σ ⊗ tr).
Proof. Let x ∈ N , then y = x∗ x is selfadjoint. Let L be the von Neumann
algebra generated by 1 and y. By functional calculus L is isomorphic to
L∞ (sp(y), µ) where the measure is induced via the trace τ
µ(A) = τ (1A (y)) .
Given a subset I ⊂ N, we will consider
X −2/p
Γj x∗j xj .
ZI =
j∈I
∗
∗
First let us note that xj xj = (x x)j = yj . For fixed ω and a finite subset I,
the element ZI (ω) ∈ N ⊗n for n ≥ max I. Moreover, y ∈ L is positive
and henceforth ZI (ω) ∈ L⊗n ∼
= L∞ (sp(y)n , µ⊗n ) is positive. According to
Lemma 2.1, the positivity and Fatou’s lemma, we see that Z{1,...,n} converges in
Lq/2 [0, 1] × sp(y)N , λ ⊗ µ⊗N
to the p/2-stable random variable Z := ZN and therefore satisfies
1/p
p q Z
p
|t| 2 dµ(t)
kZkq/2 = c ,
2 2
sp(y)
p q p q 1/p
=c ,
=c ,
τ (|y|p/2 )
kxk2p .
2 2
2 2
402
M. JUNGE
GAFA
In particular, Z{1,...,n} is a Cauchy sequence. This implies
X
2
X
m −1/p
m − p2 ∗ Γj xj ⊗ ej1 = lim Γj (x x)j lim n<m→∞
n<m→∞
q
j=n
Therefore
n
u (x) =
n
X
= 0.
q/2
j=n
−1
Γj p xj ⊗ ej1
j=1
is a Cauchy sequence converging to u(x) and satisfies
ku(x)k2q = lim kun (x)k2q = lim kZ{1,...,n} k q = kZk q = c
n
n
2
2
2
p q
2 , 2 kxkp
.
Remark 2.3. Let (rj )j∈N be a sequence of independent Rademacher variables. Using the first part in Lemma 2.1, the same argument as above
shows that for a normal
Xelement x ∈ N
−1/p
Γj rj xj = c(p, q) kxkp .
q
j
In particular, the selfadjoint part Lp (N, τ )sa is – as a Banach space or
p-Banach space over the real numbers – isometrically isomorphic to a subspace of Lq (R ⊗ M, τR ⊗ σ)sa . The linear map
X −1/p
φ(x) =
Γj rj xj
j
is the non-commutative analogue of a p-stable process. Since φ respects the
real and imaginary part of a random variable x, we deduce from Lemma 1.5
for all x ∈ Lp (N, τ )
2− p − q c(p, q) kxkp ≤ kφ(x)kq ≤ 2 p + q c(p, q) kxkp .
In particular for 1 < p < 2 this yields a 16-isomorphic embedding of
Lp (N, τ ) into the non-commutative L1 -space L1 (R⊗M, τR ⊗σ) = (R⊗M )∗ ,
the predual of a finite von Neumann algebra.
1
1
1
1
Corollary 2.4.
Let N be a hyperfinite von Neumann algebra with
a faithful normal trace and 1 < p < 2, then Lp (N, τ ) is isometrically
isomorphic to a subspace of the predual of a hyperfinite von Neumann
algebra. Moreover, Lp (R, τR ) is isometrically isomorphic to a subspace of
L1 (R ⊗ B(`2 ), τR ⊗ tr) and 16-isomorphic to a subspace of L1 (R, τR ).
Proof. If N is hyperfinite, the same holds true forSM being the closure with respect to the weak operator topology of
N ⊗n . The space
Lq ([0, 1], λ; Lq (M ⊗ B(`2 ), σ ⊗ tr)) is isometrically isomorphic to a subspace
¯ ⊗B(`
¯
of the space Lq (R ⊗ M ⊗ B(`2 ), τR ⊗ σ ⊗ tr). Since R⊗M
2 ) is hyperfinite, we proved the assertion. In order to obtain the isomorphic result in
Vol. 10, 2000
EMBEDDINGS OF NON-COMMUTATIVE Lp -SPACES
403
the case N = R, we note that the construction yields the hyperfinite factor
¯
R⊗M
which has to be isomorphic to R, see [MuN]. Since the trace on M
satisfies σ(1) = 1, this isomorphism has to preserve the trace and therefore
¯
the isomorphism between R⊗M
and R is trace preserving. In particular,
the corresponding L1 -spaces are isometrically isomorphic.
Remark 2.5. The embeddings constructed here in general do not preserve
the canonical operator space structure of Lp (N, τ ). This is studied in more
detail in [Ju].
Let us briefly outline how to derive the almost isometrical result form the
infinite dimensional result. We will need an entirely classical approximation
result from the theory of vector-valued Lp -spaces. The proof is a standard
induction argument and left to the reader.
Lemma 2.6. Let X be a q-Banach space, S ⊂ Lq ([0, 1], X) be a finite set
and ε > 0, then there exists an n ∈ N and elements (xj (s))j=1,...,n,s∈S ⊂ X
such that for all s ∈ S
n
X
s −
xj (s)1[ j−1 , j ) ≤ ε.
n n
Lq (X)
j=1
Corollary 2.7. Let 0 < q < p < 2, m ∈ N and ε > 0, then there exists
an N ∈ N and u : Spn → SpN such that for all x ∈ Mm
(1 − ε) kxkp ≤ ku(x)kq ≤ (1 + ε) kxkp .
Proof. We will restrict to the case 0 < q ≤ 1. Let
u : Lp (Mm , τm ) → Lq [0, 1]; Lq (⊗n∈N Mm ⊗ B(`2 ), σ ⊗ tr)
be the isomorphism
u(x) =
∞
X
−1
Γk p ek1 ⊗ xk
k=1
from Theorem 2.2. Let eij be the matrix units and consider yij = m1/p u(eij ).
Since the sum above is converging, we can find for δ > 0 an N ∈ N such
that
N
1 X −1
p
yij − m p
≤δ
Γ
e
⊗
(e
)
ij
k1
k
k
2
q
k=1
for all i, j = 1, . . . , m. Using Lemma 2.6, we can find n ∈ N and vectors
zi,j,l ∈ MN mN such that
n
X
yij −
z
1
l−1 l
ijl
[
, ) ≤ δ .
l=1
n
n
q
404
M. JUNGE
GAFA
In particular,
kakqp
q X
q X
X
=
aij yij ≤ aij zijl 1[ l−1 , l ) +
|aij |q δ q
q
ij
n
ijl
q
q
n
X
≤
aij zijl 1[ l−1 , l ) + m2 δ q kakqp .
n
ijl
Similarly,
ij
q
n
q
X
aij zijl 1[ l−1 , l ) ≤ m2 δ q kakqp + kakqp .
n
ijl
n
q
Hence, we get
q
X
1
1
aij zijl 1[ l−1 , l ) ≤ (1 + δ q m2 ) q kakp .
(1 − δ q m2 ) q kakp ≤ ijl
n
n
q
But clearly,
X
a
z
1
l−1
ij ijl [
, l ]
ijl
n
n
q

n−1/q P aij zij1
0
0
ij
P

−1/q
0
n
0

ij aij zij2

= 
···


···
0
0
···
···
···
n−1/q
P
ij
aij zijn

0 0





q.
Hence choosing δ small enough, we obtain a (1 + ε) isomorphic embedding
N
into SqnN m . The case q ≥ 1 is similar using the triangle inequality instead
of the fact that Lq is q-normed for q ≤ 1. Moreover, the proof can be
simplified using conditional expectations.
References
[A]
H. Araki, Some properties of modular conjugation operator of von Neumann algebras and a non-commutative Radon-Nikodym theorem with a
chain rule, Pacific J. of Math. 50 (1974), 305–354.
[Ar]
J. Arazy, Basic sequences, embeddings, and the uniqueness of the symmetric structure in unitary matrix spaces, J. Funct. Anal. 40:3 (1981),
302–340.
[BP] D. Blecher, V. Paulsen, Tensor products of operator spaces, J. Funct.
Anal. 99:2 (1991), 262–292.
[BoLM] J. Bourgain, J. Lindenstrauss, V. Milman, Approximation of
zonoids by zonotopes, Acta Math. 162:1-2 (1989), 73–141.
Vol. 10, 2000
EMBEDDINGS OF NON-COMMUTATIVE Lp -SPACES
405
[BrD] J. Bretagnolle, D. Dacunha-Castelle, Application de l’etude de
certaines formes lineaires aleatoires au plongement d’espaces de Banach
dans des espaces Lp , Ann. Sci. Ecole Norm. Sup. 4:2 (1969), 437–480.
[BrDK] J. Bretagnolle, D. Dacunha-Castelle, J.L. Krivine, Lois stable
et espace Lp , Ann. Inst. H. Poincaré 2, (1966), 231-259.
[D]
J. Dixmier, Von Neumann Algebras, North-Holland, Amsterdam-New
York-Oxford, 1981.
[FK] T. Fack, H. Kosaki, Generalized s-numbers of τ -measurable operators,
Pacific J. Math. 123:2 (1986), 269–300.
[Fe]
W. Feller, An Introduction to Probability Theory and Its Applications,
Vol. II, John Wiley & Sons, Inc., New York, N.Y., 1950.
[G]
S. Geiss, BMOψ -spaces and applications to extrapolation theory, StudiaMath. 122:3 (1997), 235–274.
[JS]
W.B. Johnson, G. Schechtman, Embedding lpm into l1n , Acta Math.
149:1-2 (1982), 71–85.
[Ju]
M. Junge, Operator subspaces which embed into the predual of a von
Neumann algebra, in preparation.
[KS1] S. Kwapien, C. Schütt, Some combinatorial and probabilistic inequalities and their application to Banach space theory, Studia Math. 82:1
(1985), 91–106.
[KS2] S. Kwapien, C. Schütt, Some combinatorial and probabilistic inequalities and their application to Banach space theory, II, Studia Math. 95:2
(1989), 141–154.
[LT]
M. Ledoux, M. Talagrand, Probability in Banach Spaces: Isoperimetry and Processes, Ergebnisse der Mathematik und ihrer Grenzgebiete (3)
23, Springer-Verlag, Berlin, 1991.
[LiT] J. Lindenstrauss, L. Tzafrir, Classical Banach Spaces I: Sequence
Spaces, New York, Springer, 1977.
[MP] M.B. Marcus, G. Pisier, Characterizations of almost surely continuous p-stable random Fourier series and strongly stationary processes, Acta
Math. 152:3-4 (1984), 245–301.
[MaP] B. Maurey, G. Pisier, Séries des variables aléatoires vectorielles
indépendentes et géométrie des espaces de Banach, Studia Math. 58 (1976),
45–90.
[MuN] F.J. Murray, J. von Neumann, On rings of operators, IV, Ann. Math.
(2) 44 (1943), 716–808.
[N]
E. Nelson, Notes on non-commutative integration, J. Funct. Anal. 15
(1974), 103–116.
[P]
K.R. Parthasarathy, An Introduction to Quantum Stochastic Calculus;
Birkhäuser Monographs in Mathematics 85, 1992.
[Pi1] G. Pisier, On the dimension of the lpn -subspaces of Banach spaces, for
1 ≤ p < 2, Trans. Amer. Math. Soc. 276:1 (1983), 201–211.
406
[Pi2]
[RS]
[T]
[Ta]
[Te]
[To]
[Y]
M. JUNGE
GAFA
G. Pisier, Probabilistic methods in the geometry of Banach spaces,
Springer Lecture Notes in Math. 1206 (1986), 167–241.
Y. Raynaud, C. Schütt, Some results on symmetric subspaces of L1 ,
Studia Math. 89:1 (1988), 27–35.
M. Takesaki, Theory of Operator Algebras I; Springer, New York, 1979.
M. Talagrand, Embedding subspaces of Lp in lpN , in “Geometric Aspects
of Functional Analysis (Israel, 1992–1994)”, Oper. Theory Adv. Appl. 77,
Birkhäuser, Basel, (1995), 311–325.
M. Terp, Lp -spaces Associated with von Neumann Algebras I and II,
Copenhagen Univ. 1981.
N. Tomczak-Jaegermann, Banach-Mazur Distances and Finitedimensional Operator Ideals, Longmann, 1988.
F. Yeadon, Non-commutative Lp -spaces, Math. Proc. Cambridge Philos.
Soc. 77 (1975), 91–102.
Marius Junge, Mathematisches Seminar der Universität Kiel, Ludewig-MeynStr. 4, 24098 Kiel, Germany
[email protected]
and
Department of Mathematics at Urbana-Champaign, 273 Altgeld Hall, 1409 West
Green Street, Urbana Illinois 61801, IL, USA
[email protected]
Submitted: September 1998
Revision: August 1999
Final version: November 1999