A Central Limit Theorem for Bose Z–Independent
Quantum Random Variables
Michael Skeide∗
Lehrstuhl fr Wahrscheinlichkeitstheorie und Statistik
Brandenburgische Technische Universitt Cottbus
Postfach 10 13 44
D–03013 Cottbus
Germany
December 1997
Abstract
We present a central limit theorem for Bose Z–independent operator-valued random variables. Furthermore, we show that the central limit distribution may be
represented by an algebra of creators and annihilators on a symmetric Fock module. As an example we recover the distribution of creators and annihilators on the
truncated Fock space, i.e. the central limit distribution of boolean independence.
1
Preliminaries
Throughout these notes Z denotes a unital C ∗ –algebra. In [5] we introduced the notion
of centered Hilbert Z–modules and centered Z–algebras. We described the construction
of the symmetric Fock module over a centered Hilbert Z–module and pointed out that
the possibility for such a construction depends essentially on the fact that the underlying
Hilbert module is centered. In [6] we defined what we understand by a Bose Z–independent
family of Z–quantum variables (in the sense of Voiculescu [10]). We pointed out that the
∗–algebra generated by all creators and annihilators on the symmetric Fock module is
∗
E-mail: [email protected]
1
a centered Z–algebra. Moreover, the algebras generated by creators and annihilators to
elements of mutually orthogonal subspaces are Bose Z–independent.
In these notes we show that a central limit theorem also holds for Bose Z–independent
Z–quantum variables. Like in the usual Bose independence, the limit distributions may
be realized as distributions of certain operators on the symmetric Fock module fulfilling
generalized canonical commutation relations in the (conditional) vacuum expectation.
In the remainder of this section we quickly recall the necessary definitions from [5, 6]
and some prerequisits about Hilbert modules; see [3, 1, 7]. All our vector spaces are
over C. All our modules are modules over unital complex algebras carrying a vector
space structure which is compatible with the unit of the algebra. The ⊗–sign stands
exclusively for the tensor product of vector spaces. The tensor product over Z is denoted
by ¯. (Recall that the tensor product of a right Z–module E and a left Z–module F is
the usual tensor product E ⊗ F divided out by the linear span of all xz ⊗ y − x ⊗ zy
(x ∈ E, y ∈ F, z ∈ Z).)
1.1 Definition [5]. A centered Z–module is a two-sided Z–Z–module E which is generated by the space
©
ª
CZ (E) = x ∈ E : xz = zx (z ∈ Z) .
CZ is called the Z–center of E.
A topological Z–Z–module over a topological algebra is centered, if it is generated
topologically by its Z–center.
In other words, any element x in a centered Z–module E may be written in the form
P
P
zi xi with suitable xi ∈ CZ (E) and zi ∈ Z. In the topological case the
xi zi =
x=
i
i
sum may be infinite.
Observe that CZ (Z) is the usual center of Z. If z ∈ CZ (Z), then zx = xz for all x ∈ E.
Clearly, the tensor product of centered modules again is centered. As the following
theorem shows centered Z–modules behave well with respect to tensor products.
1.2 Theorem [5].
Let E and F be centered Z–modules. Then there is a (unique) flip
isomorphism τ : E ¯ F → F ¯ E of two-sided modules, sending x ¯ y to y ¯ x for all
x ∈ CZ (E) and y ∈ CZ (F ).
1.3 Remark. The flip sends (z1 x) ¯ (z2 y) to z1 z2 (y ¯ x). It is important to realize that
the algebra elements z1 , z2 can be put at any place in the tensor (also at different places,
e.g. yz1 ¯ xz2 ). Only the order has to be preserved, i.e. z1 has to appear on the left of z2 .
Recall that a pre-Hilbert Z–module is a right Z–module E with a sesquilinear (linear
on the right) Z–valued inner product fulfilling the usual properties. These properties are
2
positivity (i.e. hx, xi is a positive element of Z), definiteness (i.e. hx, xi = 0 implies x = 0)
and right Z–linearity (i.e. hx, yzi = hx, yiz). Among the most obvious consequences
of this definition there are symmetry (i.e. hx, yi = hy, xi∗ ), left Z–anti–linearity (i.e.
hxz, yi = z ∗ hx, yi) and Cauchy-Schwartz inequality
hx, yihy, xi ≤ khy, yik hx, xi;
see [7]. From Cauchy-Schwartz inequality it follows that kxk =
p
khx, xik defines a norm
on E which turns it into a normed Z–module (i.e. kxzk ≤ kxk kzk). If E is complete, it
is called a Hilbert module.
We say E is a two-sided (pre-)Hilbert Z–Z–module, if E is a (pre-)Hilbert Z–module
with left action of Z such that hx, zyi = hz ∗ x, yi. If E is a two-sided Hilbert Z–Z–module
and F is a two-sided (pre-)Hilbert Z–Z–module we turn E ¯F into a two-sided pre-Hilbert
Z–Z–module by defining an inner product on E ¯ F via
­
®
hx ¯ y, x0 ¯ y 0 i = y, hx, x0 iy 0 .
(If E is only pre, then definiteness of this inner product may fail; see [1].) We see that
starting with a two-sided Hilbert module E it is possible to construct the two-sided preHilbert modules E ¯n (n ∈ N0 ). Of course, E ¯0 = Z.
If E is a centered (pre-)Hilbert Z–module, then the inner product necessarily maps the
Z–center of E to the center of Z. One easily checks that the flip τ applied to the tensor
product of centered Hilbert Z–modules is an isomteric (i.e. inner product preserving)
isomorphism.
With the help of the flip we may represent the permutation group Sn on the n–fold
P
π. The
tensor product E ¯n . We define the operator Pn : E ¯n → E ¯n as Pn = n!1
π∈Sn
symmetric n–fold tensor product E ¯s n is the range of Pn . We also use the notation
xn ¯s . . . ¯s x1 = Pn xn ¯ . . . ¯ x1 (xi ∈ E). Notice that here, unlike the Hilbert space
case, the order is important. Like in Remark 1.3 we find
(zn xn ) ¯s . . . ¯s (z1 x1 ) = zn . . . z1 (xn ¯s . . . ¯s x1 )
(1.1)
for xi ∈ CZ (E). The zi may be placed anywhere in the n–fold tensor as long as the order
is preserved. Whereas the xi in the Z–center may be written in arbitrary order.
Following [5], we define the symmetric Fock module over E
Γ(E) =
∞
M
E ¯s n
n=0
where the direct sum of (pre-)Hilbert modules is equipped with the natural operations.
To any element x ∈ E we define the creator a+ (x) by setting
√
a+ (x)xn ¯s . . . ¯s x1 = n + 1x ¯s xn ¯s . . . ¯s x1
3
and a+ (x)z = xz ∈ E ¯1 for z ∈ E ¯0 = Z. Like the creator on the usual symmetric
Fock space the creator on the symmetric Fock module has a formal adjoint a(x) called
annihilator. The creators fulfill the generalized CCR
a(x)a+ (y) − a+ (y)a(x) = hx, yi,
(1.2)
if at least one of x, y is in CZ (E); see [5] for details. Notice that the algebra element on
the right-hand side acts on Γ(E) as multiplication from the left. Also this operation has
an adjoint, namely, multiplication by hy, xi.
1.4 Example. Let G and H be Hilbert spaces. Then B(G, H) is turned into a Hilbert
B(G)–module by defining the inner product hL, M i = L∗ M . Any Hilbert Z–module may
be considered as Z–submodule of some B(G, H) where Z is identified with a C ∗ –subalgebra
of B(G) via a faithful representation; see [2, 7].
Now consider E = H ⊗ Z. Clearly, E with inner product hf ⊗ z, f 0 ⊗ g 0 i = hf, f 0 iz ∗ z 0
is a centered pre-Hilbert Z–module. If Z ⊂ B(G), we consider E as a submodule of
B(G, H ⊗ G) via identifying f ⊗ z with the mapping g 7→ f ⊗ zg. By our structure
theorem in [7] any centered Hilbert Z–module may be identified as a Z–submodule of
ˆ
ˆ denoting the complete tensor product). Observe that E ¯ E = H ⊗ H ⊗ Z
B(G, H ⊗G)
(⊗
(and similarly, E ¯n = H ⊗n ⊗ Z). It follows that the symmetric Fock module over E is a
ˆ
suitable completion of the subset Γ(H) ⊗ Z of B(G, Γ(H)⊗G).
If H is an L2 –space, then (with some attention) the elements of E may be considered
as Z–valued square-integrable functions on a measure space W . In this case the elements
of E ¯n may be identified (again with some attention) with Z–valued square-integrable
functions on W n . The elements of E ¯s n may be identified with symmetric Z–valued
square-integrable functions on W n .
Denote by La (E) the ∗–algebra of (not necessarily bounded) operators on E which
have a formal adjoint. Such operators are right Z–linear, automatically, and closeable.
Hence, if E is complete, then La (E) coincides with the ∗–algebra B a (E) of bounded
adjointable operators on E which easily is seen to be a C ∗ –algebra.
Notice that Z is a C ∗ –subalgebra of B a (E). Clearly, any element x ∈ E gives rise to a
completely positive mapping La (E) → Z via a 7→ hx, axi. In fact, any completely positive
mapping can be recovered in this way by a GNS–construction; see [3, 7]. We remark
that, if a 7→ hx, axi defines a conditional expectation, then necessarily x ∈ CZ (E). If A is
centered, then so is the A–Z–submodule of E generated by x. Let us recall Voiculescu’s
definition of a Z–random variable (restricted to the framework of ∗–algebras).
1.5 Definition [10]. A ∗–Z–algebra is a ∗–algebra A which contains Z as a ∗–subalgebra
and with 1Z = 1A = 1. A Z–quantum probability space is a ∗–Z–algebra A with a unit
4
preserving conditional expectation ϕ : A → Z (i.e. ϕ(1) = 1 and ϕ(zaz 0 ) = zϕ(a)z 0 ).
The elements of a Z–quantum probability space are called Z–random variables. The
distribution of a Z–random variable a is the restriction of ϕ to the ∗–Z–subalgebra of A
generated by a.
In other words, the distribution is the collection of all moments ϕ(z0 a# z1 a# . . . a# zn )
where a# can be a or a∗ . This is closer to Voiculescu’s original definition.
It is noteworthy that within the category of (∗–)Z–algebras it is, in general, not possible to define a natural multiplication on the tensor product. In order to overcome this
obstacle, we restrict to centered ∗–Z–algebras.
1.6 Definition [5, 6]. A (topological) ∗–Z–algebra is called centered, if it is centered as
a (topological) Z–Z–module. A Z–random variable in a Z–quantum probability space A
is called centered, if it is contained in a centered ∗–Z–subalgebra of A.
(Unfortunately, this notion is in conflict with the usual meaning of ‘centered random
variable’. For us a Z–random variable a with ϕ(a) = 0 will be called mean-zero.)
Clearly, any Z–random variable in a centered Z–quantum probability space A is centered.
On the tensor product A ¯ B of two centered ∗–Z–algebras A and B we may define
a multiplication via (a ¯ b)(a0 ¯ b0 ) = aa0 ¯ bb0 and an involution (a ¯ b)∗ = a∗ ¯ b∗ for
a, a0 ∈ CZ (A) and b, b0 ∈ CZ (B). Notice that A ¯ B and B ¯ A are isomorphic via τ ;
see [5, 6]. Obviously, A and B may be identified with the ∗–Z–subalgebras A ¯ 1 and
1 ¯ B, respectively, of A ¯ B. (This follows from the remark that the unit of a Z–algebra
is in its Z–center, automatically.) In general, nothing like this is possible, if A or B is
non-centered.
One easily checks that the ∗–Z–subalgebra of La (Γ(E)), which is generated by the
creators a+ (x) to elements x of the centered submodule of E generated by CZ (E), is a
centered ∗–Z–algebra.
¡ ¢
1.7 Definition [6]. Let Bi i∈I be a family of centered ∗–Z–subalgebras of the Z–quantum
probability space (A, ϕ) indexed by some set I. Denote ϕi = ϕ ¹ Bi . Given a finite tuple
©
ª
J
Bi and
(b1 , . . . , bn ) with b` ∈ Bi` , we define K = i` (` = 1, . . . , n) . Denote BK =
i∈K
J
ϕK =
ϕi . Finally, denote by αi (i ∈ K) the canonical embedding Bi → BK . We say
i∈K ¡ ¢
the family Bi i∈I is Bose Z–independent, if
ϕ(b1 . . . bn ) = ϕK (αi1 (b1 ) . . . αin (bn ))
¡ ¢
for all finite tuples. We say a family bi i∈I of centered (not necessarily mean-zero)
5
Z–random variables is Bose Z–independent, if the family Bi of centered ∗–Z–subalgebras
generated by bi is Bose Z–independent.
Observe that the creators and annihilators on the symmetric Fock module Γ(E ⊕ F )
to elements of E are Bose Z–independent from those to elements of F ; see [6] for details.
1.8 Example. Let H be a Hilbert space and define the Hilbert M2 –module E = H ⊗M2 .
For f ∈ H denote
Ã
Ã
!!
Ã
!
0
0
0
0
Ff+ = a+ f ⊗
= a+ (f )
1 0
1 0
Ã
and
Ff = a f ⊗
Ã
0 0
1 0
!!
= a(f )
Ã
!
0 1
0 0
where we identify a+ (f ) with a+ (f ⊗ 1). Obviously, we have Ff Fg = Ff+ Fg+ = 0 and
Ã
!
1
0
Ff Fg+ + Fg+ Ff = hf, gi
+ a+ (g)a(f ),
(1.3)
0 0
because a(f ) and a+ (g) fulfill (1.2). All monomials in operators F + and F are 0, unless
they are of alternating type, i.e.
Ff1 Ff+2 Ff3 Ff+4 . . .
or
Ff+1 Ff2 Ff+3 Ff4 . . . .
Calculating the vacuum conditional expectation h1, •1i, only the first type gives contributions different from 0. By Equation (1.3) we find
Ã
!
1
0
h1, Ff1 Ff+2 Ff3 Ff+4 . . . Ff2n−1 Ff+2n 1i =hf1 , f2 i
h1, Ff3 Ff+4 . . . Ff2n−1 Ff+2n 1i
0 0
Ã
!
1 0
= · · · = hf1 , f2 i . . . hf2n−1 , f2n i
.
0 0
In other words, the vacuum conditional expectation of a monomial in operators Ff+ and Fg
coincides up to the factor (10
00) with the vacuum expectation of the corresponding monomial
in creators and annihilators on the truncated Fock space which describe the central limit
distribution of boolean independence; see [4].
2
The central limit theorem
The proof of our central limit theorem is based on the general algebraic central limit
theorem by Speicher and von Waldenfels [9].
2.1 Theorem [9]’. Let (B, ϕ) be a Z–quantum probability space. Let J be a fixed index
(j)
set. Consider elements bi ∈ B (j ∈ J, i ∈ N), which fulfill the following assumptions.
6
³
´
(j1 )
(jn )
i) We have ϕ bσ(1) . . . bσ(n) = 0 for all n ∈ N, all (j1 , . . . , jn ) ∈ J n and all σ : {1, . . . , n} →
N with the property that there exists k ∈ N such that #σ −1 (k) = 1.
° ³
´°
°
(j1 )
(jn ) °
ii) For all n ∈ N there exists a constant Cn < ∞, such that °ϕ bσ(1)
. . . bσ(n)
° ≤ Cn
for all (j1 , . . . , jn ) ∈ J n and all (σ(1), . . . , σ(n)) ∈ Nn .
iii) We have an invariance of all second order correlations under order preserving injections, i.e.
³
ϕ
(j1 )
bϑ(σ(1))
(jn )
. . . bϑ(σ(n))
´
³
=ϕ
(j1 )
bσ(1)
(jn )
. . . bσ(n)
´
(2.1)
for all n ∈ N, all (j1 , . . . , jn ) ∈ J n , all σ : {1, . . . , n} → Im(σ) with the property
that for all k ∈ Im(σ) we have #σ −1 (k) = 2, and for all order preserving injective
mappings ϑ : Im(σ) → N.
For each N ∈ N we define
(j)
(j)
SN =
(j)
b1 + · · · + bN
√
.
N
Then we have for all n ∈ N and all (j1 , . . . , jn ) ∈ J n
¡ (j )
(j ) ¢
lim ϕ SN 1 . . . SN n = 0,
N →∞
if n is odd, and
¡ (j )
1
(j ) ¢
lim ϕ SN 1 . . . SN n = ¡ n ¢
N →∞
2
´
³
(j1 )
(jn )
,
ϕ bσ(1)
. . . bσ(n)
X
!
σ : {1,...,n}→{1,..., n
}
2
} : #σ −1 (k)=2
∀k∈{1,..., n
2
if n is even.
2.2 Remark. Replacing ‘Z–quantum probability space’ with ‘quantum probability space’,
Theorem 2.1 is more or less word by word taken from [9]. Also the proof in [9] works word
by word in our more general situation. Alternatively, one can apply the original theorem
to all states of the form ψ ◦ ϕ where ψ runs over all states on the C ∗ –algebra Z. By an
application of the principle of uniform boundedness we find the weaker result that the
¡ (j )
(j ) ¢
family lim ψ ◦ ϕ SN 1 . . . SN n of numbers determins a unique element z in the universal
N →∞
¡ (j )
(j ) ¢
enveloping von Neumann algebra of Z fulfilling ψ(z) = lim ψ ◦ ϕ SN 1 . . . SN n for all
N →∞
states ψ.
2.3 Remark. Of course, in applications of Theorem 2.1 it is supposed that the ∗–Z–algebras
(j)
Bi which are generated by bi
(j ∈ J) for a fixed
³ i ∈ N are´independent in some notion
(j1 )
(jn )
of independence. Condition (i) means that ϕ bσ(1)
. . . bσ(n)
vanishes, if at least one of
7
¡ (j) ¢
(j)
the Bi appears precisely once. In particular, it follows that ϕ bi
= 0 so that bi are
mean-zero Z–random variables.
Let M1 be a monomial in B1 and Mn the corresponding monomial in Bn . Condition
(iii) implies, in particular, that ϕ(M1 ) = ϕ(Mn ). In other words, all Bn are identically
distributed. They may be considered as copies of one and the same Z–quantum probability
space (B1 , ϕ ¹ B1 ). The full contents of Condition (iii) is that the joint distribution of
a tuple (Bi1 , . . . , Bin ) does not depend on the precise numbering as long as the order is
preserved.
In [9] it was pointed out that in the case of ‘classical’ independences (as there are
tensor independence, free independence and boolean independence) also the order does not
matter. Only some graded variants of tensor independence (for instance, q–independence
and Fermi independence) require to take care for ordering. Also in our framework we will
see that the restriction to order preserving injections ϑ in the assumptions of Theorem
2.1 is not necessary. This is so, because (see Remark 1.3 and Equation (1.1)) a change
of ordering only affects elements in the Z–center. The order of the coefficients zi is not
changed by any permutation constructed with the help of the flip τ .
We change the notions of Theorem 2.1 and bring them into a form which is more
suitable for both, to see clearly what we explained in the preceding remark and to see our
algebraic definitions at work.
¡ ¢
2.4 Definition. We say a family Bi ∈I of Bose Z–independent centered ∗–Z–algebras
in a Z–quantum probability space (A, ϕ) is identically distributed, if there is a centered
¡ ¢
Z–quantum probability space (B, ψ) and a family βi i∈I of ∗–Z–algebra isomorphisms
B → Bi (i.e. each βi is a Z–Z–linear mapping and a unital isomorphism of ∗–algebras)
such that ϕ ◦ βi = ψ for all i ∈ I. We say (B, ψ) is their marginal distribution.
In the sequel, we will concenrate on sequences, i.e. I = N. Let K ⊂ N. We use
the notation of Definition 1.7 where we, however, identify Bi with B via βi−1 . We find
J
J
BK =
B = B ¯#K and ϕK =
ψ = ψ ¯#K . Observe that both B ¯#K and ψ ¯#K only
i∈K
i∈K
depend on the number N = #K of elements in K. To avoid ambiguities we have to define
to which position i in B ¯N we assign a certain element bk ∈ Bk = B. This amounts to
choose a bijection σ : {1, . . . , N } → K such that σ −1 (k) = i. Needless to say that values
of ψ ¯N do not depend on the choice of the bijection σ which provides the identification
of BK and B ¯N . Like in Definition 1.7 we denote by αi (i = 1, . . . , N ) the canonical
embeding of B into the i–th position of B ¯N .
8
Let n ∈ N and (b(1) , . . . , b(n) ) be a tuple of elements b(j) in B and let ij ∈ K (j =
(j)
1, . . . , n). Denote bij = βij (b(j) ). By Bose Z–independence we have
³
´
¡ (1)
(n) ¢
¯N
(1)
(n)
ασ−1 (i1 ) (b ) . . . ασ−1 (in ) (b ) .
ϕ bi1 . . . bin = ψ
(2.2)
Together we the preceding consideration this is invariance of (2.1) under all injections
(j)
ϑ : N → N so that the bi
satisfy Condition (iii) from Theorem 2.1 with the finite set
J = {1, . . . , n}. Also Condition (ii) is clearly fulfilled, because by (2.2) we reduce (for
fixed and finite J) the problem of finding Cn to that of finding the maximum of a finite
set.
Now suppose that ij0 = k0 ∈ K for a certain j0 ∈ J and that ij 6= k0 , whenever j 6= j0 .
Furthermore, suppose that ψ(b(j0 ) ) = 0. We show that right-hand side of (2.2) is zero.
P
(j)
(j)
Indeed, let us decompose all b(j) into (b(j) )` z` with (b(j) )` ∈ CZ (B) and z` ∈ Z. Then
`
ψ ¯N
³
´
(1)
(n)
ασ−1 (i1 ) (b ) . . . ασ−1 (in ) (b )
³
´
X (1)
(n)
z`1 . . . z`n ψ ¯N ασ−1 (i1 ) ((b(1) )`1 ) . . . ασ−1 (in ) ((b(n) )`n ) .
=
`1 ,...,`n
Like in the tensor product of states on ∗–algebras, ψ becomes to be evaluated on (b(j0 ) )`j0
which is the only factor at position σ −1 (ij0 ) in the n–fold tensor product. We may dis(1)
(n)
(1)
(j −1)
tribute the z`1 . . . z`n in such a way that all z`1 . . . z`j 0−1 remain on the left of (b(j0 ) )`j0 ,
that all
(j +1)
z`j 0+1
0
(n)
. . . z`n
0
come to the right of (b
(j0 )
(j )
)`j0 and that all z`j 0 come together with
0
(b(j0 ) )`j0 . Clearly, the sum over `j0 gives 0. This is Condition (i) of Theorem 2.1. We just
have proved the following.
2.5 Proposition. Let b(j) (j ∈ J) be elements of B such that ψ(b(j) ) = 0 for all j ∈ J.
¡ ¢
(j)
Then, setting bi = βi b(j) (j ∈ J, i ∈ N) and using the notations of Theorem 2.1, the
¡ (j )
(j ) ¢
limit N → ∞ of ϕ SN 1 . . . SN n exists and is equal to what is stated in Theorem 2.1.
Now since we know that the central limit exists, we want to give its concrete form
as moments of creators and annihilators on a suitable symmetric Fock module. For this
aim, instead of fixing on a certain tuple (b(1) , . . . , b(n) ) of elements in B, we find it more
convenient to fix only on the order n of the monomials, but consider the expressions
appearing in Theorem 2.1 rather as Z–Z–linear mappings on B ¯n .
For each N ∈ N we define the mapping SN : B → B ¯N by
SN (b) =
α1 (b) + . . . + αN (b)
√
.
N
n
(1)
Clearly, SnN = M n ◦ S¯n
¯ . . . ¯ a(n) ) 7→ (a(1) . . . a(n) ) denotes the n–fold
N (where M : (a
multiplication in a Z–algebra) is a well-defined Z–Z–linear mapping B ¯n → B ¯N . Hence,
ψ ¯N ◦ SnN is a Z–Z–linear mapping B ¯n → B.
9
Let E be the GNS–pre–Hilbert Z–module of the conditional expectation ψ. In other
words, E is B divided by the submodule N consisting of all b for which ψ(b∗ b) = 0, and
E is equipped with the inner product hb + N, b0 + Ni = hb, b0 i. The elements of B act on
E as elements of La (E) via b(b0 + N) = bb0 + N where, of course b∗ acts as the adjoint of
b. (All these assertions follow from Cauchy-Schwartz inequality which also holds true for
non-definite inner products; see [7] for details.) In particular, E has a left action of Z.
Clearly, E is centered, because B is centered.
Let Γ(E) denote the symmetric Fock module over E. (Recall that the inner product on
Γ(E) may fail to be definite, if E is not complete.) Define the mapping A : B → La (Γ(E))
by A(b) = a+ (b) + a(b∗ ). (Here and in the sequel, we omit ‘N’.) Observe that A is
Z–Z–linear and hermitian, i.e. A(b∗ ) = A(b)∗ . Like for SN we define An = M n ◦ A¯n .
Denote by Ω the vacuum conditional expection on La (Γ(E)).
2.6 Theorem. For each n ∈ N we have
lim ψ ¯N ◦ SnN = Ω ◦ An
N →∞
pointwise on ker(B)¯n .
Proof. Since both sides are well-defined and established as Z–Z–linear mappings, its
suffices to show the statement only for elements in the Z–center. On the Z–center, however, ψ maps into the commutative ∗–algebra CZ (Z) so that we are reduced to a purely
commutative situation. The proof of our assertion follows precisely as in the well-known
case where Z = C.
2.7 Remark. Proposition 2.5 is the analogue of the central limit theorem for Z–free
Z–random variables obtained by Voiculescu in [10] where also the notion of Z–random
variables has been introduced. Theorem 2.6 is the analogue of a result by Speicher [8]
which asserts that Z–free central limit distribution may be represented by an algebra of
creators and annihilators on a full Fock module. It is clear that both results may be
recovered starting from Theorem 2.1 in a similar manner.
3
The C ∗–case
The results obtained so far are valid for algebraically centered ∗–Z–algebras. Since a
centered C ∗ –Z–algebra is only rarely centered algebraically, it is an interesting question
in how far the results extend to a C ∗ –framework. Since we are interested only in the
values of a certain conditional expectation, and since conditional expectations admit a
GNS–representation, we restrict to C ∗ –Z–algebras of operators on a Hilbert module.
10
ˆ the
3.1 Definition. Let E and F denote centered Hilbert Z–modules. Denote by E ¯F
centered Hilbert Z–module obtained by completing E ¯ F . Let A and B be centered
C ∗ –Z–subalgebras of B a (E) and B a (F ), respectively. For a ∈ A denote by a ¯ id the
ˆ ). For b ∈ B define the operator id ¯b = τ ◦ (b ¯ id) ◦ τ
operator x ¯ y 7→ ax ¯ y in B a (E ¯F
ˆ ).
in B a (E ¯F
ˆ of A and B, we mean the unital C ∗ –Z–subalgebra of
By the tensor product A¯B
ˆ ) generated by A ¯ id and id ¯B.
B a (E ¯F
3.2 Proposition. i) The mappings a 7→ a ¯ id and b 7→ id ¯b are isometric homomorˆ is a centered C ∗ –Z–algebra.
phisms of C ∗ –Z–algebras. In particular, A¯B
ii) The mapping τ : a ¯ id 7→ id ¯a for a ∈ CZ (A) and id ¯b 7→ b ¯ id for b ∈ CZ (B)
ˆ → B ¯A
ˆ of C ∗ –Z–algebras (i.e. τ is a unital
extends to a unique isomorphism A¯B
∗–algebra isomorphism and Z–Z–linear).
Proof. (i) The first mapping a 7→ a ¯ id is isometric, because Z acts faithfully on F .
The statement for the second mapping is reduced to the former by the fact that τ is an
ˆ defines
isometry E ¯ F → F ¯ E. Clearly, A ¯ B 3 a ¯ b 7→ (a ¯ id)(id ¯b) ∈ A¯B
a Z–Z–linear mapping. In particular, the generating subset (CZ (A) ¯ id)(id ¯CZ (B)) is
ˆ
contained in CZ (A¯B).
ˆ generated by CZ (A) ¯ id
(ii) Clearly, τ is isometric on the ∗–Z–subalgebra of A¯B
and id ¯CZ (B) and is a ∗–Z–algebra homomorphism. Henceforth, τ extends uniquely to
a C ∗ –Z–algebra isomorphism.
3.3 Proposition. Let E be a centered Hilbert Z–module and B a centered C ∗ –Z–subalgebra
of B a (E). Let x ∈ CZ (E) with hx, xi = 1, so that ψ(•) = hx, •xi defines a unit preserving
ˆ
ˆ
the canonical embeding and define
: B ¯n → B ¯n
conditional expectation. Denote by γ ¯n
the conditional expectation
ˆ
ˆ
ˆ
ψ ¯n (•) = hx¯n , •x¯n i
ˆ
on B ¯n
. Then
ˆ
ˆ
ψ ¯n ◦ γ ¯n = ψ ¯n .
In particular, ψ ¯n has norm 1.
Proof. Clear.
Now we are in a position to formulate all statements of Proposition 2.5 and Theorem
2.6 on the larger domain B. The rude estimate for the constants Cn in Condition (ii)
of Theorem 2.1 is improved considerably, namely, Cn = max kb(j) kn . Conditions (i)
j=1,...,n
and (iii) extend by continuity. Also the proof of Theorem 2.6 extends by a continuity
argument to the larger algebra.
11
3.4 Theorem. Proposition 2.5 and Theorem 2.6 remain true also in the case, when B is
a centered C ∗ –Z–algebra.
Proof. Consider the GNS-representation of B on a Hilbert Z–module E with a cyclic
vector x.
References
[1] Lance, E.C., Hilbert C ∗ –modules, Cambridge University Press, 1995
[2] Murphy, G.J., Positive definite kernels and Hilbert C ∗ –modules, to appear in Proc.
Edinburgh Math. Soc., 1996
[3] Paschke, W.L., Inner product modules over B ∗ –algebras, Trans. Amer. Math. Soc.
182, 443-468 (1973)
[4] Schürmann, M., Non-commutative probability on algebraic structures, Probability
measures on groups and related structures, XI (Oberwolfach, 1994), 332–356, World
Sci. Publishing, River Edge, NJ, 1995
[5] Skeide, M., Hilbert modules in quantum electro dynamics and quantum probabilty,
Commun. Math. Phys. 192 (569–604) 1998
[6] Skeide, M., A note on Bose Z–independent random variables fulfilling q–commutation
relations, Preprint, Rome, 1996
[7] Skeide, M., Generalized matrix C ∗ –algebras and representations of Hilbert modules,
Preprint, Cottbus, 1997
[8] Speicher, R., Combinatorial theory of the free product with amalgamation and
operator-valued free probability theory, Habilitation, Heidelberg, 1994, to appear
in Memoirs of the American Mathematical Society
[9] Speicher, R., von Waldenfels, W., A general central limit theorem and invariance
principle, Accardi, L. (ed.), Quantum Probability & Related Topics IX, Singapore,
New Jersey, London, Hong Kong, World Scientific, 1994
[10] Voiculescu, D., Operations on certain non-commutative operator-valued random variables, Preprint, Berkely 1992
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