A NOTE ON BOSE Z–INDEPENDENT RANDOM VARIABLES
FULFILLING Q–COMMUTATION RELATIONS
MICHAEL SKEIDE
Lehrstuhl für Wahrscheinlichkeitstheorie und Statistik,
Brandenburgische Technische Universität Cottbus,
Postfach 10 13 44, D–03013 Cottbus, Germany, E-mail:
[email protected], Homepage:
http://www.math.tu-cottbus.de/INSTITUT/lswas/ skeide.html
In Reference17 Voiculescu generalizes his notion of free independent random variables16 to the notion Z–free Z–random variables (free independence
with amalgamation over Z; see also Speicher15 ). Surprisingly, an amalgamated version of Bose independence resists to be meaningful in full generality
(roughly speaking, because there is in general no tensor product of Z–random
variables). In these short notes (a slightly revised version of the preprint10 )
we intend to do not much more than to propose a definition of Bose Z–
independence at least for so-called centered Z–random variables and to present
some examples including creators and annihlilators on the symmetric Fock
module11 and some random varibles fulfilling q–comutation relations. The
crucial notion of centered Z–random variables relies on the notion of centered
Hilbert modules11 and it can be shown that every B(G)–random variable (G
some Hilbert space) is of that type; see References13,2 . Meanwhile, we know
that every central limit distribution of Bose Z–independent Z–random variables may be represented by creators and annihilators on some symmetric
Fock module; see Reference12 .
1
Basics
Throughout these notes Z denotes a C ∗ –algebra. (Using the algebraic methods from 1 also more general ∗–algebras are possible. See Appendix C of
Reference14 for a systematic introduction to such P ∗ –algebras.) Algebras are
unital and modules are modules over algebras. All tensor products and direct sums are algebraic. The word ‘centered’ is reserved for a certain type of
two-sided module or elements in the ‘center’ of such a module (see below). A
random variable whose first moment vanishes will be called ‘mean-zero random variable’. The tensor product over Z of two Z–Z–modules E and F is
the Z–Z–module defined by E ¯ F = E ⊗ F/(xz ⊗ y − x ⊗ zy). By L and B
we denote spaces of linear and bounded linear mappings, respectively. A superscript a means (mutually) adjointable mappings. Notice that adjointable
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mappings on inner product modules are right linear automatically.
1.1 Definition. A Z–algebra is a Z–Z–module A with a Z–Z–linear multiplication M : A ¯ A → A and a Z–Z–linear unit mapping m: Z → A, such that
the associativity condition
M ◦ (M ¯ id) = M ◦ (id ¯ M )
and the unit property
M ◦ (m ¯ id) = id = M ◦ (id ¯ m)
are fulfilled. We use the notation M (a ¯ b) = ab and m(1) = 1 (i.e. 1Z =
1A = 1).
A ∗–Z–algebra is a Z–algebra with a Z–Z–anti-linear involution, i.e.
∗
(zaz 0 )∗ = z 0 a∗ z ∗ (a ∈ A; z, z 0 ∈ Z).
A Z–state is a positive normalized (i.e. ϕ(1) = 1) Z–functional (i.e. a
Z–Z–linear mapping ϕ: A → Z).
A Z–quantum probability space is a pair (A, ϕ) consisting of a Z–algebra
A and a Z–state ϕ.
For simplicity, we always assume that m is injective, so that we may
identify the subalgebra m(Z) of the algebra A with Z. In this case ϕ is
a unit-preserving conditional expectation and we recover the definitions of
Voiculescu17 .
Let (A, ϕ) be a Z–quantum probability space. Usually, a Z–random variable is a ∗–Z–algebra homomorphism j into A. By dividing out the kernel of
j we may assume that j is injective. In this case we identify the domain of j
with a ∗–Z–subalgebra of A.
1.2 Definition. A Z–random variable B is a ∗–Z–subalgebra of A.
Sometimes, we call, like in Reference15 , an element a ∈ A a Z–random
variable. Actually, we mean the ∗–Z–subalgebra Ba of A generated by a.
One main obstacle in defining Z–analogues of constructions, which work
for algebras, lies in the fact that, in general, E ¯ F has nothing in comon
with F ¯ E. In Reference11 we introduced a subcategory of the category of
Z–Z–modules which is free of this obstacle.
©
1.3 Definition. The
ª Z–center of a Z–Z–module E is the set CZ (E) = x ∈
E: xz = zx (z ∈ Z) . A Z–Z–module E is called centered, if it is generated
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by its Z–center. This means that for any x ∈ E there exist n ∈ N, xk ∈
CZ (E), zk ∈ Z (k = 1, . . . , n), such that
x=
n
X
k=1
1.4 Proposition.
center.
xk zk =
n
X
zk xk .
k=1
1. A Z–Z–linear mapping maps the Z–center into the Z–
2. Any element of a centered Z–Z–module commutes with any element of the
center of Z.
3. Consequently, a Z–Z–module over a commutative algebra Z is centered,
if and only if left and right action coincide.
4. For two centered Z–Z–modules E, F we have CZ (E)¯CZ (F ) ⊂ CZ (E¯F ).
Therefore, also E ¯ F is a centered Z–Z–module.
1.5 Theorem. Let E, F be two centered Z–Z–modules. There is a unique Z–
Z–module isomorphism F: E ¯ F → F ¯ E, called flip isomorphism, fulfilling
F (x
¯ y) = y ¯ x
for all x ∈ CZ (E) and y ∈ CZ (F ).
The proposition is obvious. The proof of the theorem can be found
in Reference11 . Within the category of centered Z–algebras (i.e. Z–algebras
whose module structure is centered) we may define the tensor product A¯B of
two Z–algebras A and B by defining the multiplication MA¯B = (MA ¯ MB ) ◦
(id¯ F ¯id) and the unit 1A¯B = 1A ¯1B . Obviously, F: A¯B → B¯A defines
a Z–algebra isomorphism. Notice also that the embeddings eA : a 7→ a ¯ 1 and
eB : b 7→ 1 ¯ b define Z–algebra homomorphisms.
1.6 Definition. Let C be a subset of a Z–algebra C. By the Z–commutant C 0
we mean the Z–subalgebra of C generated by all elements of C which commute
with all elements of C. If a Z–algebra C is its own Z–commutant C 0 , we call
it a Z–commutative Z–algebra.
Notice that C 0 is usually much bigger than the C–commutant. If C is
a centered Z–algebra and CZ (C) is a commutative subalgebra of C, then C
is Z–commutative. For instance, in Reference4 a quantum dynamical semigroup on Z is called essentially commutative, if it admits a dilation into a
Z–commutative Z–algebra.
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The following universal property is checked easily: The tensor product
of centered Z–algebras A and B is the unique centered Z–algebra A ¯ B,
such that for any pair j: A → C and k: B → C of Z–algebra homomorphisms
into a Z–algebra C, fulfilling j(A) ⊂ k(B)0 (and conversely), there exists a
unique Z–algebra homomorphism j ¯M k = M ◦ (j ¯ k): A ¯ B → C, fulfilling
(j ¯M k) ◦ eA = j and (j ¯M k) ◦ eB = k. (For C = Z we have j ¯M k = j ¯ k.)
Notice that all what we said remains true for centered ∗–Z–algebras and
∗–Z–algebra homomorphisms. In particular, we realize that j ¯ k is a ∗–Z–
algebra homomorphism, if j and k are.
1.7 Definition. A centered Z–random variable B is a centered ∗–Z–subalgebra of A.
If we are interested only in centered Z–random variables, then we may
restrict to the biggest centered Z–subalgebra Ac of A. If Ac = A, we speak of
a centered Z–quantum probability space. In this case, every a ∈ A generates
a centered ∗–Z–subalgebra Bac of A. (Observe that Bac ⊃ Ba .) Again, we call
an element a ∈ A = Ac a centered Z–random variable, but, actually, we mean
Bac . The notion of centered Z–random variable has not to be confused with
mean-zero Z–random variables, i.e. a ∈ A with ϕ(a) = 0.
1.8 Definition. The distribution ϕB of a Z–random variable B is the restriction of the Z–state
¡ ¢ ϕ on A to B.
A family Bi i∈I of centered Z–random variables in a Z–quantum probability space (A, ϕ) is called Bose Z–independent, if for an arbitrary finite
subset {i1 , . . . , in } ⊂ I
ϕ ◦ (j1 ¯M . . . ¯M jn ) = ϕBi1 ¯ . . . ¯ ϕBin ,
where ji (i ∈ I) denotes the canonical embedding Bi → A.
For Z = C this is, clearly, the usual Bose independence of quantum random variables; see e.g. Schürmann8 . We see that the mixed moments of
elements in Bose Z–independent Z–subalgebras of A equals the corresponding
moments in the tensor product of these Z–algebras in the tensor product of
their distributions. There are two other known notions of quantum independence. See Schürmann9 for a unified description of all three cases. Both of
the other notions of independence are invariant (up to algebra isomorphism)
under the exchange of their factors. However, in Reference11 we pointed out
by an example that the Z–analogue of the tensor product of algebras is not
symmetric under exchange of the factors. Actually, it does not even allow
for an obvious definition of a multiplication. This is the reason why we have
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to restrict the category under consideration. We believe that the category of
centered Z–algebras is the right one. (In this context, we neglect the rather obvious possibilities for twisted tensor products which always appear as graded
modifications of Bose independence. In such a case the modules are no longer
generated by their Z–center, but, by their even elements. In fact, the stochastic limit of the QED–Hamiltonian considered in Reference11 yields a module
of this type.)
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The symmetric Fock module
2.1 Definition. A pre-Hilbert Z–module over a C ∗ –algebra Z is a right Z–
module E with a sesquilinear inner product h•, •i: E × E → Z, such that
hx, xi ≥ 0 for x ∈ E (positivity), that hx, yzi = hx, yiz for x, y ∈ E; z ∈ Z
(right linearity), and that hx, xi = 0 implies x = 0 (strict positivity). If h•, •i
is not necessarily strictly positive, we speak of a semi-inner product and of a
semi-Hilbert Z–module. We remark that sesquilinearity and positivity imply
hx, yi = hy, xi∗ (symmetry), and that right linearity and symmetry imply
hxz, yi = z ∗ hx, yi (left anti-linearity).
A pre- (or semi-) Hilbert Z–Z–module is a two-sided Z–Z–module E which
is also a pre- (or semi-) Hilbert Z–module, such that hx, zyi = hz ∗ x, yi for
x, y ∈ E; z ∈ Z (∗–property).
For two semi-Hilbert Z–Z–modules E and F we turn the tensor prod0
0
uct
­ E ¯0 F0 ®into a semi-Hilbert Z–Z–module, by setting hx ¯ y, x ¯ y i =
y, hx, x iy .
For general reference on Hilbert modules we refer the reader to the book
of Lance5 . For an easy accessible introduction see Reference13 . See also
Reference13 for a more systematic investigation of centered semi-Hilbert Z–Z–
modules (i.e. a semi-Hilbert Z–Z–module whose module structure is centered)
also taking into account topological questions. In References13,2 we show that
a (sufficiently closed) Hilbert B(G)–B(G)–module (with a normal left multiplication) is isomorphic to a suitable closure of H ⊗ Z (see Section 3) and also
the algebra of operators on such a module is centered. Therefore, literature
(mainly dealing with B(G)) provides us with lots of centered modules.
The simple proof of the following proposition can be found in Reference13 .
2.2 Proposition. If E is a semi-Hilbert Z–Z–module, then hCZ (E), CZ (E)i
⊂ CZ (Z).
If E and F are centered semi-Hilbert Z–Z–modules, then F: E ¯F → F ¯E
is an isometry.
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2.3 Definition. Let E denote a semi-Hilbert Z–Z–module. By the full Fock
module zZ (E) over E we mean the semi-Hilbert Z–Z–module
M
E ¯n
zZ (E) =
n∈N0
(E ¯0 = Z).
On zZ (E) we define the creators `+ (x) (x ∈ E) by setting
`+ (x)xn ¯ . . . ¯ x1 = x ¯ xn ¯ . . . ¯ x1 ,
`+ (x)1 = x
and the annihilators `(x) (x ∈ E) by setting
`(x)xn ¯ . . . ¯ x1 = hx, xn ixn−1 ¯ . . . ¯ x1 ,
`(x)1 = 0.
The creator and annihilator to the same x ∈ E are adjoint elements of
Ba (zZ (E)). Moreover, we have the relations
`(x)`+ (y) = hx, yi
where the algebra element hx, yi acts as multiplication from the left.
The above definition has been introduced by Pimsner7 . However, notice
that Pimsner only considers complete modules. The first use in quantum
probability occured in Speicher15 .
The full Fock module zZ (E) is turned into a Z–algebra by setting
(xn ¯ · · · ¯ x1 )(ym ¯ · · · ¯ y1 ) = xn ¯ · · · ¯ x1 ¯ ym ¯ · · · ¯ y1
and m(b) = b ∈ E ¯0 . As a Z–algebra the full Fock module has a universal
property which parallels the universal property of the tensor algebra. See
Reference11 for details.
With the help of the flip isomorphism it is possible to define permutations
on the n–fold tensor product E ¯n of a centered Z–Z–module E in an obvious
way. Each permutation is an element of Ba (E ¯n ) with the inverse permutation
being the adjoint.
2.4 Definition. Let E be a centered semi-Hilbert Z–Z–module. We define
the number operator N in La (zZ (E)) by setting N ¹ (E ¯n ) = n. The symmetrization operator P in Ba (zZ (E)) is defined by P ¹ (E ¯n ) being the mean
over all permutations on E ¯n .
We define the symmetric Fock module ΓZ (E) over E by
√ setting ΓZ (E) =
P zZ (E). On ΓZ (E) we√define the creators a+ (x) = P N `+ (x) and the
annihilators a(x) = `(x) N P for all x ∈ E. Sometimes, we call `+ and `
the free creators and annihilators and a+ and a the symmetric creators and
annihilators, respectively.
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2.5 Remark. P is a self-adjoint projection. N is self-adjoint. Therefore,
a+ (x) and a(x) are adjoints. One easily checks P N = N P , a+ (x)P = a+ (x)
and N `+ (x) = `+ (x)(N + 1). By these relations the creators and annihilators
fulfill
a(x)a+ (y) − a+ (y)a(x) = hx, yi,
(1)
if at least one of the arguments is in the Z–center of E.
Clearly, ΓZ (E) when equipped with the multiplication
√
√
√
(P N F )(P N G) = P N (F G) (F, G ∈ zZ (E))
is a centered Z–commutative Z–algebra. As a centered Z–commutative Z–
algebra it has a universal property which parallels the universal property of
the symmetric tensor algebra. See Reference11 for details. One easily checks
the functorial property
ΓZ (E ⊕ F ) = ΓZ (E) ¯ ΓZ (F ),
by looking at elements of the center.
2.6 Theorem. The ∗–algebra A(E) of adjointable operators on ΓZ (E) generated by a+ (x) (x ∈ E) and Z is a centered Z–algebra.
Moreover, considering A(E)¯A(F ) as an algebra of operators on ΓZ (E)¯
ΓZ (F ), we have A(E ⊕ F ) = A(E) ¯ A(F ). This means, in particular, that
A(E) and A(F ) are Bose Z–independent Z–random variables in the centered
Z–quantum probability space (A(E ⊕ F ), h1, •1i).
Proof. Obviously, we have a+ (zxz 0 ) = za+ (x)z 0 (x ∈ E; z, z 0 ∈ Z). Henceforth, the ∗–algebra generated by a+ (CZ (E)) is contained in CZ (A(E)) and
generating for A(E). This proves the first assertion.
The second assertion follows by decomposition into centered elements and
by the observation that a+ (x) and a(y) commute by (1) for x ∈ CZ (E) and
y ∈ CZ (F ).
3
Examples
Suppose that z1 , z2 , Z are elements of Z which fulfill the relation
z1∗ z2 − Zz2 z1∗ = 0.
Let H be a Hilbert space and E be the centered pre-Hilbert Z–Z–module
H ⊗ Z with inner product hf ⊗ z, f 0 ⊗ z 0 i = hf, f 0 iz ∗ z 0 .
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Then for x = f ⊗ z1 and y = g ⊗ z2 we find for the Z–random variables
a+ (x) and a+ (y) in A(E) the relation
a(x)a+ (y) − Za+ (y)a(x) = hx, yi.
In the following two examples we choose for Z the C ∗ –algebra A which
underlies Woronowicz’s quantum group SUq (2) (q ∈ (−1, 1)); see Reference18 .
This algebra may be considered as the universal enveloping C ∗ –algebra generated by indeterminates
α, γ subject to all relations, which arise by requiring
´
³
α −qγ ∗
to be unitary. One finds αγ (∗) = qγ (∗) α, γγ ∗ = γ ∗ γ,
the matrix γ α∗
and 1 = α∗ α + γ ∗ γ = αα∗ + q 2 γ ∗ γ.
3.1 Example. We choose z1 = γ, z2 = α∗ and Z = q1. We find
a(x)a+ (y) − qa+ (y)a(x) = hx, yi for all f, g ∈ H. We see that it is possible to realize q–commutation relations by independent random variables.
Notice that a+ (x)a+ (y) − qa+ (y)a+ (x) = 0. Unfortunately, this example is
not symmetric under the exchange of x and y.
∗
1−γ γ
3.2 Example. We choose z1 = z2 = z = αγ and Z = q −2 1−q
2 γ ∗ γ . Then
+
+
a(x)a (y) − Za (y)a(x) = hx, yi. This example is symmetric under the
exchange x ↔ y. Hence, the whole subspace H ⊗ αγ of E classifies a family of
random variables, where an arbitrary pair fulfills Z–commutation relations,
however, with Z being not a number in C, but, an element of Z (i.e. a module
scalar). Notice also that whenever z1 = z2 = z the creators to all elements of
the subspace H ⊗ z commute among themselves.
3.3 Example. The equation z ∗ z − qzz ∗ = 0 admits no (non-trivial)psolution
z in the bounded operators, unless |q| = 1 . (If |q| 6= 1, then kzk 6= |q| kzk,
unless kzk = 0, i.e. z = 0.)
However, for q > 0 there exists an unbounded
¡ ¢ solution. Consider ak preHilbert space h with orthonormal Hamel basis ek k∈Z . Then zek = q − 2 ek−1
defines a solution.
Like indicated in Reference11 , we can generalize our notions to pre-Hilbert
modules over arbitrary operator ∗–algebras. Therefore, in this generalized
framework there exist families of random variables which fulfill the original
q–commutation relations for q > 0.
In Reference3 Bozejko and Speicher introduced a new inner product on
the full Fock space which made the creators and their formal adjoints fulfill
q–commutation relations. However, in Reference6 van Leeuwen and Maassen
showed that the ‘joint distribution’ of these ‘quantum random variables’ is not
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determined, as it should be, by their separate distributions. The authors of
Reference6 conclude from this fact that the convolution on the q–Fock space
in Reference3 is not an example for a q–convolution. Our examples show
possibilities of how to write down Bose Z–independent Z–random variables
which fulfill q–commutation releations. However, the relations are due to
the choice of elements. The convolution which arises is the generalization
of the usual convolution of states to the convolution of Z–states on centered
Z–algebras, not a q–convolution.
Recall also that Z–states map into Z. It seems, therefore, difficult to
compare our methods with other existing convolutions for states with values
in C. Notice, however, that for a Z–quantum probability space (A, ϕ) we
obtain many states on A by considering ψ ◦ ϕ, where ψ may be an arbitrary
state on Z. By (ψ◦ϕ1 )¯(ψ◦ϕ2 ) := ψ◦(ϕ1 ¯ϕ2 ) we may define a ‘convolution’ of
the states ψ ◦ ϕi (i = 1, 2). We consider it as an interesting question, wether
other convolutions can be recovered in this manner. We know that this is
possible at least for the central limit distribution of boolean independence;
see Reference12 .
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