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C*-Multipliers, crossed product algebras, and canonical commutation relations

The notion of a multiplier of a group X is generalized to that of a C*-multiplier by allowing it to have values in an arbitrary C*-algebra A. On the other hand, the notion of the action of X in A is generalized to that of a projective action of X as linear transformations of the space of continuous functions with compact support in X and with values in A. It is shown that there exists a one-to-one correspondence between C*-multipliers and projective actions. C*-multipliers have been used to define twisted group algebras. On the other hand, the projective action tau can be used to construct the crossed product algebra A x_tau X. Both constructions are unified in the present approach. The results are applicable in mathematical physics. The multiplier algebra of the crossed product algebra A x_tau X contains Weyl operators {W(x),x in X}. They satisfy canonical commutation relations w.r.t. the C*-multiplier. Quantum spacetime is discussed as an example.


Introduction
Starting point for this work is the observation that the construction of the crossed product of a C * -algebra with a group is very similar to the construction of twisted group algebras and of the C * -algebra of canonical commutation relations (CCR). The convolution product of the group algebra L 1 (X) of a locally compact group X can be deformed in two ways. In the first case the integrable functions of X are allowed to have values in a C * -algebra A, and the deformation involves a representation τ of X as homomorphisms of A. This is the basis for the definition of the crossed product algebra A × τ X.
In the other case a multiplier ξ : X × X → C is used, which leads to the notion of a twisted group algebra. By a slight generalization both constructs can be unified. The generalization of the crossed product algebra is obtained by replacing the representation τ by a projective action. The generalization of the twisted group algebra is obtained by allowing the multiplier to have values in an arbitrary C * -algebra A.
A short history of the crossed product algebra can be found in the introduction of [13]. Its physical importance was brought out in [4]. The twisted group algebra was studied in [7] and [1]. A special case of twisted group algebra, of relevance in mathematical physics, is the algebra of CCR, introduced in [9], and, independently, in [12]. An introductory treatment is found in [10].
An important generalization, orthogonal to the present one, is obtained by replacing the group X by a groupoid. See [2], or [11], Ch. II, Sec. 1. The combination of both generalizations is not considered here.
The structure of the paper is as follows. In the next section the main results are stated. Then theorem 1 is proved. In section 4 the crossed product algebra is constructed. Theorem 2 is proved in section 5. The final section discusses quantum spacetime as an application in quantum mechanics.
Definition 1 A C * -multiplier of a group X acting in a C * -algebra A is a map ξ of X × X into the unitary elements of the multiplier algebra M(A) of A satisfying the following axioms.
(M2) There exists a map σ of X into the automorphisms of A such that σ e is the identity transformation and one has σ x ξ(y, z) = ξ(x, y)ξ(xy, z)ξ(x, yz) * , x, y, z ∈ X (2) (M4) The map y ∈ X → aξ(x, y) is continuous for all x ∈ X and a ∈ A.
The maps x ∈ X → aξ(x, x −1 ) and x ∈ X → σ x a are continuous for all a ∈ A.
ξ is also called an A-multiplier of X.
If A = C then σ is trivial and (2) reduces to (1). If the *-algebra generated by the range of ξ is dense in M(A) then σ is uniquely determined by (2). In that case (M3) follows from (M2). In general, σ is not a group representation, but is twisted by means of the C * -multiplier, according to (3). For convenience, the map σ is called a twisted representation associated with ξ. In [1] the pair (ξ, σ) is called a twisting pair for X and A and is considered in a more general setting, with A an involutive Banach algebra, and with the continuity requirements (M4) replaced by measurability conditions.
Notation 1 Let C c (X) denote the space of complex continuous functions with compact support in X. Similarly, let C c (X, A) denote the space of continuous functions with compact support in X and values in the C * -algebra A.
Given a C * -multiplier ξ and an associated twisted representation σ, one can define a map τ of X into the linear transformations of C c (X, A) by It is straightforward to verify that τ satisfies the axioms of the following definition.
Definition 2 Let X be a locally compact Hausdorff group and let A be a C * -algebra. A map τ of X into the linear transformations of C c (X, A) is a projective action if it satisfies (A1) τ e is the identity transformation of C c (X, A).
The following characterization of C * -multipliers is proved.
Theorem 1 Let X be a locally compact Hausdorff group and A a C * -algebra. There is a one-to-one correspondence between twisting pairs (ξ, σ), consisting of an A-multiplier ξ of X and an associated twisted representation σ, and projective actions τ of X in C c (X, A).
Recall that a group action τ of X as automorphisms of A can be used to construct the crossed product algebra A × τ X. This is done by defining a multiplication and an involution on the linear space C c (X, A) to make it into an involutive algebra. The latter is then completed by closure in a C * -norm. This construction is given here in terms of projective actions using continuous functions instead of measurable functions. The projective action τ is shown to leave the C * -norm invariant.
Example 1 Let X = {0, 1} and let A = C. Then C c (X, A) coincides with C 2 . A projective action τ of X as linear transformations of C 2 is defined by for all a, b ∈ C, with α ∈ R fixed. It is easy to verify that τ is indeed a projective action. The multiplier ξ is given by ξ(1, 1) = e iα and ξ(x, y) = 1 otherwise. The crossed product algebra A × τ X equals C 2 with product law and involution a b * = a be −iα (8) It is an abelian algebra with unit 1 0 .

Ω
Recall that a representation π of A in a Hilbert space H is X-covariant if there exists a unitary representation x → U(x) of X such that π(σ x a) = U(x)π(a)U(x) * holds for all x ∈ X and a ∈ A. The crossed product algebra A × τ X has the basic property that there is a relation between its *-representations and the covariant *-representations of A. This property is preserved in the following form.
Theorem 2 Let X be a locally compact Hausdorff group. Let A be a C *algebra. Let ξ be an A-multiplier of X, σ a twisted representation of X associated with ξ, and let τ be the projective action determined by ξ and σ via (4). A can be identified with a sub-C * -algebra of the multiplier algebra M(A× τ X). There exists a map W of X into the unitary elements of M(A× τ X) satisfying and and The W (x) generalize the Weyl operators, see e.g. [10] and the discussion below.
Example 2 Let X = {1, i, j, k} be the Vierergruppe von Klein. One has i 2 = j 2 = k 2 = 1 and ij = ji = k. A cocycle ξ with values in C is defined by ξ(i, j) = ξ(j, k) = ξ(k, i) = i, ξ(j, i) = ξ(k, j) = ξ(i, k) = −i, and ξ = 1 otherwise (note that the symbol i is used with 2 different meanings). Take A = C. The crossed product algebra A × τ X is the algebra of complex 4-by-4 matrices of the form The generalized Weyl operators satisfy the same commutation relations as the Pauli matrices. In particular, A × τ X is isomorphic to the algebra of all complex 2-by-2 matrices. Ω Assume now that a *-representation π of A × τ X in a Hilbert space H with cyclic vector Ω is given. Then, by the previous theorem, there exists a map x → U(x) of X into the unitary operators of H such that x ∈ X, f ∈ A × τ X and and Indeed, it suffices to take U(x) = π(W (x)) for all x ∈ X. This shows that any *-representation of A× τ X with cyclic vector defines a (generalized) covariant representation of A. The obvious reason to apply the previous theorem in quantum mechanics is for the construction of covariant representations. It can also be used to construct representations of the CCR, as is shown below.
Recall that a symplectic space H, s is a real vector space H equipped with a symplectic form s, i.e., a real bilinear antisymmetric form which is non-degenerate: s(x, y) = 0 for all y ∈ H implies x = 0. A quantization of H is a map x ∈ H → W (x) of H into a C * -algebra satisfying the following CCR. W The unique C * -algebra generated by the unitary elements W (x), x ∈ X, is called the C * -algebra of CCR. The vector space H can be considered as an abelian group for the addition. It is locally compact for the discrete topology. A multiplier ξ of H with values in C is defined by The associated twisted representation σ is trivial. This shows that (11) is a generalization of the standard CCR. A simple but nontrivial example of this quantization is the quantum spacetime of [5], [6]. This example is worked out in the last section of the paper.

Proof of Theorem 1
First assume that ξ and σ are given, and let τ be defined by (4). The continuity (M4) is needed to show that τ x maps C c (X, A) into itself. The properties (A1) to (A4) can be verified by straightforward calculation. Note that, in order to prove that the function g of (A3) is continuous one needs again (M4).
The remainder of this section deals with the proof of theorem 1 in the other direction. It is assumed in subsequent subsections that a map τ of X into the linear transformations of C c (X, A), satisfying the axioms (A1) to (A4), is given. The study of its properties will lead to the proof that τ is a projective action associated with a C * -multiplier ξ.

Projective actions
Clearly, W a (λ) belongs to C c (X, A).
Fix a ∈ A and let us construct a function h for which (τ x h)(e) = a holds.
Let λ ∈ C c (X) be such that λ(x) = 1. Let Then by axiom (A3) one obtains This shows that for each a ∈ A there exists a function g ∈ C c (X, A) such that af = τ x g. Now let (u α ) α be an approximate unit of A and let g α be such that u α f = τ x g α . From the following estimate, obtained using (A4), follows that the elements (g α (y)) α converge to some function g. The support S(g) of g equals x −1 S(f ). The function g is continuous as is obvious from Hence g belongs to C c (X, A). Finally, from follows that f = τ x g. This ends the proof. Ω Proof As a consequence of the previous proposition one may assume that x = e. Now take f = W c (λ) with c ∈ A and λ ∈ C c (X). There follows λ(e)ac = λ(e)bc. Since c and λ are arbitrary there follows a = b. Ω

Existence of the C * -multiplier
The properties of τ proven in the previous subsection can be used to establish the existence of a C * -multiplier ξ. The proof that ξ satisfies all axioms defining a C * -multiplier is completed later on.
Proposition 2 There exists a map ξ : For a fixed z one can select a λ ∈ C c (X) for which λ(z) = 0. Let (u α ) α be an approximate unit of A, and define ξ(y, z) by That the limit converges follows from the following estimates. Let h ∈ C c (X, A) be such that τ yz h(e) = a (see the proof of proposition 1). Then, using (A2) and (A4), one obtains Hence (τ y W uα (λ)(yz)a) α forms a Cauchy net converging to λ(z)ξ(y, z)a.
It is straightforward to prove that ξ(y, z) belongs to M(A). From (A2) one obtains for arbitrary f ∈ C c (X, A) Note that ξ(y, z) does not depend on the choice of λ. Indeed, if λ(z) = 0 and µ(z) = 0 then for all f , because of (28). By corollary 1 this implies This shows that both candidates for the r.h.s. of (26) coincide, i.e. the definition of ξ does not depend on the choice of λ. Ω Proposition 3 The map ξ of the previous proposition satisfies Proof From (A4) and (25) follows Take a ∈ A and λ ∈ C c (X) arbitrarily, and let f be defined by Then (A3) implies that Equation (32) becomes |λ(yz)| ||ξ(y, z)a|| = |λ(yz)| ||a|| Since λ is arbitrary one concludes that ξ(y, z) is an isometry of A. Ω Note that the map ξ satisfies ξ(e, y) = ξ(x, e) = 1, x, y ∈ X This is an immediate consequence of (25) and corollary 1.

Existence of the associated twisted representation
One can identify A with a class of linear transformations of C c (X, A) in the following way.
These linear transformations will later on become elements of the multiplier algebra of the crossed product algebra A × τ X.
Let us now show that there exists a deformed representation σ of X in A.
Proposition 4 There exists a map σ of X into the *-homomorphisms of A such that σ e = 1, and and For each a ∈ A the map x ∈ X → σ x a is continuous.

Proof
Let λ ∈ C c (X) be such that λ(e) = 0. Define σ x by Note that the definition of σ x does not depend on the choice of λ. Indeed, using (A2), one obtains for any f ∈ C c (X, A) By corollary 1 this implies that the definition of σ x a does not depend on the choice of λ. Equation (41) shows at once that (39) holds.
Obviously, one has This proves that σ e is the identity transformation of A. Now take any f ∈ C c (X, A) and calculate with h such that W e (λ) = τ x −1 y h (this exists because of proposition 1). Using (A2) there follows This is (38). Because of (A3) the map x → τ x f (y) is continuous for all y ∈ X and f ∈ C c (X, A). Hence also the map is continuous. Finally, let us show that σ x is a *-homomorphism of A. Clearly, σ x is a linear transformation. Given a, b ∈ A , using (A2), one finds This shows that σ x is a homomorphism. Let g(x) = τ x W a * (λ)(e) * . Then by (A3) one has For any h ∈ C c (X, A) holds, using (A2) twice, By corollary 1 there follows that τ x g(x) = τ x W a (λ)(x). Hence (47) becomes This ends the proof of the proposition. Ω Corollary 2 For each a ∈ A and x ∈ X the map y → aξ(x, y) is continuous.
The function y → τ x W b (λ)(xy) is continuous. Hence y → aξ(x, y) is continuous on the set of y for which λ(y) = 0. Since λ is arbitrary overall continuity follows. Ω

Corollary 3
The map x → aξ(x, x −1 ) is continuous for all a ∈ A.

Proof
Let λ and b be as in the previous lemma. One obtains The l.h.s. of this expression is a continuous function of x because of (A3). Hence x → aξ(x, x −1 ) is continuous on the set of x for which λ(x −1 ) = 0. Since λ is arbitrary overall continuity follows. Ω Proposition 5 σ satisfies (2).

Proof
One calculates From proposition 1 follows that W e (λ) = τ y h for some h ∈ C c (X, A). Hence the previous expression becomes Now apply (A2) to obtain To obtain the latter, use is made of ξ(x, y) * τ x W e (λ)(x) = ξ(x, y) * (60) Ω Corollary 5 σ x is an automorphism of A for any x ∈ X.

Ω 4 Construction
Let be given a map τ of X into the linear transformations of C c (X, A). Assume τ satisfies the axioms (A1) to (A4).

Product and involution
Consider C c (X, A) as a linear space. In what follows a product law and an involution are defined which make C c (X, A) into a *-algebra. In the next subsection this algebra is completed in norm. In the final subsections an approximate unit and the enveloping C * -algebra are constructed. Define a product on C c (X, A) by Lemma 1 If f and g belong to C c (X, A) then also f g belongs to C c (X, A).

Proof
From assumption (A4) follows that the support of τ y g satisfies S(τ y g) = yS(g). Hence From the continuity of multiplication and inverse and the fact that S(f ) and S(g) are compact follows that also ∪ y∈S(f ) S(g) is compact. Hence f g has a compact support. Continuity of f g is obvious. Ω Proposition 7 Observe that, using (A2), Hence one concludes (64). Ω It is now easy to show that the product is associative. Indeed, one has Next define an involution of C c (X, A). The adjoint f * of a function f ∈ C c (X, A) is defined by x ∈ X (67) (the modular function of X is denoted ∆). f * belongs again to C c (X, A) by assumption (A3). One has with g(x) = τ x f (e) * . Using (A2) one proves that Hence f * * (x) = τ x g(e) * follows. Using (A3) this implies f * * (x) = f (x). Obviously, the involution is compatible with the linear structure of C c (X, A). It is also compatible with the product. Indeed, using (A2) and (A3), There follows This shows the compatibility of the involution and the multiplication.

Norm completion
A norm is defined on C c (X, A) by Let us show that this norm is compatible with the product and the involution of C c (X, A).
One concludes that the completion of C c (X, A) in the || · || 1 -norm is an involutive Banach algebra. It is denoted L 1 (X, A, τ ), the algebra of integrable functions of X with values in A. In many places, the notation L 1 (A, X) is used. The notation of [1] would read here L 1 (A, X; ξ, σ). It is however more natural to see the C * -algebra A as a generalization of the complex numbers and to adapt a notation compatible with L 1 (X, C).

Approximate unit
Let (u α ) α be an approximate unit of A. Select for each neighborhood v of e ∈ X a continuous non-negative function δ v with support in v and normalized to one: (W uα (δ v )) α,v is an approximate unit of C c (X, A). Indeed, one has The latter expression can be made arbitrary small (note that from (A3) follows that y → τ y g(x) is continuous; use further that τ y g is continuous with compact support). Using (4) one obtains Each of these two terms can be made arbitrary small. Hence (W uα (δ v )) α,v is also a right approximate unit.

The crossed product algebra
The crossed product of A with X is denoted It is defined as the enveloping C * -algebra of L 1 (X, A, τ ), see [3], Section 2.7.
The C * -norm is given by ||f || = sup{ω(f * f ) 1/2 : ω positive normalized functional of L 1 (X, A, τ )} (79) The proof that || · || is nondegenerate goes as follows. We first need Proposition 8 Fix f ∈ C c (X, A). Any state ω 0 of A extends to a positive linear form ω f of L 1 (X, A, τ ) by Proof Linearity is obvious. Positivity follows from It is straightforward to show that a constant K(f ) exists such that |ω f (g)| ≤ K(f )||g|| 1 holds for all g ∈ C c (X, A). Hence, by continuity ω f extends to a positive linear functional on L 1 (X, A, τ ). Ω

Proof of theorem 2
The following result shows that A can be identified with a sub-C * -algebra of M(A × τ X).
Proposition 11 ζ extends to an injection of A into M(A × τ X). One has (ζ a ) * = ζ a * for all a ∈ A.
Proof By continuity ζ extends to a linear transformation of A × τ X. A straightforward calculation shows that Hence ζ a belongs to the multiplier algebra of A × τ X. Ω From now on ζ is omitted, i.e. A is considered to be a sub-C * -algebra of M(A × τ X).

Proposition 12
The linear transformation W (x) defined by belongs to M(A × τ X).
Proof For all f, g ∈ A × τ X is Hence W(x) belongs to the multiplier algebra of A × τ X and one has Proof W (x) * W (x) = 1 follows from (83). In order to prove W (x)W (x) * = 1 calculate, using (88), Ω The remainder of the proof of theorem 2 is straightforward. Equation (10) follows immediately from (39) and the unitarity of W (x). Equation (11) follows from (25).

Application to Quantum Spacetime
Let M = R 4 , + denote Minkowski space and consider it as a locally compact group. Let Σ be the space of anti-symmetric 4-by-4 real matrices of the form satisfying |e| 2 = |m| 2 and e.m = ±1. It is locally compact for the topology induced by the supremum norm of linear transformations of M. A motivation for this particular choice of Σ is given in [5], [6]. Let C 0 (Σ, C) denote the C * -algebra of complex continuous functions of Σ vanishing at infinity. A bicharacter ξ : It is easy to verify that ξ is a C * -multiplier of the group M, with values in A = C 0 (Σ, C), and with associated action σ which is trivial, i.e. σ k a = a for all k ∈ M and a ∈ A. The crossed product algebra A × τ M, with τ the projective action associated with ξ, is an alternative for the algebra constructed in [6].
Let ω be a state of A × τ M for which the maps λ ∈ R → ω(g * τ λk f ) are continuous for all f, g ∈ A × τ M and k ∈ M (this is a so-called Weylstate). Let (π, H, Ω) be the GNS-representation induced by ω. Then for any k ∈ M, the map λ ∈ R → π(W (λk)) is a strongly continuous oneparameter group of unitary operators. Hence, by Stone's theorem, there exists a self-adjoint operator Q(k) which is the generator of this group. Let e µ , µ = 0, 1, 2, 3 be unit vectors of R 4 . Let Q µ ≡ Q(e µ ). On a dense domain one has Q(k) = 3 µ=0 k µ Q µ . Now calculate exp(ik µ Q µ ) exp(ik ν Q ν ) = π(W (k µ e µ )W (k ν e ν )) = π(ξ(k µ e µ , k ν e ν )W (k µ e µ + k ν e ν )) = π(ξ(k µ e µ , k ν e ν )) exp(i(k µ Q µ + k ν Q ν ) (93) Using Stone's theorem one shows that there exist self-adjoint operators R µ,ν satisfying π(ξ(k µ e µ , k ν e ν )) = e ikµkν Rµ,ν and R µ,ν = −R ν,µ Comparing (93)  The Q µ can be interpreted as the component operators of the position of a relativistic particle. They do not commute, but rather satisfy the commutation relations (96), which have been proposed in [5,6] as a simplified model for quantum spacetime. The example is rather simple in that both the group M and the C *algebra A are commutative. The present theory allows for non-commutative groups and algebras. Hence there are many possibilities for generalization. The example shows further that it is necessary to allow for C * -algebras without unit. Indeed, the continuity of k ′ → f ξ(k, k ′ ) is only true for functions f vanishing at infinity. Note that the Weyl-operators W (k) have been introduced here taking into account the usual topology of M, instead of the discrete topology. Common complaints against the algebra of the CCR are the necessity of considering the discrete topology on the underlying symplectic space, and the fact that relevant physical observables do not belong to it -see e.g. [8], IV.3.7 and the notes of IV.3.5. An alternative for the algebra of the CCR is the crossed product algebra. It takes the proper topology into account, and contains the Weyl-operators in its multiplier algebra.