This paper extends the Stone-von Neumann Theorem to $ C^{st} $-dynamical systems involving abelian groups and compact operators, using Hilbert $ C^{st} $-modules and introducing new commutation relations.
Contribution
It introduces a covariant version of the Stone-von Neumann Theorem for abelian group actions on $ C^{st} $-algebras of compact operators, utilizing Hilbert $ C^{st} $-modules.
Findings
01
Proves a uniqueness theorem for $ C^{st} $-dynamical systems with abelian groups.
02
Develops a representation of Weyl relations on Hilbert $ C^{st} $-modules.
03
Simplifies the proof of Takai-Takesaki Duality using new Hilbert $ C^{st} $-module results.
Abstract
In this paper, we formulate and prove a version of the Stone-von Neumann Theorem for every C∗-dynamical system of the form (G,K(H),α), where G is a locally compact Hausdorff abelian group and H is a Hilbert space. The novelty of our work stems from our representation of the Weyl Commutation Relation on Hilbert K(H)-modules instead of just Hilbert spaces, and our introduction of two additional commutation relations, which are necessary to obtain a uniqueness theorem. Along the way, we apply one of our basic results on Hilbert C∗-modules to significantly shorten the length of Iain Raeburn's well-known proof of Takai-Takesaki Duality.
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Full text
The Covariant Stone-von Neumann Theorem for Actions of Abelian Groups on C∗-Algebras of Compact Operators
Leonard Huang
Leonard Huang, Department of Mathematics & Statistics, University of Nevada, Reno, 1664 N. Virginia Street, Reno, NV 89557
In this paper, we formulate and prove a version of the Stone-von Neumann Theorem for every C∗-dynamical system of the form (G,K(H),α), where G is a locally compact Hausdorff abelian group and H is a Hilbert space. The novelty of our work stems from our representation of the Weyl Commutation Relation on Hilbert K(H)-modules instead of just Hilbert spaces, and our introduction of two additional commutation relations, which are necessary to obtain a uniqueness theorem. Along the way, we apply one of our basic results on Hilbert C∗-modules to significantly shorten the length of Iain Raeburn’s well-known proof of Takai-Takesaki Duality.
2010 Mathematics Subject Classification:
46L08, 47L55, 46L60, 81S05
1. Introduction
One of the most famous mathematical results in quantum mechanics is the Stone-von Neumann Theorem. Informally, the theorem establishes the physical equivalence of Werner Heisenberg’s matrix mechanics and Erwin Schrödinger’s wave mechanics, which was seen by Heisenberg to be an outstanding problem in the early days of quantum mechanics ([7]). The theorem was an attempt to prove that any pair (A,B) of self-adjoint unbounded operators on a Hilbert space H that satisfies the Heisenberg Commutation Relation on a common dense invariant subset D of their domain, i.e.,
[TABLE]
is unitarily equivalent to a direct sum of copies of (X,Pℏ), which are self-adjoint unbounded operators on L2(R) defined as follows:
[TABLE]
We recall that W1,2(R) denotes the space of weakly-differentiable square-integrable functions on R whose weak derivative is also square-integrable. This statement about unbounded operators is not true in general, where complications arise from domain issues — a well-known counterexample involving the Hilbert space of L2-functions on a two-sheeted Riemann surface is given in [12].
The Stone-von Neumann Theorem was first given a rigorous formulation by Marshall Stone in 1930 ([19]), and it was this formulation that
John von Neumann proved in 1931 ([20]). The exponentiated form of the Heisenberg Commutation Relation, called the Weyl Commutation Relation, is investigated in these papers, because it involves only one-parameter unitary groups. More precisely, a pair (R,S) of strongly-continuous one-parameter unitary groups on a Hilbert space H satisfies the Weyl Commutation Relation if and only if
[TABLE]
von Neumann proved that any such pair is unitarily equivalent to a direct sum of copies of (U,V), where U and V are strongly-continuous one-parameter unitary groups on L2(R) defined by
[TABLE]
Basically, U acts by translations, and V acts by phase modulations.
The statement of the Stone-von Neumann Theorem has undergone major revisions in the decades since its initial formulation. George Mackey appears to have been the first to recognize its generalization to second-countable locally compact Hausdorff abelian groups, in [10]. Nowadays, his generalization is treated as part of his theory of induced representations of locally compact Hausdorff groups, and is generally considered the standard modern formulation of the Stone-von Neumann Theorem, which we now state.
Theorem 1**.**
Let G be a locally compact Hausdorff abelian group. If R and S are strongly-continuous unitary representations of G and G, respectively, on a Hilbert space H that satisfy the Weyl Commutation Relation, i.e.,
[TABLE]
then (H,R,S) must be unitarily equivalent to a direct sum of copies of (L2(G),UG,VG), where UG denotes the unitary representation of G on L2(G) by left translations, and VG denotes the unitary representation of G on L2(G) by phase modulations, i.e.,
[TABLE]
for all x∈G, φ∈G, and f∈L2(G).
The work of Marc Rieffel in [16, 17] has revealed that Theorem 1 is actually a statement about the Morita equivalence of the C∗-algebras C and C∗(G,C0(G),lt), where lt denotes the strongly-continuous action of G on C0(G) by left translations. The theorem thus acquires a more algebraic flavor. This Morita equivalence is a special case of a more general result known as Green’s Imprimitivity Theorem, which we actually need to prove our covariant generalization of Theorem 1.
Several generalizations of the Stone-von Neumann Theorem can be found in the literature. For example, [4] extends the theorem to measurable unitary representations of G and G on a Hilbert space, and [13] extends the theorem to Hecke pairs using the machinery of non-abelian duality. Although these generalizations are non-trivial and interesting, their use of only Hilbert-space representations is a common limiting feature.
In this paper, we provide not another incremental generalization of the Stone-von Neumann Theorem, but a complete paradigm shift that significantly augments the theorem’s range of applicability. By leaving the realm of Hilbert spaces and working with representations on Hilbert C∗-modules, we show that the Stone-von Neumann Theorem is not really about representations of locally compact Hausdorff abelian groups on Hilbert spaces, but is really about representations of C∗-dynamical systems on Hilbert C∗-modules. More precisely, for every C∗-dynamical system of the form (G,K(H),α), where G is a locally compact Hausdorff abelian group and H is a Hilbert space, our covariant generalization classifies up to unitary equivalence all quadruples (X,ρ,R,S) with the following properties:
•
X is a non-trivial Hilbert K(H)-module.
•
R and S are strongly-continuous unitary representations of G and G, respectively, on X that satisfy the Weyl Commutation Relation.
•
ρ is a non-degenerate ∗-representation of K(H) on X that obeys the following commutation relations:
[TABLE]
for all x∈G, φ∈G, and a∈A. These relations are also called covariance relations.
Using results on non-abelian duality, one could very well generalize our covariant version of the Stone-von Neumann Theorem to non-abelian C∗-dynamical systems, or even quantum-group dynamical systems, but such an undertaking would take us too far afield, so we content ourselves with presenting only the abelian case, which we feel is already a significant advance. Further generalizations will be explored in a sequel.
This paper is organized as follows:
•
Section 2 is a short preliminary section that recalls some concepts and results about C∗-crossed products that we need. In particular, we show how to associate a Hilbert C∗-module to a C∗-dynamical system in a canonical way. This Hilbert C∗-module is featured in Green’s Imprimitivity Theorem, and is crucial to a formulation of our covariant generalization of the Stone-von Neumann Theorem.
•
Section 3 introduces Heisenberg modular representations and the Schrödinger modular representation of an abelian C∗-dynamical system (G,A,α). These concepts allow an efficient formulation of our covariant generalization of the Stone-von Neumann Theorem. We construct an injective map from the class of all Heisenberg modular representations of (G,A,α) to the class of all covariant modular representations of (G,C0(G,A),lt⊗α), which is a C∗-dynamical system that plays a pivotal role in Iain Raeburn’s proof of Takai-Takesaki Duality. A basic result in this section allows us to significantly shorten his proof.
•
Section 4 provides an overview of the properties of Hilbert K(H)-modules that have been established in [2, 3]. Hilbert K(H)-modules obviously generalize Hilbert spaces, yet they behave very much like Hilbert spaces, which makes them very desirable to work with.
•
Section 5 contains our main result: the Covariant Stone-von Neumann Theorem. We give a statement of Green’s Imprimitivity Theorem, and explain its relevance to the main result.
•
Section 6 poses some open questions that this paper was unable to answer. It also suggests new avenues of research that would be of interest to both mathematicians and physicists.
•
Finally, an appendix contains proofs of two results, stated in the main body of this paper, that would be considered folklore, but for which we were unable to locate adequate references.
We assume that the reader has a reasonable working knowledge of C∗-algebras, C∗-dynamical systems, and Hilbert C∗-modules. Throughout this paper, we adopt the following notations and conventions:
•
N denotes the set of positive integers, and for each n∈N, let [n]=dfN≤n.
•
For a set I, let Fin(I) denote the set of finite subsets of I.
•
For a locally compact Hausdorff abelian group G, let G denote its Pontryagin dual.
•
For a locally compact Hausdorff space X and a normed vector space V, let ⋄:C0(X)×V→C0(X,V) be defined by
[TABLE]
Note that ⋄ takes Cc(X)×V to Cc(X,V).
•
For a Hilbert space H and vectors v,w∈H, let ∣v⟩⟨w∣ denote the rank-one operator on H defined by
[TABLE]
•
For a Hilbert space H and a closed subspace K of H, let ProjH,K denote the orthogonal projection of H onto K.
•
For a C∗-algebra A, let A∼ denote its minimal unitization.
•
For a C∗-algebra A and a∈A, let σA(a) denote the spectrum of a.
•
For a C∗-algebra A, let ≤A denote the usual partial order on the cone A≥ of positive elements of A.
•
For a C∗-algebra A, and Hilbert A-modules X and Y, the set of adjointable/compact/unitary operators from X to Y is denoted by L(X,Y)/K(X,Y)/U(X,Y). If X=Y, then we write L(X)/K(X)/U(X).
•
For a C∗-algebra A and a Hilbert C∗-module X (not necessarily over A), a ∗-representation of A on X is a C∗-homomorphism ρ:A→L(X), which is then said to be non-degenerate if and only if
[TABLE]
•
For a locally compact Hausdorff group G and a Hilbert C∗-module X, a unitary representation of G on X is a group homomorphism R from G to the group U(X) of unitary adjointable operators on X, which is then said to be strongly continuous if and only if the map
[TABLE]
is continuous for each ζ∈X.
2. Preliminaries
As C∗-crossed products will be used extensively in this paper, let us recall some concepts in this area.
Throughout this section, we shall fix an arbitrary C∗-dynamical system (G,A,α), with G not assumed to be abelian, and we shall fix a Haar measure μ on G.
Recall that the C-vector space Cc(G,A) can be given a convolution ⋆G,A,α and an involution ∗G,A,α by
[TABLE]
where ΔG denotes the modular function of G.
Definition 1**.**
A (G,A,α)-covariant modular representation is a triple (X,ρ,R) with the following properties:
(1)
X is a Hilbert C∗-module (not necessarily over A).
2. (2)
ρ is a non-degenerate ∗-representation of A on X.
3. (3)
R is a strongly-continuous unitary representation of G on X.
4. (4)
R(x)ρ(a)=ρ(αx(a))R(x) for all x∈G and a∈A.
Covariant modular representations are used in the construction of C∗-crossed products. Given a (G,A,α)-covariant modular representation (X,ρ,R), we can define an algebraic ∗-homomorphism ΠX,ρ,R, called the integrated form of (X,ρ,R), from the convolution ∗-algebra (Cc(G,A),⋆G,A,α,∗G,A,α) to L(X) by
[TABLE]
The full crossed productC∗(G,A,α) is defined as the C∗-algebraic completion of (Cc(G,A),⋆G,A,α,∗G,A,α) with respect to the universal norm ∥⋅∥(G,A,α),u given by
[TABLE]
for all f∈Cc(G,A). This norm is well-defined as it is dominated by the L1-norm on Cc(G,A).
We let η(G,A,α) denote the canonical dense linear embedding of Cc(G,A) into C∗(G,A,α), and if A=C, in which case α is necessarily trivial, we simply write ηG.
For a (G,A,α)-covariant modular representation (X,ρ,R), we denote by ΠX,ρ,R the extension of ΠX,ρ,R to a C∗-homomorphism from C∗(G,A,α) to L(X).
Lemma 1**.**
Let x∈G and a∈A. Then we can find nets (fi)i∈I and (gj)j∈J in Cc(G,A) such that for any (G,A,α)-covariant modular representation (X,ρ,R), the associated nets (ΠX,ρ,R(fi))i∈I and (ΠX,ρ,R(gj))j∈J strongly converge in L(X), respectively, to R(x) and ρ(a).
A proof of this lemma will be provided in the appendix.
To (G,A,α), one can associate a special Hilbert A-module, denoted by L2(G,A,α), in a canonical manner. Observe that Cc(G,A) is a pre-Hilbert A-module, whose right A-action ∙ and A-valued pre-inner product [⋅∣⋅]:Cc(G,A)×Cc(G,A)→A are defined as follows:
•
ϕ∙a=df{Gx→↦Aϕ(x)αx(a)} for all a∈A and ϕ∈Cc(G,A).
•
[ϕ∣ψ]=df∫Gαx−1(ϕ(x)∗ψ(x))dμ(x) for all ϕ,ψ∈Cc(G,A).
Define L2(G,A,α) to be the Hilbert A-module obtained by completing Cc(G,A) with respect to the norm induced by [⋅∣⋅]. Let q(G,A,α):Cc(G,A)↪L2(G,A,α) denote the canonical dense linear embedding, and if no confusion can arise, we will omit the subscript and simply write q.
This Hilbert A-module is the linchpin of our formulation of the covariant Stone-von Neumann Theorem.
We will use q(G,A,α) when defining operators on L2(G,A,α) as a way of emphasizing that unless A=C, the elements of L2(G,A,α) are generally not functions of any sort from G to A.
3. Modular Representations
Throughout this section, we shall fix an arbitrary C∗-dynamical system (G,A,α) with G abelian. We shall also fix a Haar measure μ on G.
Definition 2**.**
A (G,A,α)-Heisenberg modular representation is a quadruple (X,ρ,R,S) with the following properties:
(1)
X is a full Hilbert A-module.
2. (2)
ρ is a non-degenerate ∗-representation of A on X.
3. (3)
R is a strongly continuous representation of G on X.
4. (4)
S is a strongly continuous representation of G on X.
5. (5)
S(φ)R(x)=φ(x)⋅R(x)S(φ) for all x∈G and φ∈G.
Hence, (R,S) satisfies the Weyl Commutation Relation for G on X.
6. (6)
R(x)ρ(a)=ρ(αx(a))R(x) for all x∈G and a∈A.
Hence, (X,ρ,R) is a (G,A,α)-covariant modular representation.
7. (7)
S(φ)ρ(a)=ρ(a)S(φ) for all φ∈G and a∈A.
Hence, (X,ρ,S) is a (G,A,ι)-covariant modular representation, with ι denoting the trivial action of G on A.
With the aim of producing an example of a (G,A,α)-Heisenberg modular representation, let us first equip L2(G,A,α), as defined in the Preliminaries, with the following structural data:
•
A ∗-representation M(G,A,α) of A on L2(G,A,α) such that for all a∈A and ϕ∈Cc(G,A),
[TABLE]
•
A unitary representation U(G,A,α) of G on L2(G,A,α) such that for all x∈G and ϕ∈Cc(G,A),
[TABLE]
•
A unitary representation V(G,A,α) of G on L2(G,A,α) such that for all φ∈G and ϕ∈Cc(G,A),
[TABLE]
Proving that these representations are well-defined is a routine exercise. We refer the reader to Chapter 4 of [22] for details. We will omit supscripts and simply write M, U, and V if no confusion arises from doing so.
Definition 3**.**
(L2(G,A,α),M,U,V) is called the (G,A,α)-Schrödinger modular representation.
Proposition 1**.**
(L2(G,A,α),M,U,V)* is a (G,A,α)-Heisenberg modular representation.*
Proof.
We will verify the various axioms in Definition 2.
The fullness of L2(G,A,α) as a Hilbert A-module
Fix an a∈A. Using Urysohn’s Lemma, find a φ∈Cc(G,R≥0) such that ∫Gφ(x)dμ(x)=1. Let (eλ)λ∈Λ be an approximate identity for A. Define ϕ∈Cc(G,A) and a net (ψλ)λ∈Λ in Cc(G,A) by
[TABLE]
Then for all λ∈Λ,
[TABLE]
Hence, λ∈Λlim⟨q(ϕ)∣q(ψλ)⟩L2(G,A,α)=λ∈Λlimaeλ=a. As a∈A is arbitrary, we get
[TABLE]
which proves that L2(G,A,α) is a full Hilbert A-module.
The non-degeneracy of M
Fix ϕ∈Cc(G,A). As Range(ϕ)⊆ϕ[Supp(ϕ)]∪{0A}, and as Supp(ϕ) is a compact subset of G, we see that Range(ϕ) is contained in a compact subset of A. Compact subsets of metric spaces are separable, and subsets of separable subsets of metric spaces are separable, so in particular, Range(ϕ) is a separable subset of A. Let D be a countable dense subset of Range(ϕ). If B denotes the C∗-subalgebra of A generated by Range(ϕ), then B is also the C∗-subalgebra of A generated by D. Hence, B is a separable C∗-algebra, which means that it possesses a sequential approximate identity (en)n∈N norm-bounded by 1. Now, for all n∈N,
[TABLE]
Next, notice for all n∈N and x∈G that
[TABLE]
Hence, ({Gx→↦R≥0∥ϕ(x)−enϕ(x)∥A2})n∈N is dominated by the integrable function {Gx→↦R≥04∥ϕ(x)∥A2}, and as it converges pointwise to 0G→R≥0, the Lebesgue Dominated Convergence Theorem yields
[TABLE]
Finally, an 3ϵ-argument shows that for any Φ∈L2(G,A,α) and any ϵ>0, there exists an a∈A such that ∥Φ−[M(a)](Φ)∥L2(G,A,α)<ϵ. Therefore, M is non-degenerate.
The strong continuity of U
Fix ϕ∈Cc(G,A) and ϵ>0. We will show that there exists an open neighborhood W of eG in G such that
[TABLE]
Observe for all x∈G that
[TABLE]
Let K be a compact neighborhood of eG in G, and let x∈K. Then
[TABLE]
for the following reasons:
•
Supp(ϕ)⊆KSupp(ϕ), so ϕ(y)=0A for all y∈G∖KSupp(ϕ).
•
For all y∈G∖KSupp(ϕ), we have x−1y∈G∖Supp(ϕ), so ϕ(x−1y)=0A; if x−1y∈Supp(ϕ), then y∈xSupp(ϕ)⊆KSupp(ϕ), which is a contradiction.
It follows that for all x∈K,
[TABLE]
Find G-indexed families (Uy)y∈G and (Vy)y∈G of open subsets of G with the following properties:
•
Uy is an open neighborhood of eG in G contained in K for each y∈G.
•
Vy is an open neighborhood of y in G for each y∈G.
•
For each y∈G, we have
[TABLE]
As KSupp(ϕ) is a compact subset of G, and as {Vy∣y∈G} covers KSupp(ϕ), there exists a finite subset F of G such that {Vy∣y∈F} covers KSupp(ϕ). Let x∈W=dfy′∈F⋂Uy′⊆K, and let y∈KSupp(ϕ). Then (x,y)∈Uy′×Vy′ for some y′∈F, so
[TABLE]
As y∈KSupp(ϕ) is arbitrary, this implies that
[TABLE]
Hence, for all x∈W,
[TABLE]
An 3ϵ-argument shows that for any Φ∈L2(G,A,α) and any ϵ>0, there exists an open neighborhood O of eG in G such that ∥Φ−[U(x)](Φ)∥L2(G,A,α)<ϵ for all x∈O. Therefore, U is strongly continuous.
The strong continuity of V
Fix ϕ∈Cc(G,A) and ϵ>0. We will show that there exists an open neighborhood W of eG in G such that
[TABLE]
Observe for all φ∈G that
[TABLE]
The topology on G is the compact-open topology, i.e., is given by uniform convergence on compact subsets of G, so we can pick an open neighborhood W of eG such that for all φ∈W and x∈Supp(ϕ),
[TABLE]
Then we have for all φ∈W that
[TABLE]
Hence, ∥q(ϕ)−[V(φ)](q(ϕ))∥L2(G,A,α)<ϵ for all φ∈W.
An 3ϵ-argument shows that for any Φ∈L2(G,A,α) and any ϵ>0, there exists an open neighborhood O of eG in G such that ∥Φ−[V(φ)](Φ)∥L2(G,A,α)<ϵ for all φ∈O. Therefore, V is strongly continuous.
(U,V) satisfies the Weyl Commutation Relation for G on L2(G,A,α)
Observe for all x∈G, φ∈G, and ϕ∈Cc(G,A) that
[TABLE]
so by continuity, V(φ)U(x)=φ(x)⋅U(x)V(φ) for all x∈G and φ∈G.
(L2(G,A,α),M,U) is a (G,A,α)-covariant modular representation
Observe for all x∈G, a∈A, and ϕ∈Cc(G,A) that
[TABLE]
so by continuity, U(x)M(a)=M(αx(a))U(x) for all x∈G and a∈A.
(L2(G,A,α),M,V) is a (G,A,ι)-covariant modular representation
Observe for all φ∈G, a∈A, and ϕ∈Cc(G,A) that
[TABLE]
so by continuity, V(φ)M(a)=M(a)V(φ) for all φ∈G and a∈A.
∎
The ultimate goal of this section is to establish the following proposition, which we presently state in an imprecise form.
Proposition 2**.**
There is an injective map from the class of (G,A,α)-Heisenberg modular representations to the class of (G,C0(G,A),lt⊗α)-covariant modular representations, where lt denotes the action of G on C0(G,A) by left translations, and lt⊗α denotes the action of G on C0(G,A) defined by
[TABLE]
The proposition is imprecisely stated because we have not yet specified what the injective map is, but this will be explicated in due course.
The main tool for proving the proposition is a C∗-algebra-valued version of the Fourier transform, which we will introduce soon. In order to show that this generalized Fourier transform is well-defined, the following approximation lemma is indispensable.
Lemma 2**.**
Let X be a locally compact Hausdorff space, V a normed vector space, and D a dense subset of V. Then for any f∈Cc(X,V) and ϵ>0, there exist φ1,…,φn∈Cc(X) and v1,…,vn∈D such that
[TABLE]
If ν is a regular Borel measure on X, then for any f∈Cc(X,V) and ϵ>0, there exist φ1,…,φn∈Cc(X) and v1,…,vn∈D such that
[TABLE]
This is a folklore result that can be straightforwardly proven using partitions of unity. To avoid disrupting the flow of this paper, we will provide a proof of it in the appendix.
Definition 4**.**
The A-valued generalized Fourier transform for G with respect to a Haar measure ν on G is the map FνA:Cc(G,A)→C0(G,A) defined by
[TABLE]
We proceed to demonstrate the consistency of this definition.
When A=C, it is not at all obvious why the image of FνA should be in C0(G,A). To see this, let us pick f∈Cc(G,A). For every x∈G, the integrand of ∫Gx(φ)⋅f(φ)dν(φ) belongs to Cc(G,A), so the integral exists. Furthermore, for all x∈G,
[TABLE]
so FνA(f) is a function from G to A that is pointwise-bounded by ∥f∥ν,1. To check that it is also continuous, fix x∈G and ϵ>0. As Supp(f) is a compact subset of G with respect to the compact-open topology on C(G), the Arzelà-Ascoli Theorem says that Supp(f) is an equicontinuous subset of C(G), so there exists an open neighborhood U of x in G such that for all y∈U and φ∈Supp(f),
[TABLE]
Consequently, for all y∈U,
[TABLE]
As x∈G is arbitrary, this proves that FνA(f) is continuous, so the image of FνA is contained in Cb(G,A).
Now, given an f∈Cc(G) and an a∈A, we have for all x∈G that
[TABLE]
As we already know that FνC(f)∈C0(G), we get FνA(f⋄a)=FνC(f)⋄a∈C0(G,A). Hence, as f∈Cc(G) and a∈A are arbitrary, we obtain
[TABLE]
Let f∈Cc(G,A). Lemma 2 makes it possible to find a sequence (fn)n∈N in Span(Cc(G)⋄A) such that n→∞lim∥f−fn∥ν,1=0. Then because FνA(f−fn)Cb(G,A)≤∥f−fn∥ν,1 for all n∈N, we get
[TABLE]
However, as seen above, FνA(fn)∈C0(G,A) for all n∈N, so because C0(G,A) is complete with respect to the supremum norm, it follows that FνA(f)∈C0(G,A). As f∈Cc(G,A) is arbitrary, we have proven that FνA maps Cc(G,A) to C0(G,A).
Knowing now that FνA:Cc(G,A)→C0(G,A) is well-defined, our next step is to show the following.
Proposition 3**.**
FνA* extends to a C∗-isomorphism FνA:C∗(G,A,ι)→C0(G,A).*
Proof.
It is routine to check that FνA:(Cc(G,A),⋆G,A,ι,∗⋆G,A,ι)→C0(G,A) is a ∗-homomorphism. As we know that FνA is contractive with respect to ∥⋅∥ν,1, the theory of C∗-crossed products says that FνA extends to a C∗-homomorphism FνA:C∗(G,A,ι)→C0(G,A).
By Lemma 2.73 of [22] and the theory of C∗-tensor products, we have the series of C∗-isomorphisms
[TABLE]
which are implemented as follows: For all f1,…,fn∈Cc(G) and a1,…,an∈A,
[TABLE]
However, FνA(i=1∑nfi⋄ai)=i=1∑nFνC(fi)⋄ai, so FνA agrees with some C∗-isomorphism from C∗(G,A,ι) to C0(G,A) on a dense subset. It is therefore precisely that C∗-isomorphism.
∎
Definition 5**.**
For a (G,A,ι)-covariant modular representation (X,ρ,S), let
[TABLE]
which is a non-degenerate ∗-representation of C0(G,A) on X.
Finally, let us tackle the main objective of this section.
Fixing a Haar measure ν on G, we divide the proof into two parts.
Defining the desired class map
Observe for all x,y∈G and f∈Cc(G,A) that
[TABLE]
Hence, (lt⊗α)x(FνA(f))=FνA(x−1⋅(αx∘f)) for all x∈G and f∈Cc(G,A).
Given a (G,A,α)-Heisenberg modular representation (X,ρ,R,S), we will exploit the computation above to show that (X,πνX,ρ,S,R) is a (G,C0(G,A),lt⊗α)-covariant modular representation.
Firstly, ρ is a non-degenerate ∗-representation of A on X, so ΠX,ρ,S is a non-degenerate ∗-representation of C∗(G,A,ι) on X, which, in turn, means that πνX,ρ,S is a non-degenerate ∗-representation of C0(G,A) on X. Secondly, we have for all x∈G and f∈Cc(G,A) that
[TABLE]
As the image of FνA is dense in C0(G,A), it follows from continuity that for all x∈G and g∈C0(G,A),
[TABLE]
Hence, (X,πνX,ρ,S,R) is a (G,C0(G,A),lt⊗α)-covariant modular representation. We can thus define a map from the class of (G,A,α)-Heisenberg modular representations to the class of (G,C0(G,A),lt⊗α)-covariant modular representation according to the rule
[TABLE]
Injectivity of the class map
Let (X1,ρ1,R1,S1) and (X2,ρ2,R2,S2) be (G,A,α)-Heisenberg modular representations such that
[TABLE]
Clearly, X1=X2, R1=R2, and πνX1,ρ1,S1=πνX2,ρ2,S2. Hence,
[TABLE]
which yields ΠX1,ρ1,S1=ΠX2,ρ2,S2. By Lemma 1, ρ1=ρ2 and S1=S2. Therefore, the proposed class map is indeed injective.
∎
Actually, one can show that the image of the class map above is the class of (G,C0(G,A),lt⊗α)-covariant modular representations whose underlying Hilbert C∗-module is a full Hilbert A-module. However, we will have no need of this fact.
4. Hilbert K(H)-Modules
In this section, we take a brief excursion into Hilbert K(H)-modules. The initial material can be found in [2, 3], but we have decided to supply our own proofs, some of which are simpler than the original ones.
Throughout this section, we shall fix a non-trivial Hilbert space H.
Lemma 3**.**
Let P be a rank-one projection on H. Then there exists a unique linear functional f on K(H) such that PSP=f(S)⋅P for all S∈K(H).
Proof.
Let K denote the range of P, which is a one-dimensional subspace of H. Given S∈K(H), the map (PSP)∣K is a linear operator on K, so it is a unique scalar multiple of IdK=P∣K. Hence, there is a unique function f:K(H)→C such that (PSP)∣K=f(S)⋅P∣K for all S∈K(H). Then
[TABLE]
for all S∈K(H) and v∈H, which means that PSP=f(S)⋅P. Finally, observe for all S,T∈K(H) and λ∈C that
[TABLE]
As f(S+λ⋅T) is the unique κ∈C for which P(S+λ⋅T)P=κ⋅P, we get f(S+λ⋅T)=f(S)+λf(T). Therefore, f is the unique linear functional on K(H) such that PSP=f(S)⋅P for all S∈K(H).
∎
Let P1 denote the set of all rank-one projections on H. Lemma 3 says that there exists a P1-indexed family (fP)P∈P1 of linear functionals on H such that
[TABLE]
These linear functionals play a pivotal role in the next result.
Theorem 2**.**
Let X be a non-trivial Hilbert K(H)-module, and P a rank-one projection on H. Then X∙P is a non-trivial closed subspace of X∙P that has the structure of a Hilbert space, whose inner product ⟨⋅∣⋅⟩X∙P is given by
[TABLE]
Furthermore, the norm on X∙P induced by ⟨⋅∣⋅⟩X∙P coincides with the restriction of ∥⋅∥X to X∙P.
Proof.
It is clear that X∙P is a subspace of X. As X is non-trivial, Span(⟨X∣X⟩X)K(H) is a non-trivial ideal of K(H), but as K(H) is a simple C∗-algebra, we have Span(⟨X∣X⟩X)K(H)=K(H). Hence,
[TABLE]
which implies that X∙P is a non-trivial subspace of X.
To see that X∙P is a closed subspace of X, suppose that (ζn)n∈N is a sequence in X∙P that converges to some η∈X. Then because ζn∙P=ζn for all n∈N, we have
[TABLE]
Hence, η∈X∙P, which proves that X∙P is a closed subspace of X.
Clearly, ⟨⋅∣⋅⟩X∙P is a sesquilinear form on X∙P, so it remains to see that it is positive definite and complete. Let ζ∈X∙P. Then ⟨ζ∣ζ⟩X is positive in K(H), which means that
[TABLE]
is positive in K(H) as well. As IdH−P is not invertible in K(H)∼, we have 1∈σK(H)(P). Hence,
[TABLE]
which demonstrates that ⟨⋅∣⋅⟩X∙P is at least positive semidefinite. Next, observe for all ζ,η∈X∙P that
[TABLE]
Consequently, if ⟨ζ∣ζ⟩X∙P=0 for some ζ∈X∙P, then ⟨ζ∣ζ⟩X=0K(H), which yields ζ=0X=0X∙P. This proves that ⟨⋅∣⋅⟩X∙P is positive definite. Incidentally, this also proves that ∥ζ∥X∙P=∥ζ∥X for all ζ∈X∙P. As X∙P is a closed subspace of X, it is a Banach space with respect to the restriction of ∥⋅∥X to X∙P, and is thus a Banach space with respect to ∥⋅∥X∙P. Therefore, X∙P is a Hilbert space whose inner product is given by ⟨⋅∣⋅⟩X∙P, and the Hilbert-space norm on X∙P is precisely the restriction of ∥⋅∥X to X∙P.
∎
Theorem 3**.**
Let X be a Hilbert K(H)-module, Y a K(H)-submodule of X that is not necessarily closed, and P a rank-one projection on H. Then the closed K(H)-linear span of Y∙P in X is the closure YX of Y in X.
Proof.
As Y is a K(H)-submodule of X, we can see that Y∙P⊆Y and that the K(H)-linear span of Y∙P is contained in Y. Hence, the closed K(H)-linear span of Y∙P in X is contained in YX. It thus remains to establish the reverse inclusion.
Let ζ∈Y, and let F be a rank-n operator on H. We claim that ζ∙F belongs to the K(H)-linear span of Y∙P. Let (vi)i∈[n] be an orthonormal basis of Range(F), and v a unit vector in Range(P). Then
[TABLE]
Hence,
[TABLE]
As ∣vi⟩⟨v∣ and ∣v⟩⟨vi∣F are finite-rank operators on H for all i∈[n], they belong to K(H). It follows that ζ∙∣vi⟩⟨v∣∈Y for all i∈[n] because Y is a K(H)-submodule of X, so ζ∙F belongs to the K(H)-linear span of Y∙P, as claimed.
As F is an arbitrary finite-rank operator on H, we see that ζ∙T is in the closed K(H)-linear span of Y∙P in X for any T∈K(H), as T is the limit in K(H) of some sequence of finite-rank operators on H.
Let (Ei)i∈I be an approximate identity in K(H). By the argument above, ζ∙Ei is in the closed K(H)-linear span of Y∙P in X for all i∈I. As (ζ∙Ei)i∈I converges to ζ, it follows that ζ is in the closed K(H)-linear span of Y∙P in X. As ζ∈Y is arbitrary, Y is thus contained in the closed K(H)-linear span of Y∙P in X.
Finally, by the definition of closure, YX is contained in the closed K(H)-linear span of Y∙P in X.
∎
The next theorem is the main result of [3], and it explains why Hilbert K(H)-modules behave like Hilbert spaces. It says that the C∗-algebra of adjointable operators on a Hilbert K(H)-module X is isomorphic to the C∗-algebra of bounded operators on the Hilbert space X∙P, for any rank-one projection P on H. At first sight, this seems rather astonishing because X∙P is generally a much smaller space that X itself, and it would be hard to imagine why it should have much to say about X. However, having seen in Theorem 3 that X∙P generates a dense submodule of X, one can start to understand why the theorem holds.
The proof given in [3] relies on concepts from an earlier paper [2], but the proof that we give here is very direct and only depends on the previous definitions and results of this section.
Theorem 4**.**
Let X be a non-trivial Hilbert K(H)-module, and P a rank-one projection on H. Then X∙P is an invariant subspace for each T∈L(X), and the map
[TABLE]
is a C∗-isomorphism, where X∙P is viewed as a Hilbert space. Furthermore, the restriction of this map to K(X) yields a C∗-isomorphism from K(X) to K(X∙P).
Proof.
For each T∈L(X), the K(H)-linearity of T implies that X∙P is an invariant subspace of T.
It is easy to check that {L(X)T→↦B(X∙P)T∣X∙P} is at least a C∗-homomorphism. To see that it is injective, let S,T∈L(X) satisfy S∣X∙P=T∣X∙P. Then by the K(H)-linearity and continuity of both S and T, they must agree on the closed K(H)-linear span of X∙P, which is equal to X by Theorem 3. Hence, S=T.
Surjectivity is trickier to prove. Let L∈B(X∙P), and let (εi)i∈I be an orthonormal basis of X∙P. For each J∈Fin(I), let TJ=dfi∈J∑ΘL(εi),εi∈K(X) and LJ=dfi∈J∑∣L(εi)⟩⟨εi∣∈K(X∙P); then for all ζ∈X∙P,
[TABLE]
which implies that TJ∣X∙P=LJ. Next, for all ζ∈X∙P,
[TABLE]
so (LJ)J∈Fin(I) is a net (partially ordered by ⊆) that strongly converges to L. Also, for all J∈Fin(I),
[TABLE]
which gives, by the first part, ∥TJ∥L(X)=∥LJ∥B(X∙P)≤∥L∥B(X∙P). Notice now that (TJ)J∈Fin(I) is a norm-bounded net in L(X) that strongly converges on the K(H)-linear span of X∙P, which is dense in X. By an 3ϵ-argument, (TJ)J∈Fin(I) strongly converges everywhere to some T∈B(X). Likewise, (TJ∗)J∈Fin(I) strongly converges everywhere to some S∈B(X). As ⟨TJ(ζ)∣η⟩X=⟨ζ∣TJ∗(η)⟩X for all ζ,η∈X and J∈Fin(I), taking limits yields ⟨T(ζ)∣η⟩X=⟨ζ∣S(η)⟩X. Therefore, T∈L(X) and T∣X∙P=L, which finishes our proof that the restriction map is a C∗-isomorphism from L(X) to B(X∙P).
For the final part of the proof, observe for all ζ,η∈X∙P, S,T∈K(H), and ξ∈X∙P that
[TABLE]
which means that Θζ∙S,η∙T∣X∙P∈K(X∙P). Let ζ,η∈X. Then by Theorem 3, we can find
[TABLE]
so that i=1∑mζi∙Si and j=1∑nηj∙Tj are arbitrarily close to ζ and η, respectively, which ensures that
[TABLE]
is arbitrarily close to Θζ,η in K(X). Hence, by the continuity of the restriction C∗-isomorphism,
[TABLE]
is arbitrarily close to Θζ,η∣X∙P, which says that Θζ,η∣X∙P∈K(X∙P). Therefore, the image of K(X) under the restriction C∗-isomorphism is a non-trivial ideal in K(X∙P). As K(X∙P) is a simple C∗-algebra, this image is precisely K(X∙P). The proof is now complete.
∎
Using Theorem 4, we can show that every closed submodule of a Hilbert K(H)-module has an orthogonal complement. The complementability of Hilbert K(H)-modules has been known for a long while ([11]), but Theorem 4 appears to provide an expedient proof.
Theorem 5**.**
Let Y be a closed submodule of a Hilbert K(H)-module X. Then X=Y⊕Y⊥.
Proof.
Let P be a rank-one projection on H. Then Y∙P is a closed subspace of X∙P. By Theorem 4, there is a projection Q∈L(X) such that Q∣X∙P=ProjX∙P,Y∙P. If we can show that Range(Q)=Y, then we are done, for a closed submodule of a Hilbert C∗-module is complementable if it is the range of an adjointable operator (Corollary 15.3.9 of [21]). Indeed, as Q is a projection, Range(Q) is a closed submodule of X, and
Let X be a non-trivial Hilbert K(H)-module, and P a rank-one projection on H. Then there is a ζ∈X such that ⟨ζ∣ζ⟩X=P.
Proof.
By Theorem 2, we can find a non-zero η∈X∙P, so ⟨η∣η⟩X∙P>0. As
[TABLE]
we see that ζ=df⟨η∣η⟩X∙P1⋅η satisfies ⟨ζ∣ζ⟩X=P.
∎
Definition 6**.**
Let A be a non-trivial C∗-algebra, and X a non-trivial Hilbert A-module. We say that K(X)acts irreducibly on X if and only if the only closed submodules of X that are invariant under the left action of K(X) are {0X} and X.
Proposition 4**.**
Let X be a non-trivial Hilbert K(H)-module. Then K(X) acts irreducibly on X.
Proof.
Let P be a rank-one projection on H. By Theorem 2, X∙P is a non-trivial Hilbert space. Suppose that K(X) does not act irreducibly on X, i.e., there is a non-trivial proper closed submodule Y of X that is invariant under the left action of K(X). By Theorem 2 again, Y∙P is a non-trivial closed subspace of X∙P. We will then have a contradiction if we can show that Y∙P is a non-trivial and proper closed subspace of X∙P that is invariant under the left action of K(X∙P) (note that the C∗-algebra of compact operators on a non-trivial Hilbert space must act irreducibly on the Hilbert space).
If Y∙P were not a proper subspace of X∙P, then Y∙P=X∙P, so the closed K(H)-linear span of Y∙P in X would be that of X∙P in X. By Theorem 3, we would obtain Y=X, which violates the earlier assumption that Y is a proper submodule of X. Hence, Y∙P⊊X∙P.
Let T∈K(X∙P). Then T=S∣X∙P for some S∈K(X), so
[TABLE]
proving that Y∙P is invariant under the left action of K(X∙P). This produces the desired contradiction, which completes the proof of this proposition.
∎
Proposition 5**.**
Let X and Y be non-trivial Hilbert K(H)-modules. If Φ is a non-degenerate ∗-representation of K(X) on Y, then (Y,Φ) is unitarily equivalent to a direct sum of copies of (X,iK(X)↪L(X)).
Proof.
Our proof is an adaptation of Arveson’s proof of Theorem 1.4.4 in [1]. Fix a rank-one projection P on H, and consider
[TABLE]
where K(X∙P)⟶≅K(X) and L(Y)⟶≅B(Y∙P) come from Theorem 4. By definition, the non-degeneracy of (Φ,Y) means that
[TABLE]
which yields
[TABLE]
Hence, Ψ is a non-degenerate ∗-representation of K(X∙P) on the Hilbert space Y∙P, and so the first part of Arveson’s proof says that there is a rank-one projection Q∈K(X∙P) such that Ψ(Q)=0K(Y∙P).
By Theorem 4, there is a projection E∈K(X) such that Q=E∣X∙P. As Ψ(Q)=0K(Y∙P), it must be that Φ(E)=0L(Y), so E=0K(X). By Lemma 3, there exists a linear functional fP:K(X∙P)→C satisfying
[TABLE]
Define a linear functional g:K(X)→C by g(T)=dffP(T∣X∙P) for all T∈K(X); then
[TABLE]
By Theorem 4 again, we may conclude that ETE=g(T)⋅E for all T∈K(X).
Consider the K(H)-submodule E[X] of X, which is non-trivial as E=0K(X), and closed as E is a projection. Similarly, Φ(E)[Y] is a non-trivial closed K(H)-submodule of Y. Hence, by Lemma 4, there exist ζ∈E[X] and η∈Φ(E)[Y] such that ⟨ζ∣ζ⟩X=P=⟨η∣η⟩Y. We now claim that the map
[TABLE]
defined by
[TABLE]
for all T1,…,Tn∈K(X) and A1,…,An∈K(H) is well-defined by virtue of being an isometry. Indeed,
[TABLE]
and a nearly-identical computation using ⟨ζ∣ζ⟩X=P also yields
[TABLE]
Therefore, U is a surjective isometry. By continuity, U extends to a surjective isometry U:X′→Y′, where
[TABLE]
and
[TABLE]
Note that X′ is a K(X)-invariant closed submodule of X, and is non-trivial as ζ∈X′. Also, Y′ is a Φ[K(X)]-invariant closed submodule of Y, and is non-trivial as η∈Y′. Hence, X′=X by Proposition 4, so U:X→Y′ is a surjective isometry that, moreover, is K(H)-linear. We may thus apply Theorem 3.5(i) of [9] to deduce that U∈U(X,Y′).
Next, we claim that UT=Φ(T)∣Y′U for all T∈K(X). Fix T∈K(X). Then for all T1,…,Tn∈K(X) and A1,…,An∈K(H), we have
[TABLE]
By the density of Span({T(ξ∙A)∣T∈K(X),a∈K(H)}) in X, we obtain UT=Φ(T)∣Y′U as expected.
Now, define a poset (P,⊑) with the following properties:
•
S is an element of P if and only if the following hold:
–
S consists of pairs of the form (Z,V), where Z is a non-trivial Φ[K(X)]-invariant closed submodule of Y, and V∈U(X,Z) with T=V−1[Φ(T)∣Z]V for all T∈K(H).
–
If (Z1,V1) and (Z2,V2) are distinct elements of S, then Z1⊥Z2.
•
For all S1,S2∈P, we have S1⊑S2 if and only if S1⊆S2.
If C is a chain in (P,⊑), then ⋃C is an upper bound for C in (P,⊑), so by Zorn’s Lemma, there exists a maximal element M of (P,⊑). We claim that Y=(Z,V)∈M⨁Z, where the direct sum is internal. If this were not true, then (Z,V)∈M⨁Z⊊Y. Letting Z′=df(Z,V)∈M⨁Z⊥, Theorem 5 says that Z′ is a non-trivial closed submodule of Y. A routine verification reveals that Φ(T)[Z′]⊆Z′ for each T∈K(H), and that
[TABLE]
is a non-degenerate ∗-representation of K(X) on Z′. We may thus apply the first part of the proof to Z′ to obtain (Z′′,W), where
•
Z′′ is a non-trivial Φ[K(X)]-invariant closed submodule of Z′ (and hence of Y), and
•
W∈U(X,Z′′) with T=W−1[Φ(T)∣Z′′]W for all T∈K(H).
As M⊊{(Z′′,W)}∪M∈P, this contradicts the maximality of M. Therefore, Y=(Z,V)∈M⨁Z indeed, so
[TABLE]
The proof is finally complete.
∎
5. The Covariant Stone-von Neumann Theorem
In [16], Marc Rieffel applied a special instance of the next theorem — Green’s Imprimitivity Theorem — to derive the classical Stone-von Neumann Theorem. According to him, the classical Stone-von Neumann Theorem is really a statement about the Morita equivalence of the C∗-algebra C with the crossed product C∗(G,C0(G),lt). This gives us a more algebraic way of seeing things, and it is precisely this point of view that guided our search for the covariant Stone-von Neumann Theorem in the beginning. As we are dealing with Hilbert C∗-modules instead of just Hilbert spaces, the full strength of Green’s Imprimitivity Theorem is required.
Theorem 6** (Green’s Imprimitivity Theorem).**
Let (G,A,α) be a C∗-dynamical system. Then L2(G,A,α) is a (C∗(G,C0(G,A),lt⊗α),A)-imprimitivity bimodule with the following properties:
•
If Ξ denotes the non-degenerate ∗-representation of C0(G,A) on L2(G,A,α) uniquely determined by
[TABLE]
then (L2(G,A,α),Ξ,U) is a (G,C0(G,A),lt⊗α)-covariant modular representation, and the left action of C∗(G,C0(G,A),lt⊗α) on L2(G,A,α) is ΠL2(G,A,α),Ξ,U.
•
The C∗(G,C0(G,A),lt⊗α)-valued inner product on L2(G,A,α) is uniquely determined by
[TABLE]
for all ϕ,ψ∈Cc(G,A).
•
Both the right A-action and the A-valued inner product on L2(G,A,α) are precisely the ones that define L2(G,A,α) as a Hilbert A-module.
Complete proofs of Green’s Imprimitivity Theorem may be found in [5, 22].
Proposition 6**.**
Let (G,A,α) be a C∗-dynamical system with G abelian. Recalling the (G,A,α)-Schrödinger modular representation (L2(G,A,α),M,U,V), and letting ν be any Haar measure on G, we have that
[TABLE]
is an injective C∗-homomorphism and that Range(ΠL2(G,A,α),πνL2(G,A,α),M,V,U)=K(L2(G,A,α)).
Proof.
Firstly, we show that πνL2(G,A,α),M,V=Ξ. Let f∈Cc(G,A) and ϕ∈Cc(G,A). Then
[TABLE]
The last integral looks like it should be q(FνA(f)ϕ), and indeed it is, but we have to exercise some caution in justifying our guess. By Fubini’s Theorem, we have for all ψ∈Cc(G,A) that
[TABLE]
This establishes the validity of our guess. Hence, πνL2(G,A,α),M,V(FνA(f))=Ξ(FνA(f)) for all f∈Cc(G,A), and as the range of FνA is dense in C0(G,A), we conclude that πνL2(G,A,α),M,V=Ξ.
It now follows from Green’s Imprimitivity Theorem and Proposition 3.8 of [15] that
[TABLE]
is an injective C∗-homomorphism whose range is K(L2(G,A,α)).
∎
Definition 7**.**
Let (G,A,α) and (G,A,β) be C∗-dynamical systems with G abelian. Let (X,ρ,R,S) be a (G,A,α)-Heisenberg modular representation; (Y,σ,T,U) a (G,A,β)-Heisenberg modular representation. We say that (X,ρ,R,S) is unitarily equivalent to (Y,σ,T,U) if and only if there exists a W∈U(X,Y) such that
[TABLE]
for all x∈G, φ∈G, and a∈A, in which case we write (X,ρ,R,S)∼W(Y,σ,T,U).
Definition 8**.**
Let (G,A,α) be a C∗-dynamical system with G abelian. We say that (G,A,α) has the von Neumann Uniqueness Property if and only if each (G,A,α)-Heisenberg modular representation is unitarily equivalent to a direct sum of copies of the (G,A,α)-Schrödinger modular representation.
We are now ready to prove the main result of this paper.
Proposition 7** (The Covariant Stone-von Neumann Theorem).**
Every C∗-dynamical system of the form (G,K(H),α), with G abelian, has the von Neumann Uniqueness Property.
Proof.
Let (X,ρ,R,S) be a (G,K(H),α)-Heisenberg modular representation. According to Proposition 2, (X,πνX,ρ,S,R) is a (G,C0(G,A),lt⊗α)-covariant modular representation, so ΠX,πνX,ρ,S,R is a non-degenerate ∗-representation of C∗(G,C0(G,A),lt⊗α) on X. However, Proposition 6 says that
[TABLE]
is a C∗-isomorphism, so it follows from Proposition 5 that
[TABLE]
for some index set I and some W∈U(X,i∈I⨁L2(G,A,α)). We thus have
[TABLE]
for all T∈K(L2(G,A,α)), or equivalently,
[TABLE]
for all F∈C∗(G,C0(G,A),lt⊗α). It follows from Lemma 1 that
[TABLE]
for all x∈G and g∈C0(G,A). However, as
[TABLE]
we find that
[TABLE]
for all f∈C∗(G,A,ι). Another application of Lemma 1 yields
[TABLE]
for all φ∈G and a∈A. The covariant Stone-von Neumann Theorem is hereby established.
∎
Our method of proof in no way depended on the classical Stone-von Neumann Theorem, so it is a proper generalization in every way, as expressed by the corollary below.
Corollary 1**.**
The classical Stone-von Neumann Theorem is precisely the case when H=C (any strongly-continuous action of a locally compact Hausdorff group on C is necessarily trivial).
6. The Non-Triviality of the Covariant Stone-von Neumann Theorem
One may now ask, “Does the covariant Stone-von Neumann Theorem really say anything new? Is there a unitary transformation that reduces it to the case of the trivial action of G on K(H)?” The following result makes this question an extremely valid one.
Proposition 8**.**
Let (G,A,α) be a C∗-dynamical system with G not assumed to be abelian. Then there is a Hilbert A-module isomorphism Ω:L2(G,A,α)→L2(G,A,ι) that satisfies
[TABLE]
Proof.
This is an easy verification that we leave to the reader.
∎
Even though L2(G,A,α) is isomorphic to L2(G,A,ι), note that the covariant Stone-von Neumann Theorem is not a statement about the unitary equivalence of Hilbert C∗-modules, but a statement about the unitary equivalence of Heisenberg modular representations. Having said this, the next two results give a complete answer to the question above.
Proposition 9**.**
Let (G,A,α) and (G,A,β) be C∗-dynamical systems, with G abelian and α=β. Then a direct sum of copies of the (G,A,α)-Schrödinger modular representation cannot be unitarily equivalent to a direct sum of copies of the (G,A,β)-Schrödinger modular representation.
Proof.
By way of contradiction, suppose that there are index sets I and J such that
[TABLE]
for some W∈Ui∈i⨁L2(G,A,α),j∈J⨁L2(G,A,β). Then we have for all x∈G and a∈A that
[TABLE]
so it follows that
[TABLE]
which yields M(G,A,β)(βx(a))=M(G,A,β)(αx(a)). Hence, for all x∈G, a∈A, and ϕ∈Cc(G,A),
[TABLE]
from which we get (αx(a)−βx(a))ϕ(y)=0A for all y∈G. As we can choose ϕ to assume any value at any point, we obtain αx(a)=βx(a) for all x∈G and a∈A, which contradicts α=β.
∎
Corollary 2**.**
Let (G,K(H),α) and (G,K(H),β) be C∗-dynamical systems, with G abelian, H a non-trivial Hilbert space, and α=β. Then any (G,K(H),α)-Heisenberg modular representation cannot be unitarily equivalent to any (G,K(H),β)-Heisenberg modular representation.
Corollary 2 should remind physicists of Haag’s Theorem in quantum field theory (QFT), which posits the failure of the uniqueness of the canonical commutation relations within QFT in general ([6]).
We finally arrive at a discussion of Takai-Takesaki Duality.
where α^ denotes the dual action of G on C∗(G,A,α).
In his proof of Takai-Takesaki Duality in [14], Iain Raeburn first showed that
[TABLE]
He then formed a C∗-isomorphism C∗(G,C∗(G,A,α),α^)≅K(L2(G))⊗A as a composition of a series of C∗-isomorphisms shown below, each requiring a lengthy justification except for the last one:
[TABLE]
His “untwisting” of α is thus performed at the level of C∗-crossed products, with the last C∗-isomorphism being given by the classical Stone-von Neumann Theorem, which relies on Green’s Imprimitivity Theorem. However, by taking full advantage of Green’s Imprimitivity Theorem, we can derive a shorter proof of this C∗-isomorphism, which “untwists” α at the level of Hilbert C∗-modules:
Proposition 10**.**
Let (G,A,α) be a C∗-dynamical system. Then C∗(G,C0(G,A),lt⊗α)≅K(L2(G))⊗A.
Proof.
Using Proposition 6, Proposition 8, and basic results about Hilbert C∗-modules, we offer a one-line proof:
[TABLE]
7. Conclusions
We would like to present here some questions and thoughts that naturally arose while writing this paper:
(1)
Is there a C∗-algebra A not C∗-isomorphic to K(H) for a Hilbert space H such that any C∗-dynamical system of the form (G,A,α) has the von Neumann Uniqueness Property? As C∗-subalgebras of K(H) are C∗-isomorphic to a direct sum i∈I⨁K(Hi), where the Hi’s are Hilbert spaces, we think that a series of technical extensions can be made to accommodate the Covariant Stone-von Neumann Theorem for such C∗-algebras.
2. (2)
The results of this paper suggest that quantum mechanics could be developed using Hilbert C∗-modules as state spaces, in which case the expectations of observables would assume values in a C∗-algebra. Can this idea be developed further?
3. (3)
As mentioned in the introduction, we suspect that the covariant Stone-von Neumann Theorem could be generalized to actions of non-abelian groups using techniques of non-abelian duality.
While interesting in a purely-mathematical context, the Covariant Stone-von Neumann Theorem has a rich interpretation from the perspective of quantum mechanics. By including representations of C∗-dynamical systems, it allows for the consideration of time-dependence of observables in addition to time-dependence of states. To contrast, recall that a time-independent quantum system is modeled by a Hilbert space H and a Hamiltonian H^ whose corresponding one-parameter unitary family, (e−(it/ℏ)⋅H^)t∈R, determines the time evolution of the state space via
[TABLE]
The time evolution of the state space determined by H^ can also be viewed as time evolution of the algebra B(H) of bounded observables via {Rt→↦B(H)e(it/ℏ)⋅H^Te−(it/ℏ)⋅H^}, for all T∈B(H). From this perspective, one may state the time-independent version of Ehrenfest’s Theorem:
[TABLE]
As the Covariant Stone-von Neumann Theorem applies to C∗-dynamical systems of the form (G,K(H),α), and as all ∗-automorphisms of K(H) are implemented via conjugation by unitaries, we make a convenient but natural restriction in the case when G=R to the action αC, where C∈B(H) is self-adjoint, and
[TABLE]
The covariance conditions present in the definition of an (R,K(H),αC)-Heisenberg modular representation (X,ρ,R,S) then reduce to commutation relations between C and the infinitesimal generators of R and S. It is in this context that we are able to get an infinitesimal version of the Covariant Stone-von Neumann Theorem, which will appear in a sequel to this article.
As mentioned in the introduction, a catalyst for the Stone-von Neumann Theorem was to investigate the uniqueness of pairs (A,B) of self-adjoint Hilbert-space operators satisfying the Heisenberg Commutation Relation. Nelson’s counterexample [12] shows that uniqueness fails in general, and decades of research have been devoted to identifying sufficient conditions for (A,B) that imply that (eis⋅A)s∈R and (eit⋅B)t∈R satisfy the Weyl Commutation Relation.
In the sequel, we follow the strategy in [8] — which takes place in the Hilbert-space setting — to provide necessary and sufficient conditions for when a pair (A,B) of unbounded self-adjoint operators on a Hilbert K(H)-module yield one-parameter unitary groups that satisfy the Weyl Commutation Relation.
Let N and O be neighborhood bases of x and eG in G respectively, directed by reverse inclusion. By Urysohn’s Lemma, we can find nets (ϕU)U∈N and (ψV)V∈O in Cc(G,R≥0) with the following properties:
•
Supp(ϕU)⊆U for each U∈N, and Supp(ψV)⊆V for each V∈O.
•
∫GϕU(y)dμ(y)=1=∫GψV(y)dμ(y) for all U∈N and V∈O.
Also, let (eλ)λ∈Λ be an approximate identity for A norm-bounded by 1.
Let ζ∈X and ϵ>0. As R(y) converges strongly in L(X) to R(x) as y→x, we can find a U0∈N such that for all y∈U0,
[TABLE]
As ρ is non-degenerate, (ρ(eλ))λ∈Λ converges strongly in L(X) to IdX, so we can find a λ0∈Λ such that
[TABLE]
We then have for all y∈U0 and λ∈Λ≥λ0 that
[TABLE]
It thus follows for all U∈N⊆U0 and λ∈Λ≥λ0 that
[TABLE]
As ζ∈X and ϵ>0 are arbitrary, the net (ΠX,ρ,R(ϕU⋄eλ))(U,λ)∈N×Λ converges strongly in L(X) to R(x).
Let ζ∈X and ϵ>0. As R(y) converges strongly in L(X) to R(eH)=IdX as y→eH, we can find a V0∈O such that
[TABLE]
so
[TABLE]
It follows for all V∈O⊆V0 that
[TABLE]
As ζ∈X and ϵ>0 are arbitrary, the net (ΠX,ρ,R(ψV⋄a))V∈O converges strongly in L(X) to ρ(a).
∎
Fix f∈Cc(X,V) and ϵ>0. Let K=dfSupp(f), and let (Up)p∈K be a K-indexed sequence of open subsets of X with the following properties:
•
Up is an open neighborhood of p in X for each p∈K.
•
∥f(x)−f(p)∥V<3ϵ for all x∈Up.
Clearly, {Up}p∈K covers K, and as K is compact, there is a finite subset F of K such that {Up}p∈F also covers K, and Up=Up′ for distinct p,p′∈F. As X is locally compact and Hausdorff, there is a partition of unity (φp)p∈F for K that is subordinate to {Up}p∈F, i.e.,
•
φp∈Cc(X,[0,1]) and Supp(φp)⊆Up for each p∈F, and
•
p∈F∑φp(x)≤1 for all x∈X, with equality holding for all x∈K.
Define P∈Cc(X,V) by
[TABLE]
For all x∈X, we have f(x)=p∈F∑φp(x)⋅f(x) (if x∈K, then p∈F∑φp(x)=1; otherwise, f(x)=0V), so
[TABLE]
As D is a dense subset of V, we can find an F-indexed sequence (vp)p∈F in D such that ∥f(p)−vp∥V<3ϵ for each p∈F. Define Q∈Cc(X,V) by
[TABLE]
Then
[TABLE]
Therefore, by the Triangle Inequality,
[TABLE]
As Q has the desired form, the first part of the theorem is therefore established.
Let U be an open neighborhood of K whose closure is compact (such a neighborhood exists because X is locally compact and Hausdorff). Then U is a locally compact Hausdorff space and f∣U∈Cc(U,V), so we may apply the first part of the theorem to find φ1,…,φn∈Cc(U) and v1,…,vn∈D such that
[TABLE]
Let φ1,…,φn denote the respective extensions of φ1,…,φn to X by 0V. Then φ1,…,φn∈Cc(X), and
[TABLE]
The proof of the second part of the theorem is therefore established.
∎
Acknowledgments
The first author wishes to thank the second author for her warm hospitality and collaborative energy when he visited the University of Nebraska-Lincoln. He would also like to thank his former PhD advisor, Professor Albert Sheu of the University of Kansas, for cultivating his deep-seated interest in the Stone-von Neumann Theorem in graduate school.
The second author would like to thank the first author for his invitation to collaborate on this project. His passion and breadth of knowledge have been a privilege to work alongside. The second author is also grateful to her advisors, Allan Donsig and David Pitts, as well as Ruy Exel, for their thoughtful comments and questions.
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