aa r X i v : . [ m a t h . OA ] J u l CCR flows associated to closed convex cones
Anbu Arjunan and S. SundarJuly 12, 2019
Abstract
Let P be a closed convex cone in R d which we assume to be spanning andpointed i.e. P − P = R d and P ∩ − P = { } . In this article, we consider CCRflows over P associated to isometric representations that arise out of P -invariantclosed subsets, also called as P -modules, of R d . We show that for two P -modulesthe associated CCR flows are cocycle conjugate if and only if the modules aretranslates of each other. AMS Classification No. :
Primary 46L55; Secondary 46L99.
Keywords : E -semigroups, CCR flows and Groupoids. Contents
The theory of E -semigroups initiated by Powers and further developed extensively byArveson is approximately three decades old. We refer the reader to the beautiful mono-graph [5] for the history, the development and the literature on E -semigroups. In this1ong introduction, we explain the problem considered in this paper, collect the prelimi-naries required and explain the techniques behind the proof of our main theorem. Let H be an infinite dimensional separable Hilbert space. We denote the algebra of boundedoperators on H by B ( H ). By an E -semigroup on B ( H ), one means a 1-parametersemigroup α := { α t } t ≥ of unital normal ∗ -endomorphisms of B ( H ). However nothingprevents us from considering semigroups of endomorphisms on B ( H ) indexed by moregeneral semigroups.The authors in collobaration with others ([13], [2]) have considered E -semigroupsover closed convex cones. In this paper, we analyse the basic examples of Arveson’stheory i.e. CCR flows associated to modules over cones. We hope that the reader will beconvinced by the end of this paper that this is not merely for the sake of generalisation andthere are some interesting connections to groupoid C ∗ -algebras and in particular to thegroupoid approach to topological semigroup C ∗ -algebras which was first systematicallyexplored by Muhly and Renault in [11].We fix notation that will be used throughout this paper. The norm that we use on R d is always the usual Euclidean norm. Let P ⊂ R d be a closed convex cone. We assume P is pointed i.e. P ∩ − P = { } . By restricting ourselves to the vector space generatedby P , there is no loss of generality in assuming that P is spanning i.e. P − P = R d . Theinterior of P will be denoted by Ω. Then Ω is dense in P . For a proof of this, we referthe reader to Lemma 3.1 of [14]. It is also clear that Ω spans R d . For x, y ∈ R d , we write x ≥ y if x − y ∈ P and write x > y if x − y ∈ Ω. We have the following
Archimedeanprinciple : Given x ∈ R d and a ∈ Ω, there exists a positive integer n such that n a > x .For a proof of this fact, we refer the reader to Lemma 3.1 of [13]Let us review the definitions of E -semigroups and some results from [2]. Let H bean infinite dimensional separable Hilbert space. By an E -semigroup, over P , on B ( H ),we mean a family α := { α x } x ∈ P of normal unital ∗ -endomorphisms of B ( H ) such that α x ◦ α y = α x + y satisfying the following continuity condition: For A ∈ B ( H ) and ξ, η ∈ H ,the map P ∋ x → h α x ( A ) ξ | η i is continuous.We consider two E -semigroups acting on different Hilbert spaces to be isomorphic if they are unitarily equivalent. The precise definition is as follows. Let K be an infinitedimensional separable Hilbert space and U : H → K be a unitary. We denote the map B ( H ) ∋ T → U T U ∗ ∈ B ( K ) by Ad ( U ). Let α := { α x } x ∈ P and β := { β x } x ∈ P be E -semigroups acting on B ( H ) and B ( K ) respectively. We say that α is conjugate to β ifthere exists a unitary U : H → K such that for every x ∈ P , β x = Ad ( U ) ◦ α x ◦ Ad ( U ) ∗ .Let α := { α x } x ∈ P be an E -semigroup on B ( H ). By an α -cocycle , we mean a strongly2ontinuous family of unitaries { U x } x ∈ P such that U x α x ( U y ) = U x + y . If U := { U x } x ∈ P isan α -cocycle, it is straightforward to check that { Ad ( U x ) ◦ α x } x ∈ P is an E -semigroup.Such an E -semigroup is called a cocycle perturbation of α . Let β be an E -semigroupacting on a possibly different Hilbert space say K . We say that β is cocycle conjugate to α if a conjugate of β is a cocycle perturabation of α .One natural operation that one can do with E -semigroups is the tensor productoperation. Let α and β be E -semigroups on B ( H ) and B ( K ) respectively. Then thereexists a unique E -semigroup, denoted α ⊗ β , on B ( H⊗K ) such that for x ∈ P , A ∈ B ( H )and B ∈ B ( K ), ( α ⊗ β ) x ( A ⊗ B ) = α x ( A ) ⊗ β x ( B ) . For a proof of this fact, we refer the reader to the paragraph preceeding Remark 4.8of [2]. It is routine to verify that if β is cocycle conjugate to γ then α ⊗ β is cocycleconjugate to α ⊗ γ .As with any mathematical structures, the first question is to know whether there areenough examples and, if possible, how to classfiy them. It is beyond the scope of thepresent paper to offer a complete classification of E -semigroups. This question is stillopen even in the 1-dimensional case. We present here a class of examples that we call the CCR flows associated to P -modules and we classify them completely.First let us recall the notion of Weyl operators on the symmetric Fock space of H .Let Γ( H ) be the symmetric Fock space. For u ∈ H , let e ( u ) := ∞ X n =0 u ⊗ n √ n ! . The set of vectors { e ( u ) : u ∈ H} is called the set of exponential vectors. We have thefollowing.(1) For u, v ∈ H , h e ( u ) | e ( v ) i = e h u | v i .(2) The set { e ( u ) : u ∈ H} is total in Γ( H ).(3) Any finite subset of { e ( u ) : u ∈ H} is linearly independent.For u ∈ H , there exists a unique unitary, W ( u ) on Γ( H ), whose action on the exponentialvectors is given by the following formula: W ( u ) e ( v ) := e − || u || −h u | v i e ( u + v ) . { W ( u ) : u ∈ H} are called the Weyl operators . The Weyl operators satisfythe following canonical commutation relation. For u, v ∈ H , W ( u ) W ( v ) = e − iIm h u | v i W ( u + v )where Im h u | v i denotes the imaginary part of h u | v i . The linear span of the Weyl operators { W ( u ) : u ∈ H} forms a unital ∗ -subalgebra of B (Γ( H )) whose strong closure is B (Γ( H )).For a unitary U on H , there exists a unique unitary, denoted Γ( U ), on Γ( H ), whoseaction on the exponential vectors is given byΓ( U ) e ( v ) := e ( U v ) . The unitary Γ( U ) is called the second quantisation of U . For a unitary U on H and u ∈ H , we have the relation Γ( U ) W ( u )Γ( U ) − = W ( U u ).Let H and H be Hilbert spaces. The mapΓ( H ⊕ H ) ∋ e ( u ⊕ u ) → e ( u ) ⊗ e ( u ) ∈ Γ( H ) ⊗ Γ( H )extends to a unitary. Via this unitary, we always identify Γ( H ⊕H ) with Γ( H ) ⊗ Γ( H ).Under this identification, we have the equality W ( u ⊕ u ) = W ( u ) ⊗ W ( u ) for u ∈ H and u ∈ H . For proofs of all the assertions made so far, we refer the reader to the book[16]. Definition 1.1
By a strongly continuous isometric representation of P on H , we meana map V : P → B ( H ) such that(1) for x, y ∈ P , V x V y = V x + y ,(2) for x ∈ P , V x is an isometry, and(3) for ξ ∈ H , the map P ∋ x → V x ξ ∈ H is continuous. Let V : P → B ( H ) be a strongly continuous isometric representation. Then thereexists a unique E -semigroup on B (Γ( H )) denoted α V := { α x } x ∈ P such that for x ∈ P and u ∈ H , α x ( W ( u )) = W ( V x u ) . For the existence of the E -semigroup α V , we refer the reader to Prop. 4.7 of [2]. We call α V the CCR flow associated to the isometric representation V . The association V → α V converts the direct sum of isometric representations to tensor product of E -semigroups.4hat is, let V and V be isometric representations of P on Hilbert spaces H and H respectively. Then V ⊕ V := (( V ) x ⊕ ( V ) x ) x ∈ P is an isometric representation of P on H ⊕ H . It is clear that if V and V are strongly continous then V ⊕ V is stronglycontinous. Under the identification Γ( H ⊕ H ) ∼ = Γ( H ) ⊗ Γ( H ), we have the equality α V ⊕ V = α V ⊗ α V (See Remark 4.8 of [2]).What are the examples of isometric representations of P ? One obvious isometricrepresentation is the “left” regular representation of P on L ( P ). One can also considerthe “left” regular representation with multiplicity. A slight generalisation of the aboveis as follows. Let A ⊂ R n be a proper closed subset which is invariant under translationby elements of P i.e. A + x ⊂ A for x ∈ P . Such a subset is called a P -module . Thenotion of P -modules in the discrete setting was first considered by Salas in [20]. Let k ∈ { , , · · · , } ∪ {∞} be given and let K be a Hilbert space of dimension k . Considerthe Hilbert space L ( A, K ). For x ∈ P , let V x be the isometry on L ( A, K ) defined bythe equation: V x ( f )( y ) := f ( y − x ) if y − x ∈ A, y − x / ∈ A (1.1)for f ∈ L ( A, K ). Then V := { V x } x ∈ P is a strongly continous isometric representationof P on L ( A, K ). We call V the isometric representation associated to the P -module A of multiplicty k . We call the associated CCR flow the CCR flow corresponding to the P -module A of multiplicity k and denote it by α ( A,k ) .Let A be a P -module and z ∈ R d be given. Set B := A + z . Then B is clearly a P -module. It is clear that for k ∈ { , , · · · , } ∪ {∞} , the CCR flow α ( A,k ) is conjugate to α ( B,k ) . This is due to the fact that the associated isometric represenations are unitarilyequivalent. Thus there is no loss of generality in assuming that our P -modules containthe origin 0, which we henceforth assume. The goal of this paper is to prove the followingtheorem. Theorem 1.2
Let A and A be P -modules and k , k ∈ { , , · · · } ∪ {∞} be given.Then the following are equivalent.(1) The CCR flow α ( A ,k ) is conjugate to α ( A ,k ) .(2) The CCR flow α ( A ,k ) is cocycle conjugate to α ( A ,k ) .(3) There exists z ∈ R d such that A + z = A and k = k . ⇒ (2) and (3) = ⇒ (1) are obvious. The difficult part lies inestablishing the implication (2) = ⇒ (3).The above theorem in the 1-dimensional case was proved by Arveson. To see this,observe that when d = 1 i.e. when R d is 1-dimensional the only possible choices of P are [0 , ∞ ) or ( −∞ , , ∞ ) and [ −∞ ,
0) are isomorphic, we can assume that P = [0 , ∞ ). Also note that up to a translate, the only P -module is P = [0 , ∞ )-itself.Then α ([0 , ∞ ) ,k ) is nothing but the usual 1-dimensional CCR flow of index k and the indexis a complete invariant of such CCR flows. For a proof of this well known fact, we referthe reader to [5]. The classification of 1-dimensional CCR flows relies heavily on the factthat the 1-dimensional CCR flows have units in abundance. Though we do not need thefollowing fact , we must mention here that the situation in the multi-dimensional case isdifferent and there are not enough units. However another invariant, the gauge group ofan E -semigroup, comes to our rescue.Since the classification of the 1-dimensional CCR flows is complete, we no longerconcentrate on the 1-dimensional case and we assume from now on that the dimensionof R d i.e. d ≥
2. We now explain the techniques behind the proof of Theorem 1.2. Asalready mentioned the gauge group of an E -semigroup plays a key role in establishingthe proof of Theorem 1.2. Let us recall the notion of the gauge group associated to an E -semigroup.Let α := { α x } x ∈ P be an E -semigroup on B ( H ) where H is an infinite dimensionalseparable Hilbert space. An α -cocycle is called a gauge cocycle if it leaves α invariant.To be precise, let U := { U x } x ∈ P be an α -cocycle. Then U is called a gauge cocycle of α if for x ∈ P and A ∈ B ( H ), U x α x ( A ) U ∗ x = α x ( A ). The set of gauge cocycles of α , knownas the gauge group of α and denoted G ( α ), is a topological group. For U := { U x } x ∈ P and V := { V x } x ∈ P ∈ G ( α ) the multiplication U V is given by
U V := { U x V x } x ∈ P . Theinverse of U is given by U ∗ := { U ∗ x } x ∈ P . The topology on G ( α ) is the topology of uniformconvergence on compact subsets of P , where the topology that we impose on the unitarygroup U ( H ) is the strong operator topology.The main result obtained in [2], which we recall now, is the description of the gaugegroup of a CCR flow associated to a strongly continous isometric representation whichis pure. Let V : P → B ( H ) be an isometric representation. We say that V is pure iffor a ∈ Ω V ∗ ta converges strongly to zero as t tends to infinity (Recall that the Ω is theinterior of P ). It is proved in Prop 4.6 of [2] that isometric representations associatedto P -modules are pure. In what follows, let V : P → B ( H ) be a strongly continousisometric representation which is pure. For a ∈ P , we denote the range projection of V a E a . The orthogonal complement of E a i.e. 1 − E a will be denoted by E ⊥ a . Let M be the commutant of the von Neumann algebra generated by { V x : x ∈ P } . Denote theunitary group of M by U ( M ). We endow U ( M ) with the strong operator topology.Let ξ : P → H be a map and denote the image of x under ξ by ξ x . We say that ξ isan additive cocycle of V in case ξ satisfies the following three conditions:(1) ξ x + V x ξ y = ξ x + y , x, y ∈ P ,(2) V ∗ x ξ x = 0, x ∈ P , and(3) ξ is continuous with respect to the norm topology on H .Let A ( V ) denote the set of additive cocycles of V . We endow A ( V ) with the topologyof uniform convergence on compact subsets of P where H is given the norm topology.The main theorem obtained in [2] (Thm. 7.2) is stated below. Theorem 1.3
The map R d × A ( V ) × U ( M ) ∋ ( λ, ξ, U ) → { e i h λ | x i W ( ξ x )Γ( U E ⊥ x + E x ) } x ∈ P ∈ G ( α ) is a homeomorphism. Now we explain the contents and the organisation of this paper.Let A be a P -module and V be the isometric representation associated to A ofmultiplicity k where k ∈ { , , · · · , } ∪ {∞} . Denote the CCR flow associated to V by α ( A,k ) . In section 2, we show that V admits no non-zero additive cocycle. (Recall thatwe have assumed that the dimension of the vector space R d i.e. d ≥ α ( A,k ) is isomorphicto R d × U ( M ) where M is the commutant of the von Neumann algebra generated by { V x : x ∈ P } and U ( M ) is the unitary group of M endowed with the strong operatortopology. We must mention here that in [2] this result was obtained for a few examplesof R -modules using barehand techniques.In section 3, we compute the commutant M of the von Neumann algebra generatedby { V x : x ∈ P } . It is not difficult to see that it suffices to compute the commutant M when the isometric representation V is of multiplicity 1. Let V : P → B ( L ( A )) be theisometric representation associated to the P -module A of multiplicity 1 and let M bethe commutant of the von Neumann algebra generated by { V x : x ∈ P } . Let G A := { z ∈ R d : A + z = A } .
7t is clear that G A forms a subgroup of R d . We call G A the isotropy group of the P -module A . For z ∈ G A , let U z : L ( A ) → L ( A ) be the unitary defined by the equation U z f ( x ) = f ( x − z )for f ∈ L ( A ). We show that M is generated by { U z : z ∈ G A } . Here is where we employgroupoid techniques. The second author has constructed in [21] a “universal groupoid”which encodes all isometric representations with commuting range projections. We mustmention here that the results obtained in [21] owes a lot to the papers [11], [15] and [9].In Section 4, we prove our main theorem i.e. Theorem 1.2.Let us end this introduction by thanking a few people who has helped us immenselyby sharing their knowledge on Mathematics whenever we were faced with a difficultproblem. First and foremost, we thank R. Srinivasan for introducing us to the beautifultheory of E -semigroups and also for illuminating discussions on the subject. We wouldlike to thank Murugan for useful conversations about the symmetric Fock space, theexponential vectors and the Weyl operators. We thank Prof. Ramadas for directing ustowards the theory of distributions in proving Prop.2.4. We thank Prof. V.S. Sunder forsowing the seeds for the proof of Prop. 4.4. Last but not the least, the second author ishugely indebted to Prof. Renault for passionately sharing his knowledge on groupoidswithout which this paper would not have materialsed.We dedicate this paper in the memory of Prof. Arveson whose ideas have not onlyinspired us but also many others. First we collect a few topological and measure theoretical aspects of P -modules. Let A be a P -module which is fixed for the rest of this section. Recall that we always assumethat 0 ∈ A . We denote the interior of A by Int ( A ) and the boundary of A by ∂ ( A ). Fora proof of the following Lemma, we refer the reader to Lemma II.12 of [9]. Lemma 2.1
We have the following.(1) The interior of A , Int ( A ) is dense in A , and(2) The boundary of A , ∂ ( A ) has measure zero. We need the following topological fact in the sequel.8 emma 2.2
The Interior of A , Int ( A ) and A are connected.Proof. Note that A + Ω ⊂ Int ( A ). Let x, y ∈ Int ( A ) be given. Since Ω spans R d , thereexists b, a ∈ Ω such that x − y = b − a i.e. x + a = y + b . Observe that { x + ta } t ∈ [0 , is a path in Int ( A ) connecting x and x + a . Similarly { y + tb } t ∈ [0 , is a path in Int ( A )connecting y and y + b . Since x + a = y + b , it follows that x is connected to y by apath in Int ( A ). This proves that Int ( A ) is path connected and hence connected. Since Int ( A ) = A , it follows that A is connected. This completes the proof. ✷ We collect in the following proposition a few facts regarding the topology of A andits boundary ∂ ( A ). For E, F ⊂ R d , we denote the complement of F in E by E \ F . Proposition 2.3
Let a ∈ Ω be given. We have the following.(1) The map ∂ ( A ) × (0 , ∋ ( x, s ) → x + sa ∈ Int ( A ) \ ( A + a ) is continuous and isonto.(2) The sequence { A + na } n ≥ is a decreasing sequence of closed subsets which decreasesto the empty set i.e. T ∞ n =1 ( A + na ) = ∅ .(3) The map ∂ ( A ) × (0 , ∞ ) ∋ ( x, s ) → x + sa ∈ Int ( A ) is a homemorphism.(4) The boundary ∂ ( A ) is connected.(5) The boundary ∂ ( A ) is unbounded.(6) The set Int ( A ) \ ( A + a ) has infinite Lebesgue measure.Proof. Let ( x, s ) ∈ ∂ ( A ) × (0 ,
1) be given. Since A + Ω is an open subset of R d containedin A , it follows that A + Ω ⊂ Int ( A ). This implies that x + sa ∈ Int ( A ). Nowsuppose x + sa ∈ A + a . Then there exists y ∈ A such that x + sa = y + a , i.e. x = y + (1 − s ) a ∈ A + Ω ⊂ Int ( A ). This contradicts the fact that x ∈ ∂ ( A ). Hence x + sa / ∈ A + a . Thus we have shown that the map prescribed in (1) is meaningful. Thecontinuity of the prescribed map is obvious.Now let z ∈ Int ( A ) \ ( A + a ) be given. Let E := { t ∈ [0 ,
1] : z − ta ∈ Int ( A ) } . Note that E contains 0 and is an open subset of [0 , E by s .Since E is open in [0 ,
1] and contains 0, it follows that s >
0. Note that z − a / ∈ A and A is a closed subset of R d . Thus for t sufficiently close to 1, z − ta / ∈ A . This proves that9 <
1. Hence 0 < s <
1. As E is open in [0 , s / ∈ E i.e. z − sa / ∈ Int ( A ).Choose a sequence s n ∈ E such that s n → s . Then z − s n a ∈ Int ( A ) ⊂ A and z − s n a → z − sa . Since A is closed in R d , it follows that z − sa ∈ A . As a consequence,we have z − sa ∈ ∂ ( A ). Now note that z = ( z − sa ) + sa . This proves that the map ∂ ( A ) × (0 , ∋ ( x, s ) → x + sa ∈ Int ( A ) \ ( A + a )is onto. This proves (1).Since A + P ⊂ A , it is clear that { A + na } n ≥ is a decreasing sequence of closedsubsets. Suppose x ∈ T ∞ n =1 ( A + na ). This implies that x − na ∈ A for every n ≥
1. Let y ∈ R d be given. By the Archimedean principle there exists a positive integer n such that n a − ( x − y ) ∈ Ω. As A +Ω ⊂ A , it follows that y = ( x − n a )+( n a − ( x − y )) ∈ A +Ω ⊂ A .This proves that y ∈ A for every y ∈ R d which is a contradiction since A is a properclosed subset of R d . This proves (2).The well-definedness of the map in (3) is clear as A + Ω ⊂ Int ( A ). Let y ∈ Int ( A )be given. By (2), there exists n ≥ y / ∈ A + na . Now by (1), applied tothe interior point na , it follows that there exists s ∈ (0 ,
1) and x ∈ ∂ ( A ) such that y = x + s ( na ). This proves that the map ∂ ( A ) × (0 , ∞ ) ∋ ( x, s ) → x + sa ∈ Int ( A )is onto. Let x , x ∈ ∂ ( A ) and s , s ∈ (0 , ∞ ) be such that x + s a = x + s a . We claimthat x = x and s = s . It suffices to show that s = s . Suppose not. Without loss ofgenerality, we can assume s > s . Then x = x + ( s − s ) a ∈ A + Ω ⊂ Int ( A ) whichcontradicts the fact that x ∈ ∂ ( A ). This proves our claim. In other words, the map ∂ ( A ) × (0 , ∞ ) ∋ ( x, s ) → x + sa ∈ Int ( A )is an injection. It is clear that the above map is continuous. Let ( x n , s n ) ∈ ∂ ( A ) × (0 , ∞ )be a sequence and ( x, s ) ∈ ∂ ( A ) × (0 , ∞ ) be such that x n + s n a → x + sa . We claim that x n → x and s n → s . It is enough to prove that s n → s . Suppose s n s . Then thereexists ǫ > s n / ∈ ( s − ǫ, s + ǫ ) for infinitely many n . Suppose s n ≤ s − ǫ forinfinitely many n . Choose a subsequence s n k such that s n k ∈ (0 , s − ǫ ]. By passing to asubsequence if necessary we can assume that s n k converges, say to, t . Then t < s . Nownote that x n k → x + ( s − t ) a ∈ A + Ω ⊂ Int ( A ). This is a contradiction since x n k ∈ ∂ ( A )and ∂ ( A ) is a closed subset of R d which is disjoint from Int ( A ).Now suppose that s n ≥ s + ǫ for infinitely many n . Choose a subsequence ( s n k ) such10hat s n k ≥ s + ǫ . Write s n k = t n k + s + ǫ with t n k ≥
0. Note that x n k + t n k a + ǫa = ( x n k + s n k a ) − sa → x But x n k + t n k a + ǫa ∈ A + ǫa and A + ǫa is a closed subset of R d . This implies that x ∈ A + ǫa ⊂ A + Ω ⊂ Int ( A ). This contradicts the fact that x ∈ ∂ ( A ). Thesecontradictions imply that our assumption s n s is wrong and hence s n → s . Hence themap ∂ ( A ) × (0 , ∞ ) ∋ ( x, s ) → x + sa ∈ Int ( A )is a homeomorphism. This proves (3). It is immediate that (3) implies (4).Suppose ∂ ( A ) is bounded. Since 0 ∈ A and A + Ω ⊂ Int ( A ), it follows that Ω ⊂ Int ( A ). Let b ∈ Ω be given. By (3), there exists a sequence ( s n ) ∈ (0 , ∞ ) and x n ∈ ∂ ( A )such that nb − s n a = x n . Since ∂ ( A ) is bounded, it follows that b − s n n a = x n n →
0. Inother words, s n n a → b . Note that { sa : s ≥ } is a closed subset of R d . Hence thereexists s ≥ b = sa . Since Ω − Ω = R d , it follows that the linear span of a is R d . This implies that d = 1 which is a contradiction to our assumption that d ≥
2. Thiscontradiction implies that ∂ ( A ) is unbounded.Let E := { x ∈ R d : 0 < x < a } . Then E is a non-empty (as a ∈ E ) open and boundedset. For a proof of this fact, we refer the reader to the first line of the proof of Prop. I.1.8in [8]. Let M := sup {|| x || : x ∈ E } where the norm on R d is the usual Euclidean norm.Since a ∈ E , it follows that M ≥ || a || . Thus M >
0. The unboundedness of ∂ ( A ) impliesthat there exists a sequence { x n } n ≥ in ∂ ( A ) such that || x n − x m || ≥ M if n = m . Weclaim the following.(i) For n ≥ E + x n ⊂ Int ( A ) \ ( A + a ), and(ii) the sequence { E + x n } n ≥ forms a disjoint family of non-empty open subsets of R d .Let n ≥ E ⊂ Ω. Since Ω + A ⊂ Int ( A ), it follows that E + x n iscontained in the interior of A . Suppose the intersection ( E + x n ) ∩ ( A + a ) = ∅ . Thenthere exists e ∈ E , y ∈ A such that e + x n = y + a . Hence x n = y + ( a − e ) ∈ A + Ω ⊂ Int ( A ) . This implies that x n ∈ Int ( A ) which contradicts the fact that x n ∈ ∂ ( A ). This contradic-tion implies that the intersection ( E + x n ) ∩ ( A + a ) = ∅ . Hence E + x n ⊂ Int ( A ) \ ( A + a ).This proves ( i ). 11et m, n ≥ m = n . Suppose that the intersection ( E + x n ) ∩ ( E + x m )is non-empty. Then there exists e , e ∈ E such that e + x n = e + x m . Now observethat 3 M ≤ || x n − x m || = || e − e || ≤ || e || + || e || ≤ M which is a contradiction since M >
0. This proves that the intersection ( E + x n ) ∩ ( E + x m )is empty. This proves ( ii ).Let λ be the Lebesgue measure on R d . Since E is a non-empty open subset of R d , itfollows that λ ( E ) >
0. Now calculate as follows to observe that ∞ = ∞ X n =1 λ ( E )= ∞ X n =1 λ ( E + x n )= λ (cid:0) ∞ a n =1 E + x n (cid:1) ≤ λ ( Int ( A ) \ ( A + a )) . Hence
Int ( A ) \ ( A + a ) has infinite Lebesgue measure. This proves (6) and the proofs arenow complete. ✷ The next proposition shows that the isometric representation associated to A ofmultiplicity 1 admits no non-zero additive cocycles. Proposition 2.4
Let V : P → B ( L ( A )) be the isometric representation associated tothe P -module A of multiplicity . Suppose that { ξ x } x ∈ P is an additive cocycle of V . Thenfor every x ∈ P , ξ x = 0 .Proof. Fix a ∈ Ω. Since V ∗ a ξ a = 0, it follows that ξ a ( x ) = 0 for almost all x ∈ A + a .Without loss of generality, we can assume that ξ a ( x ) = 0 for all x ∈ A + a . Also A \ ( A + a )and Int ( A ) \ ( A + a ) differ by a set of measure zero. For, the boundary ∂ ( A ) has measurezero. Thus without loss of generality, we can assume that ξ a ( x ) = 0 for x ∈ ∂ ( A ).Let U := Int ( A ) \ ( A + a ). By (1) and (4) of Prop.2.3, it follows that U is a non-emptyopen connected subset of R d . Note that the complex conjugate of ξ a , i.e. ξ a ∈ L ( U ) ⊂ L loc ( U ). Thus, we view ξ a as a distribution on U . Let φ : U → R be a smooth functionsuch that supp ( φ ) is compact and supp ( φ ) ⊂ U . Denote the support of φ by K . We view φ as a smooth function on R d by declaring its value on the complement of U to be zero.We denote its i th partial derivative of φ by ∂ i φ . For x ∈ R d , let ∇ φ ( x ) be the gradient12f φ i.e. ∇ φ ( x ) = ( ∂ φ ( x ) , ∂ φ ( x ) , · · · , ∂ n φ ( x )) . Let M := sup x ∈ R d ||∇ φ ( x ) || . Fix b ∈ Ω such that || b || = 1. We claim that there exists δ > < t < δ then K ∩ ( A \ ( A + tb )) = ∅ . Suppose not. Then there exists a sequence ( x n ) ∈ K anda sequence of positive real numbers t n → x n ∈ A \ ( A + t n b ). By passing toa subsequence, if necessary, we can assume that x n converges say to x ∈ K . Note that K ⊂ Int ( A ) and x n − t n b → x ∈ Int ( A ). Hence eventually x n ∈ Int ( A ) + t n b ⊂ A + t n b which is a contradiction to the fact that x n ∈ A \ ( A + t n b ). This proves our claim. Choosesuch a δ .Let δ > δ < δ and K + B (0 , δ ) ⊂ U . Let L := K + B (0 , δ ). Note that L is a compact subset of U . Let ( t n ) be a sequence of positive numbers such that t n < δ and t n →
0. Note that by the mean value inequality, we have for x ∈ R d and n ≥ (cid:12)(cid:12)(cid:12) φ ( x + t n b ) − φ ( x ) t n (cid:12)(cid:12)(cid:12) ≤ M L ( x ) . (2.2)Note that since K ∩ ( A \ ( A + t n b )) = ∅ , the inner product h ξ t n b | φ i = 0. Now calculate asfollows to observe that Z U φ ( x + t n b ) − φ ( x ) t n ξ a ( x ) dx = 1 t n (cid:16) h ξ a | V ∗ t n b φ i − h ξ a | φ i (cid:17) = 1 t n (cid:16) h V t n b ξ a | φ i − h ξ a | φ i (cid:17) = 1 t n (cid:16) h ξ t n b + V t n b ξ a | φ i − h ξ a | φ i (cid:17) = 1 t n (cid:16) h ξ t n b + a | φ i − h ξ a | φ i (cid:17) = 1 t n (cid:16) h ξ a + V a ξ t n b | φ i − h ξ a | φ i (cid:17) = 1 t n h ξ t n b | V ∗ a φ i = 0 ( since φ vanishes on A + a ) . Thus we obtain, for n ≥
1, the equation Z U φ ( x + t n b ) − φ ( x ) t n ξ a ( x ) dx = 0 (2.3)13or x ∈ U and n ≥
1, Eq.2.2 implies that (cid:12)(cid:12)(cid:12)(cid:16) φ ( x + t n b ) − φ ( x ) t n (cid:17) ξ a ( x ) (cid:12)(cid:12)(cid:12) ≤ M L ( x ) | ξ a ( x ) | . The function U ∋ x → L ( x ) | ξ a ( x ) | ∈ [0 , ∞ ) is integrable. Thus letting n → ∞ in Eq.2.3and applying the dominated convergence theorem, we obtain Z U h∇ φ ( x ) | b i ξ a ( x ) = 0 . Since t Ω = Ω for every t >
0, it follows that for every b ∈ Ω, Z U h∇ φ ( x ) | b i ξ a ( x ) = 0 . Since Ω is spanning, it follows that for every z ∈ R d , R U h∇ φ ( x ) | z i ξ a ( x ) = 0 . Let e , e , · · · , e d be the standard orthonormal basis of R d . Then for every i = 1 , , · · · , d , Z U ∂ i φ ( x ) ξ a ( x ) dx = Z U h∇ φ ( x ) | e i i ξ a ( x ) = 0 . Hence each partial derivative of ξ a , in the distribution sense, vanishes. By Theorem 6.3-4of [6], it follows that there exists a complex number c a such that ξ a ( x ) = c a for almost all x ∈ U . Since ξ a ∈ L ( A ) and U has infinite measure by Prop.2.3, it follows that c a = 0.Hence ξ a = 0 for every a ∈ Ω. The density of Ω in P and the continuity of the map P ∋ ξ → ξ x ∈ L ( A ) implies that ξ x = 0 for every x ∈ P . This completes the proof. ✷ Remark 2.5
Let k ∈ { , , · · · } ∪ {∞} be given. For each i , let H i be a Hilbert space.Denote the direct sum L H i by H . For a vector ξ ∈ H , we denote its i th -componentby ξ i . For each i , let V i := { V ix } x ∈ P be an isometric representation of P on H i . Let V := L V i be the direct sum. Then cleary V is an isometric representation of P on H .If each V i is strongly continous then V is strongly continous.If ξ := { ξ x } x ∈ P is an additive cocycle of V , then ξ i := { ξ ix } x ∈ P is an additive cocycleof V i for each i . Thus if each V i admits no non-trivial additive cocycle then V admitsno non-trivial additive cocycle. Now the following is an immediate corollary of Prop.2.4 and Remark 2.5.
Corollary 2.6
Let k ∈ { , , · · · , } ∪ {∞} and K be a Hilbert space of dimension k . Let V : P → B ( L ( A ) ⊗ K ) be the isometric representation associated to the P -module A ofmultiplicity k . Suppose that ξ := { ξ x } x ∈ P is an additive cocycle of V . Then ξ x = 0 forevery x ∈ P . k ∈ { , , · · · , } ∪ {∞} and K be a Hilbert space of dimension k . Let V bethe isometric representation of P on the Hilbert space L ( A ) ⊗ K associated to the P -module A of multiplicity k . For x ∈ P , we denote the range projection of V x by E x . Theorthogonal complement of E x i.e. 1 − E x will be denoted by E ⊥ x . Denote the commutantof the von Neumann algebra generated by { V x : x ∈ P } by M . Denote the unitary groupof M by U ( M ). We endow U ( M ) with the strong operator topology. Let α ( A,k ) be theCCR flow associated to the isometric representation V and denote the gauge group of α ( A,k ) by G ( α ( A,k ) ). Theorem 2.7
With the foregoing notation, the map R d × U ( M ) ∋ ( λ, U ) → { e i h λ | x i Γ( U E ⊥ x + E x ) } x ∈ P ∈ G ( α ( A,k ) ) is a homeomorphism where the topology on R d × U ( M ) is the product topology. Let A be a P -module, k ∈ { , , · · · }∪{∞} and K be a Hilbert space of dimension k . Let V : P → B ( L ( A ) ⊗ K ) be the isometric representation associated to A of multiplicity k . The goal of this section is to compute the commutant of the von Neumann algebragenerated by { V x : x ∈ P } . First we compute the commutant when the multiplicity k = 1. This relies heavily on the groupoid approach developed in [21] to study C ∗ -algebras arising out of Ore semigroup actions.We must mention here that the results obtained in [21] are due to the deep insightof Muhly and Renault in using groupoids to understand the Wiener-Hopf C ∗ -algebras.This is achieved in their seminal paper [11]. This view was further developed by Nica in[15] and Hilgert and Neeb in [9]. The results obtained in [21] also owes a lot to [9]. Forcompleteness, we review the basics of groupoid C ∗ -algebras. For a quick introduction tothe theory of groupoids and the associated C ∗ -algebras, we either recommend the firsttwo sections of [10] or the second section of [11]. For a more detailed study of groupoids,we refer the reader to [18]. We recall here the basics of C ∗ -algebras associated to atopological groupoid.Let G be a topological groupoid with a left Haar system. We assume that G is locallycompact, Hausdorff and second countable. The unit space of G will be denoted by G (0) and let r, s : G → G (0) be the range and source maps. For x ∈ G (0) , let G ( x ) = r − ( x ).Fix a left Haar system ( λ ( x ) ) x ∈G (0) . Let C c ( G ) be the space of continuous complex valued15unctions defined on G . The space C c ( G ) forms a ∗ -algebra where the multiplication andthe involution are defined by the following formulas f ∗ g ( γ ) = Z f ( η ) g ( η − γ ) dλ ( r ( γ )) ( η ) f ∗ ( γ ) = f ( γ − )for f, g ∈ C c ( G ). We obtain bounded representations of the ∗ -algebra C c ( G ) as follows.Fix a point x ∈ G (0) . Consider the Hilbert space L ( G ( x ) , λ ( x ) ). For f ∈ C c ( G ) and ξ ∈ L ( G ( x ) , λ ( x ) ), let π x ( f ) ξ ∈ L ( G ( x ) , λ ( x ) ) be defined by the formula( π x ( f ) ξ )( γ ) := Z f ( γ − γ ) ξ ( γ ) dλ ( x ) ( γ )for γ ∈ G ( x ) . Then π x : C c ( G ) → B ( L ( G ( x ) , λ ( x ) ) is a non-degenerate ∗ -representation.Moreover π x is continuous when C c ( G ) is given the inductive limit topology. For f ∈ C c ( G ), let || f || red := sup x ∈G (0) || π x ( f ) || . Then || || red is well defined and is a C ∗ -norm on C c ( G ). The completion of C c ( G ) withrespect to the norm || || red is called the reduced C ∗ -algebra of G and denoted C ∗ red ( G ).There is also a universal C ∗ -algebra associated to G and denoted C ∗ ( G ). However wedo not need the universal one as the groupoids that we consider are amenable and foramenable groupoids C ∗ red ( G ) and C ∗ ( G ) coincide. Fix x ∈ G (0) . We denote the extensionof π x to C ∗ ( G ) by π x itself. The representation π x is called the representation of C ∗ ( G ) induced at the point x .Let γ ∈ G be such that s ( γ ) = x and r ( γ ) = y . Let U γ : L ( G ( y ) , λ ( y ) ) → L ( G ( x ) , λ ( x ) )be defined by the formula U γ ξ ( γ ) = ξ ( γγ )for ξ ∈ L ( G ( y ) , λ ( y ) ). The fact that ( λ ( x ) ) x ∈G (0) is a left Haar system implies that U γ is aunitary. Moreover it is routine to verify that U γ intertwines the representations π x and π y , i.e. for f ∈ C c ( G ), U γ π y ( f ) = π x ( f ) U γ .We need the following two facts.(1) Let x ∈ G (0) be given. Denote the isotropy group at x by G xx i.e. G xx := { γ ∈ G : r ( γ ) = s ( γ ) = x } . Note that G xx is a group. The commutant of { π x ( f ) : f ∈ C c ( G ) } is generated by { U γ : γ ∈ G xx } . 162) For x, y ∈ G (0) , the representations π x and π y are non-disjoint if and only if thereexists γ ∈ G such that s ( γ ) = x and r ( γ ) = y .Connes proved the above two facts in his paper [7]. In the appendix, we offer a proof for(1) and (2) for Deaconu-Renault groupoids considered by the second author and Renaultin [17] which is all we need. We believe that the appendix is interesting on its own rightas the proof uses notions like groupoid equivalence and Rieffel’s notion of strong Moritaequivalence.Let us recall the Deaconu-Renault groupoid considered in [17]. Let X be a compactmetric space. By an action of P on X , we mean a continuous map X × P ∋ ( x, t ) → x + t ∈ X such that x +0 = x and ( x + s )+ t = x +( s + t ) for x ∈ X and s, t ∈ P . We assumethat the action of P on X is injective i.e. for every t ∈ P , the map X ∋ x → x + t ∈ X is injective. Let X ⋊ P := { ( x, t, y ) ∈ X × R d × X : ∃ r, s ∈ P such that t = r − s and x + r = y + s } . The set X ⋊ P has a groupoid structure where the groupoid multiplication and theinversion are given by ( x, s, y )( y, t, z ) = ( x, s + t, z ) , and( x, s, y ) − = ( y, − s, x ) . We call X ⋊ P the Deaconu-Renault groupoid determined by the action of P on X . Theset X ⋊ P is a closed subset of X × R d × X . When endowed with the subspace topology, X ⋊ P becomes a topological groupoid. The map X ⋊ P ∋ ( x, t, y ) → ( x, t ) ∈ X × R d isan embedding and the range of the prescribed map is a closed subset of X × R d . Fromhere on, we always consider X ⋊ P as a subspace of X × R d .For x ∈ X , let Q x := { t ∈ R d : ( x, t ) ∈ X ⋊ P } . Note that for x ∈ X , Q x is a closed subset of R d containing the origin 0 and Q x + P ⊂ Q x .By Lemma 4.1 of [17], it follows that Int ( Q x ) is dense in Q x and the boundary ∂ ( Q x )has Lebesgue measure zero.For x ∈ X , let λ ( x ) be the measure on X ⋊ P defined by the following formula: For f ∈ C c ( X ⋊ P ), Z f dλ ( x ) = Z f ( x, t )1 Q x ( t ) dt. (3.4)Here dt denotes the usual Lebesgue measure on R d . The groupoid X ⋊ P admits a Haarsystem if and only if the map X × Ω ∋ ( x, s ) → x + s ∈ X is open. In such a case,17 λ ( x ) ) x ∈ X forms a left Haar system. When X ⋊ P admits a Haar system, we use only theHaar system described above.Assume that X ⋊ P has a Haar system. Then the action of P on X can be dilatedto an action of R d on a locally compact space Y . More precisely, there exists a locallycompact Hausdorff space Y , an action of R d on Y , Y × R d ∋ ( y, t ) → y + t , an embedding i : X → Y such that(1) the embedding i : X → Y is P -equivariant,(2) the set X := i ( X ) + Ω is open in Y , and(3) the set Y = [ t ∈ P ( i ( X ) − t ) = [ t ∈ Ω ( X − t ).The space Y is uniquely determined by conditions (1), (2) and (3) up to an R d -equivariantisomorphism. We suppress the notation i and simply identify X as a subspace of Y . Wecall the pair ( Y, R d ) the dilation associated to the pair ( X, P ). Moreover the groupoid X ⋊ P is merely the reduction of the transformation groupoid Y ⋊ R d onto X , i.e. X ⋊ P := ( Y ⋊ R d ) | X . Also the groupoids X ⋊ P and Y ⋊ R d are equivalent in thesense of [12]. Since Y ⋊ R d is amenable, it follows from Theorem 2.2.17 of [1] that X ⋊ P is amenable. For proofs and details of the facts about Deaconu-Renault groupoids(mentioned in the previous paragraphs), we refer the reader to [17].Let X be a compact metric space and X × P ∋ ( x, t ) → x + t ∈ X be an action of P on X . Assume that X ⋊ P has a Haar system. Denote the dilation associated to ( X, P )by ( Y, R d ). Let G := X ⋊ P and H := Y ⋊ R d . The range and source maps of both G and H will be denoted by r and s respectively. Fix x ∈ X and let Q x := { t ∈ R d : x + t ∈ X } . Note that G ( x ) := r − ( x ) := { x } × Q x . Thus the Hilbert space L ( G ( x ) , λ ( x ) ) can beidentified with L ( Q x ) = L ( Q x , dt ) where dt denotes the Lebesgue measure on R d . Let G xx be the isotropy group of G at x i.e. G xx := { t ∈ R d : x + t = x } . Note that for t ∈ R d , t ∈ G xx if and only if ( x, t, x ) ∈ X ⋊ P . For s ∈ G xx , let U s be the unitary on L ( Q x )defined by the following formula ( U s ξ )( t ) = ξ ( t − s )for ξ ∈ L ( Q x ). Theorem 3.1
With the foregoing notation, we have the following.
1) For x ∈ X , let π x be the representation of C ∗ ( G ) induced at x . Then the commutantof { π x ( f ) : f ∈ C c ( G ) } is the von Neumann algebra generated by { U s : s ∈ G xx } .(2) Let x, y ∈ X be given. Then π x and π y are non-disjoint if and only if there exists t ∈ R d such that x + t = y i.e. ( x, t, y ) ∈ G . We provide a proof of the above theorem in the appendix.Let V : P → B ( H ) be a strongly continous isometric representation with commutingrange projections. More precisely, let E x := V x V ∗ x for x ∈ P . We say that V has commuting range projections if { E x : x ∈ P } is a commuting family of projections. For z ∈ R d , write z = x − y with x, y ∈ P and let W z := V ∗ y V x . Then W z is well-definedand is a partial isometry for every z ∈ R d . Also { W z } z ∈ R d forms a strongly continuousfamily of partial isometries. We refer the reader to Prop.3.4 of [21] for proofs of theabove mentioned facts. For f ∈ C c ( R d ), let W f := Z f ( z ) W z dz. For f ∈ C c ( R d ), W f is called the Wiener-Hopf operator with symbol f . Lemma 3.2
With the foregoing notation, the von Neumann algebra generated by the setof Wiener-Hopf operators { W f : f ∈ C c ( R d ) } and the von Neumann algebra generatedby { V x : x ∈ P } coincide.Proof. It is clear that the von Neumann algebra generated by { W f : f ∈ C c ( R d ) } iscontained in the von Neumann algebra generated by { V x : x ∈ P } . Let z ∈ R d be given.For n ≥
1, let B ( z , n ) be the open ball centred at z and of radius n . For n ≥
1, choosea function f n ∈ C c ( R d ) such that f n ≥ R f n ( z ) dz = 1 and supp ( f n ) ⊂ B ( z , n ). Weclaim that R f n ( z ) W z dz → W z weakly.Let ξ, η ∈ H and ǫ > { W z } z ∈ R d is strongly continuous, it followsthat there exists N ≥ |h W z ξ | η i − h W z ξ | η i| ≤ ǫ for every z ∈ B ( z , N ). Let19 ≥ N be given. Calculate as follows to observe that (cid:12)(cid:12) h (cid:0) Z f n ( z ) W z dz (cid:1) ξ | η i − h W z ξ | η i (cid:12)(cid:12) = (cid:12)(cid:12) Z f n ( z ) h W z ξ | η i dz − Z f n ( z ) h W z ξ | η i dz (cid:12)(cid:12) = (cid:12)(cid:12) Z f n ( z )( h W z ξ | η i − h W z ξ | η i ) dz (cid:12)(cid:12) ≤ Z z ∈ B ( z , N ) f n ( z ) (cid:12)(cid:12) h W z ξ | η i − h W z ξ | η i (cid:12)(cid:12) dz ≤ ǫ Z f n ( z ) dz ≤ ǫ This proves that R f n ( z ) W z dz → W z . Now it is immediate that the von Neumannalgebra generated by { V x : x ∈ P } is contained in the von Neumann algebra generatedby { W f : f ∈ C c ( R d ) } . This completes the proof. ✷ Next we recall the universal groupoid constructed in [21]. Denote the set of closedsubsets of R d by C ( R d ). Let X u := { A ∈ C ( R d ) : 0 ∈ A, − P + A ⊂ A } . Consider L ∞ ( R d ) as the dual of L ( R d ) and endow L ∞ ( R d ) with the weak ∗ -topology.The map X u ∋ A → A ∈ L ∞ ( R d ) is injective. Via this injection, we view X u as a subsetof L ∞ ( R d ) and endow X u with the subspace topology inherited from the weak ∗ -topologyon L ∞ ( R d ). The space X u is a compact metric space. The map X u × P ∋ ( A, x ) → A + x ∈ X u provides us with an injective action of P on X u . Moreover the map X u × Ω ∋ ( A, x ) → A + x ∈ X u is open. See Prop.4.4 and Remark 4.5 of [21] for a proof of this fact.Consequently, the Deaconu-Renault groupoid X u ⋊ P has a Haar system. We denotethe Deaconu-Renault groupoid X u ⋊ P by G u . The range and source maps of G u will bedenoted by r and s respectively. We need the following two facts about the groupoid G u .For proofs, we refer the reader to Remark 4.5 and Prop.2.1 of [21].(1) For A ∈ X u , let Q A := { z ∈ R d : ( A, z ) ∈ X u ⋊ P } . Then Q A = − A for every A ∈ X u .(2) For f ∈ C c ( R d ), let e f ∈ C c ( G u ) be defined by the equation e f ( A, z ) := f ( z ) (3.5)for ( A, z ) ∈ G u . Then C ∗ ( G u ) is generated by { e f : f ∈ C c ( R d ) } .20et ( λ ( A ) ) A ∈ X u be the Haar system on G u defined by the equation 3.4. Fix A ∈ X u . Then G Au := r − ( A ) = { A } × Q A = { A } × − A . We identify L ( G Au , λ A ) with L ( − A ).Let A be a P -module and V : P → B ( L ( A )) be the isometric representation asso-ciated to A of multiplicity 1. Let { W z } z ∈ R d be the partial isometries, described in theparagraph following Theorem 3.1, associated to the isometric representation V . Notethat − A ∈ X u . Denote the representation of C ∗ ( G u ) induced at − A by π A . Proposition 3.3 . With the foregoing notation, we have π A ( e f ) = Z f ( − z ) W z dz for every f ∈ C c ( R d ) . Here for f ∈ C c ( R d ) , e f ∈ C c ( G u ) is as defined in Equation 3.5. We omit the proof of the above proposition as it is similar to the calculations carried outin the two paragraphs following Remark 5.3 of [17].Fix a P -module say A for the rest of this section. Let V : P → B ( L ( A )) be theisometric representation associated to A of multiplicity 1. Let G A be the isotropy groupof A , i.e. G A := { z ∈ R d : A + z = A } . For z ∈ G A , let U z be the unitary defined on L ( A ) by the equation U z f ( x ) := f ( x − z ) (3.6)for f ∈ L ( A ).The following corollary is an immediate consequence of Theorem 3.1, Prop. 3.3,Lemma 3.2 and the fact that { e f ∈ C c ( G u ) : f ∈ C c ( R d ) } generates C ∗ ( G u ). Corollary 3.4
With the foregoing notation, we have that the commutant of the vonNeumann algebra generated by { V x : x ∈ P } coincides with the von Neumann algebragenerated by { U z : z ∈ G A } . Let k ∈ { , , · · · , } ∪ {∞} be given and let K be a Hilbert space of dimension k . Let V : P → B ( L ( A ) ⊗ K ) be the isometric representation associated to A of multiplicity k . Denote the isometric representation associated to A of multiplicity 1 by e V . Then it isclear that for x ∈ P , V x = e V x ⊗
1. Let N be the von Neumann algebra on L ( A ) generatedby { e V x : x ∈ P } . Denote the commutant of N by M i.e. M = N ′ . Corollary 3.4 impliesthat M is generated by { U z : z ∈ G A } . In particular, M is abelian. Denote the CCRflow associated to the isometric representation V by α ( A,k ) . Let G ( α ( A,k ) ) denote thegauge group of α ( A,k ) . The following corollary is now immediate.21 orollary 3.5 With the foregoing notation, the commutant of the von Neumann algebragenerated by { V x : x ∈ P } is M ⊗ B ( K ) . The gauge group of α ( A,k ) i.e. G ( α ( A,k ) ) isisomorphic to R d × U ( M ⊗ B ( K )) , where U ( M ⊗ B ( K )) is the unitary group of M ⊗ B ( K ) endowed with the strong operator topology.Proof. The von Neumann algebra generated by { V x : x ∈ P } is N ⊗ { T ⊗ T ∈ N } .Hence the commutant of N ⊗ M ⊗ B ( K ). The fact that G ( α ( A,k ) ) is isomorphic to R d × U ( M ⊗ B ( K )) follows from Theorem 2.7. This completes the proof. ✷ In this section, we prove
Theorem 1.2 . We need a basic fact regarding the represen-tation theory of the unitary group of an n -dimensional Hilbert space. We start with acombinatorial lemma. Fix ℓ ≥
2. Denote the permutation group on { , , · · · , ℓ } by S ℓ .For i, j ∈ { , , · · · , ℓ } and i = j , the permutation that interchanges i and j and leavesthe rest fixed will be denoted by ( i, j ). For σ ∈ S ℓ − , let b σ ∈ S ℓ be defined by b σ ( i ) = σ ( i )for 1 ≤ i ≤ ℓ − b σ ( ℓ ) = ℓ . Via the embedding S ℓ − ∋ σ → b σ ∈ S ℓ , we view S ℓ − asa subgroup of S ℓ . For m := ( m , m , · · · , m ℓ ) ∈ Z ℓ and σ ∈ S ℓ , let m σ := ( m σ (1) , m σ (2) , · · · , m σ ( ℓ ) ) . Lemma 4.1
Let ℓ ≥ and m := ( m , m , · · · , m ℓ ) ∈ Z ℓ be given. Suppose that thereexists i, j ∈ { , , · · · , ℓ } such that i = j and m i = m j . Then the cardinality of the set { m σ : σ ∈ S ℓ } is at least ℓ .Proof. We prove this by induction on ℓ . The base case i.e. when ℓ = 2 is clearlytrue. Fix ℓ ≥ ℓ −
1. Choose i, j ∈ { , , · · · , ℓ } such that i = j and m i = m j . Replacing m by m σ for a suitable σ ifnecessary, we can without loss of generality assume that i = 1. Case 1: ≤ j ≤ ℓ −
1. Then by the induction hypothesis, the cardinality of the set { m b σ : σ ∈ S ℓ − } is at least ℓ −
1. Since m = m j either m = m ℓ or m j = m ℓ . Hencethere exists i ∈ { , j } such that m i = m ℓ . Then { m ( i,ℓ ) } is disjoint from { m b σ : σ ∈ S ℓ − } .This proves that the cardinality of the set { m σ : σ ∈ S ℓ } is at least ℓ . Case 2: j = ℓ . If there exists k ∈ { , , · · · , ℓ − } such that m = m k then byCase 1, we have that the cardinality of the set { m σ : σ ∈ S ℓ } is at least ℓ . Now assumethat m = m k for every k ∈ { , , · · · , ℓ − } . Then note that the cardinality of the set22 m ( i,ℓ ) : 1 ≤ i ≤ ℓ } is ℓ and hence the cardinality of the set { m σ : σ ∈ S ℓ } is at least ℓ .This completes the proof. ✷ Consider the n -dimensional Hilbert space C n with the usual Euclidean inner prod-uct. For i = 1 , , · · · , n , let e i ∈ C n be the vector which has 1 in the i th -coordinateand zero elsewhere. Denote the unitary group of C n endowed with the norm topologyby U ( n ). The special unitary group i.e. the set of unitary operators with determinantone will be denoted by SU ( n ). Let T n be the subgroup of U ( n ) consisting of diago-nal matrices. For λ ∈ T n and 1 ≤ i ≤ n , denote the ( i, i ) th -entry of λ by λ i . Given λ , λ , · · · , λ n ∈ T , the diagonal matrix with diagonal entries λ , λ , · · · , λ n will be de-noted by diag ( λ , λ , · · · , λ n ). Here by T , we mean the unit circle of the complex plane.For σ ∈ S n , let U σ be the unitary on C n such that U σ ( e i ) = e σ ( i ) for i ∈ { , , · · · , n } .For λ := diag ( λ , λ , · · · , λ n ) ∈ T n and σ ∈ S n , let λ σ := diag ( λ σ (1) , λ σ (2) , · · · , λ σ ( n ) ).Note that for σ ∈ S n and λ ∈ T n , U σ λU ∗ σ = λ σ − . The following proposition may beknown to experts. We use the notation introduced in the preceding two paragraphs inthe proof of the following proposition. Proposition 4.2
Let n ≥ , H be a Hilbert space and ρ be a strongly continuous unitaryrepresentation of U ( n ) on H . Suppose that the dimension of H is strictly less than n .Then ρ ( U ) = 1 for every U ∈ SU ( n ) . Here denotes the identity operator on H .Proof. For m := ( m , m , · · · , m n ) ∈ Z n , let H m := { v ∈ H : for every λ ∈ T n , ρ ( λ ) v = λ m λ m · · · λ m n n v } . Restrict ρ to the compact abelian group T n . Then the Hilbert space H decomposes as H = M m ∈ Z n H m .Fix m := ( m , m , · · · , m n ) ∈ Z n . Suppose H m = 0. Then m i = m j for every i, j ∈ { , , · · · , n } . Suppose not. Then there exists i, j ∈ { , , · · · , n } such that i = j and m i = m j . Let v ∈ H m , λ := diag ( λ , λ , · · · , λ n ) ∈ T n and σ ∈ S n be given.Calculate as follows to observe that ρ ( U σ ) ρ ( λ ) ρ ( U σ ) ∗ v = ρ ( U σ λU ∗ σ ) v = ρ ( λ σ − ) v = λ m σ − (1) λ m σ − (2) · · · λ m n σ − ( n ) v = λ m σ (1) λ m σ (2) · · · λ m σ ( n ) n v. The calculation implies that ρ ( U σ ) ∗ maps H m into H m σ . This implies in particular that H m σ = 0 for every σ ∈ S n . Note that for m ′ , m ′′ ∈ Z n if m ′ = m ′′ then H m ′ is orthogonal23o H m ′′ . This together with the fact that the cardinality of { m σ : σ ∈ S n } is at least n (Lemma 4.1) implies that the dimension of H is at least n which contradicts thehypothesis. Hence if H m = 0 then m i = m j for every i, j ∈ { , , · · · , n } . This has theconsequence that if λ ∈ T n ∩ SU ( n ) then ρ ( λ ) = 1.Let U ∈ SU ( n ) be given. Then there exists λ ∈ T n and V ∈ U ( n ) such that V λV ∗ = U . Since U ∈ SU ( n ), it follows that the determinant of λ is one. Hence ρ ( U ) = ρ ( V λV ∗ ) = ρ ( V ) ρ ( λ ) ρ ( V ) ∗ = ρ ( V ) ρ ( V ) ∗ = ρ ( V V ∗ ) = ρ (1) = 1 . This completes the proof. ✷ Let G be a compact group, let H be a separable Hilbert space and let π : G → B ( H )be a strongly continuous unitary representation of G on H . Let { ( H α , π α ) } α ∈ Λ be thecomplete list of irreducible subrepresentations occuring in ( H , π ). For α ∈ Λ, H α is finitedimensional. For α ∈ Λ, let n α be the multiplicity of ( H α , π α ) in ( H , π ). For α ∈ Λ, let ℓ n α be a Hilbert space of dimension n α . Up to a unitary equivalence, we can write H = M α ∈ Λ ( H α ⊗ ℓ n α ) π ( g ) = M α ∈ Λ ( π α ( g ) ⊗ g ∈ G . Let M α ∈ Λ B ( H α ) := (cid:8) ( T α ) α ∈ Λ : sup α ∈ Λ || T α || < ∞ (cid:9) . For T := ( T α ) α ∈ Λ ∈ M α ∈ Λ B ( H α ), define || T || := sup α ∈ Λ || T α || . Then (cid:16) M α ∈ Λ B ( H α ) , || || (cid:17) is a C ∗ -algebra. For T = ( T α ) α ∈ Λ ∈ M α ∈ Λ B ( H α ), define e T ∈ B ( H ) by the formula e T := M α ∈ Λ ( T α ⊗ . The map M α ∈ Λ B ( H α ) ∋ T → e T ∈ B ( H ) is an injective ∗ -homomorphism. The proof ofthe following proposition is elementary. Thus we omit its proof. Proposition 4.3
With the foregoing notation, we have π ( G ) ′′ = { e T : T ∈ M α ∈ Λ B ( H α ) } . H , H , K , K be non-zero separable Hilbert spaces. For i ∈ { , } , let M i be aunital commutative von Neumann algebra acting on H i . Fix i ∈ { , } . Consider thetensor product von Neumann algebra M i ⊗ B ( K i ) acting on H i ⊗ K i . Denote the unitarygroup of M i ⊗ B ( K i ) endowed with the strong operator topology by U ( M i ⊗ B ( K i )).Suppose that dim ( K ) < dim ( K ). Let f K be a finite dimensional subspace of K suchthat dim ( K ) < dim ( f K ). Denote the unitary group of f K by U ( f K ) and endow U ( f K )with the norm topology. Write K = f K ⊕ f K ⊥ . We denote the special unitary group of f K by SU ( f K ), i.e. SU ( f K ) := { U ∈ U ( f K ) : det( U ) = 1 } . For U ∈ U ( f K ), define e U ∈ B ( K ) by e U = U ⊕
1. Observe that the map U ( f K ) ∋ U → ⊗ e U ∈ U ( M ⊗ B ( K ))is a topological embedding and is also a group homomorphism. We use the preceedingnotation in the statement and the proof of the following proposition. Proposition 4.4
Let
Φ : R d × U ( M ⊗ B ( K )) → R d × U ( M ⊗ B ( K )) be a continuousgroup homomorphism. Then { (0 , ⊗ e U ) : U ∈ SU ( f K ) } is contained in the kernel of Φ .In particular, Φ is not - .Here, for i ∈ { , } , the topology on R d × U ( M i ⊗ B ( K i )) is the product topology andthe group structure on R d × U ( M i ⊗ B ( K i )) is that of cartesian product.Proof. Let π : R d ×U ( M ⊗ B ( K )) → R d and π : R d ×U ( M ⊗ B ( K )) → U ( M ⊗ B ( K ))be the first and second co-ordinate projections. Define Φ = π ◦ Φ and Φ = π ◦ Φ. Let L := H ⊗ K and π : U ( f K ) → B ( L ) be the strongly continuous unitary representationdefined by the equation π ( U ) := Φ (0 , ⊗ e U ) . Let { ( L α , π α ) } α ∈ Λ be the complete list of irreducible subrepresentations of U ( f K ) occuringin ( L , π ). Note that for each α ∈ Λ, L α is finite dimensional. For α ∈ Λ, let n α be themultiplicity of ( L α , π α ) in ( L , π ). We use/apply the notation explained before Prop. 4.3to the compact group U ( f K ) and the representation ( L , π ). Claim: dim ( L α ) ≤ dim ( K ) for every α ∈ Λ. Let α ∈ Λ be fixed. For S ∈ B ( L α ),let S := ( S α ) α ∈ Λ ∈ M α ∈ Λ B ( L α ) be defined by the following equation S α := S if α = α , α = α . B ( L α ) ∋ S → S ∈ M α ∈ Λ B ( L α ) is an injective ∗ -homomorphism. Let P ∈ B ( L α ) be a non-zero element.Since π ( U ( f K )) ⊂ U ( M ⊗ B ( K )), it follows that the von Neumann algebra generatedby π ( U ( f K )), i.e. π ( U ( f K )) ′′ , is contained in M ⊗ B ( K ). Note that e P ∈ M ⊗ B ( K )is non-zero. Treating M as a C ∗ -algebra, we see that there exists a character χ of M such that ( χ ⊗ e P ) = 0. This shows that map B ( L α ) ∋ S → ( χ ⊗ e S ) ∈ B ( K ) is anon-zero ∗ -homomorphism. Since L α is finite dimensional, it follows that B ( L α ) has nonon-zero ideal. As a consequence, we conclude that the map B ( L α ) ∋ S → ( χ ⊗ e S ) ∈ B ( K ) is an injection. Hence dim ( L α ) ≤ dim ( K ). This proves our claim.Thanks to Prop. 4.2 and to the fact that dim ( L α ) < dim ( f K ) for every α ∈ Λ, weobtain that π ( U ) = 1 for every U ∈ SU ( f K ). This implies that Φ (0 , ⊗ e U ) = 1 forevery U ∈ SU ( f K ). Note that the map Φ : R d × U ( M ⊗ B ( K )) → R d is continuous.Consequently { Φ (0 , ⊗ e U ) : U ∈ SU ( f K ) } is a compact subgroup of R d . But the onlycompact subgroup of R d is the trivial one i.e. { } . Hence Φ (0 , ⊗ e U ) = 0 for every U ∈ SU ( f K ). As a consequence, we obtain that { (0 , ⊗ e U ) : U ∈ SU ( f K ) } is containedin the kernel of Φ. This completes the proof. ✷ The following corollary is immediate from Prop. 4.4 and Corollary 3.5.
Corollary 4.5
Let A be a P -module and k , k ∈ { , , · · · , } ∪ {∞} . For i ∈ { , } ,denote the CCR flow associated to the P -module A of multiplicity k i by α ( A,k i ) . Then α ( A,k ) is cocycle conjugate to α ( A,k ) if and only if k = k . Let H , H , K be non-zero separable Hilbert spaces. Assume that K is infinite di-mensional. For i ∈ { , } , let M i ⊂ B ( H i ) be a unital commutative von Neumannalgebra. For i ∈ { , } , consider the tensor product von Neumann algebra M i ⊗ B ( K )acting on H i ⊗ K . The unitary groups of M , M ⊗ B ( K ) and M ⊗ B ( K ) are endowedwith the corresponding strong operator topologies and we will denote them by U ( M ), U ( M ⊗ B ( K )) and U ( M ⊗ B ( K )) respectively. We denote the identity element of variousunitary groups involved by 1. The identity element of R d will be denoted by 0. Lemma 4.6
With the foregoing notation, the topological groups R d × U ( M ⊗ B ( K )) and R d × U ( M ) × U ( M ⊗ B ( K )) are not isomorphic.Proof. Suppose that there exists a map, say,Φ : R d × U ( M ⊗ B ( K )) → R d × U ( M ) × U ( M ⊗ B ( K ))26uch that Φ is a topological group isomorphism. Denote the second co-ordinate projectionfrom R d × U ( M ) × U ( M ⊗ B ( K )) onto U ( M ) by π . Define e Φ : U ( M ⊗ B ( K )) → U ( M )as follows: for U ∈ U ( M ⊗ B ( K )), let e Φ( U ) = π ◦ Φ(0 , U ). Note that e Φ is a continuousgroup homomorphism.We claim that for x ∈ U ( M ), e Φ( x ⊗
1) = 1. Let x ∈ U ( M ) be given. Choose anorthonormal basis, say, { ξ , ξ , ξ , · · · } of K . Let N := { , , , · · · } . For n ∈ N , let E n bethe orthogonal projection onto the 1-dimensional subspace of K spanned by { ξ n } . For m, n ∈ N , let U m,n be a unitary on K such that U m,n E n U ∗ m,n = E m . For n ∈ N , define T n : = x ⊗ E n + 1 ⊗ (1 − E n ) S n : = T T · · · T n Note that for n ∈ N , S n = x ⊗ ( P nk =1 E k ) + 1 ⊗ (1 − P nk =1 E k ). Note that the sequence { T n } n ∈ N converges strongly to 1 and the sequence { S n } n ∈ N converges strongly to x ⊗ m, n ∈ N . Note that (1 ⊗ U m,n ) T n (1 ⊗ U m,n ) − = T m . The fact that U ( M )is abelian implies that e Φ( T n ) = e Φ( T m ). Hence the sequence { e Φ( T n ) } n ∈ N is a constantsequence. Since { T n } → e Φ is continuous, it follows that e Φ( T n ) = 1 for every n ∈ N .The fact that e Φ is a group homomorphism implies that e Φ( S n ) = 1 for every n ∈ N . But { S n } → x ⊗ e Φ is continuous. As a consequence, we obtain that e Φ( x ⊗
1) = 1. Thisproves our claim.Since Φ is a group isomorphism, it follows that Φ maps the center of the topologicalgroup R d × U ( M ⊗ B ( K )), which is { ( λ, x ⊗
1) : λ ∈ R d , x ∈ U ( M ) } , onto the center of R d × U ( M ) × U ( M ⊗ B ( K )), which is { ( µ, y, z ⊗
1) : µ ∈ R d , y ∈ U ( M ) , z ∈ U ( M ) } .Consider the element (0 , − , ⊗ ∈ R d × U ( M ) × U ( M ⊗ B ( K )) which is in the centerof R d × U ( M ) × U ( M ⊗ B ( K )). Hence there exists λ ∈ R d and x ∈ U ( M ) such thatΦ( λ, x ⊗
1) = (0 , − , ⊗ , − , ⊗
1) has order 2. Since Φ is a groupisomorphism, it follows that ( λ, x ⊗
1) has order 2. This implies that λ = 0. Consequently,we have e Φ( x ⊗
1) = − e Φ( x ⊗
1) = 1. Hencethe proof. ✷ Let A , A be P -modules and k , k ∈ { , , · · · , } ∪ {∞} . Fix i ∈ { , } . Let K i be a Hilbert space of dimension k i . Let V ( i ) : P → B ( L ( A i ) ⊗ K i ) be the isometricrepresentation associated to the P -module A i of multiplicity k i . Denote the isometricrepresentation associated to the P -module A i of multiplicity 1 by g V ( i ) . Set H i := L ( A i ) ⊗K i , H := H ⊕ H and V := V (1) ⊕ V (2) . Let { W ( i ) z } z ∈ R d be the family of partialisometries, described in the paragraph following Theorem 3.1, associated to the isometricrepresentation V ( i ) , and let { g W ( i ) z } z ∈ R d and { W z } z ∈ R d be the family of partial isometries27ssociated to the isometric representations g V ( i ) and V respectively. Note that W ( i ) z = g W ( i ) z ⊗
1. We use the notation developed in the paragraphs between Lemma 3.2 andProposition 3.3.Let G A i be the isotropy group of A i , i.e. G A i := { z ∈ R d : A i + z = A i } . For z ∈ G A i ,let U ( i ) z be the unitary on L ( A i ) defined by the formula U ( i ) z f ( x ) := f ( x − z )for f ∈ L ( A i ). Let M i be the von Neumann algebra generated by { U ( i ) z : z ∈ G A i } acting on L ( A i ). Denote the CCR flow associated to the P -module A i of multiplicity k i by α ( A i ,k i ) . Proposition 4.7
Assume that A is not a translate of A , i.e. for every z ∈ R d , A + z = A . With the foregoing notation, the gauge group of α ( A ,k ) ⊗ α ( A ,k ) isisomorphic to R d × U ( M ⊗ B ( K )) × U ( M ⊗ B ( K )) .Proof. Note that α ( A ,k ) ⊗ α ( A ,k ) is the CCR flow, denoted α V , associated to theisometric representation V . Since V (1) and V (2) admits no non-zero additive cocycles,by Remark 2.5, it follows that V admits no non-zero additive cocycle. By Theorem2.7, it follows that the gauge group of α V is isomorphic to R d × U ( M ) where M is thecommutant of the von Neumann algebra generated by { V x : x ∈ P } .By Corollary 3.5, for i ∈ { , } , the commutant of the von Neumann algebra generatedby { V ( i ) x : x ∈ P } is M i ⊗ B ( K i ). We write operators acting on H = H ⊕ H in termsof block matrices. We claim that M = n T T ! : T ∈ M ⊗ B ( K ) , T ∈ M ⊗ B ( K ) o Once the above claim is established, thanks to Theorem 2.7, the conclusion followsimmediately. It is clear that n T T ! : T ∈ M ⊗ B ( K ) , T ∈ M ⊗ B ( K ) o iscontained in M . Let T := T T T T ! ∈ M be given. It is routine to verify that T W (2) z = W (1) z T for z ∈ R d . Let f ∈ C c ( R d ) be given. Calculate as follows to observe28hat T ( π A ( e f ) ⊗
1) = T (cid:16)(cid:16) Z f ( − z ) ] W (2) z dz (cid:17) ⊗ (cid:17) = T (cid:16) Z f ( − z ) W (2) z dz (cid:17) = Z f ( − z ) T W (2) z dz = Z f ( − z ) W (1) z T dz = (cid:16) Z f ( − z ) W (1) z dz (cid:17) T = (cid:16) Z f ( − z )( ] W (1) z ⊗ dz (cid:17) T = ( π A ( e f ) ⊗ T . Since { e f : f ∈ C c ( R d ) } generates C ∗ ( G u ), it follows that T intertwines the represen-tation ( π A ( . ) ⊗ , H ) and the representation ( π A ( . ) ⊗ , H ). Theorem 3.1 and thehypothesis A + z = A for every z ∈ R d implies that π A and π A are disjoint. Thisimplies that π A ( . ) ⊗ π A ( . ) ⊗ T = 0. In a sim-ilar fashion, we conclude that T = 0. It is now clear that T ∈ M ⊗ B ( K ) and T ∈ M ⊗ B ( K ). This proves our claim. Hence the proof. ✷ Now we prove Theorem 1.2. We use the notation developed in the two paragraphsthat precede Proposition 4.7 and we write α ∼ = β to indicate that α and β are cocycleconjugate. Proof of Theorem 1.2.
As mentioned in the introduction, it is clear that (3) = ⇒ (1) = ⇒ (2). Suppose that (2) holds. Then the gauge group of α ( A ,k ) is isomorphicto the gauge group of α ( A ,k ) . By Corollary 3.5, it follows that R d × U ( M ⊗ B ( K ))is isomorphic to R d × U ( M ⊗ B ( K )). Since M and M are abelian, it follows fromProposition 4.4 that k = k . With no loss of generality, we can assume that K = K .Suppose, on the contrary, assume that A and A are not translates of each other.Since α ( A ,k ) is cocycle conjugate to α ( A ,k ) , it follows that α ( A ,k +1) ∼ = α ( A , ⊗ α ( A ,k ) ∼ = α ( A , ⊗ α ( A ,k ) . Hence α ( A ,k +1) and α ( A , ⊗ α ( A ,k ) have isomorphic gauge groups. Corollary 3.5 andProposition 4.7 together imply that the topological groups R d × U ( M ⊗ B ( f K )) and R d × U ( M ) × U ( M ⊗ B ( K )) are isomorphic, where f K is a Hilbert space of dimension k + 1. Let Φ : R d × U ( M ⊗ B ( f K )) → R d × U ( M ) × U ( M ⊗ B ( K )) be a topologicalgroup isomorphism. 29uppose k = k is finite. Let π : R d × U ( M ) × U ( M ⊗ B ( K )) → R d × U ( M ) and π : R d × U ( M ) × U ( M ⊗ B ( K )) → R d × U ( M ⊗ B ( K )) be defined by the followingformulas π ( x, Y, Z ) = ( x, Y ) π ( x, Y, Z ) = ( x, Z )for ( x, Y, Z ) ∈ R d × U ( M ) × U ( M ⊗ B ( K )). Let Φ = π ◦ Φ and Φ = π ◦ Φ.Proposition 4.4 implies that { (0 , ⊗ U ) : U ∈ SU ( f K ) } is contained in the kernel ofboth Φ and Φ . This implies that { (0 , ⊗ U ) : U ∈ SU ( f K ) } is contained in thekernel of Φ, which is a contradiction. This implies that k = k = ∞ . Hence f K and K are infinite dimensional separable Hilbert spaces. The fact that Φ is a topological groupisomorphism is a contradiction to Lemma 4.6. These contradictions are due to our initialassumption that A and A are not translates of each other. Hence there exists z ∈ R d such that A + z = A . The proof of the implication (2) = ⇒ (3) is now complete. ✷ Here we provide a proof of Theorem 3.1. The proof is an application of Rieffel’s theory ofMorita equivalence. Rieffel’s theorem, Theorem 6.23 of [19], asserts that if A and B areMorita equivalent C*-algebras then the category of representations of A and that of B areequivalent. As the Deaconu-Renault groupoid X ⋊ P is equivalent to a transformationgroupoid Y ⋊ R d , it follows from [12] that the C ∗ -algebras C ∗ ( X ⋊ P ) and C ∗ ( Y ⋊ R d ) ∼ = C ( Y ) ⋊ R d are Morita equivalent. Consequently it suffices to prove the result for atransformation groupoid.Let us begin by reviewing the basics of Rieffel’s notion of Morita equivalence. Let B be a C ∗ -algebra. For a Hilbert B -module E , we denote the C ∗ -algebra of adjointableoperators on E by L B ( E ) and the C ∗ -algebra of compact operators by K B ( E ). TheHilbert module E is said to be full if the linear span of {h x | y i : x, y ∈ E } is dense in B .Let A and B be C ∗ -algebras. By an A - B imprimitivity bimodule , we mean a pair ( E, φ ),where E is a full Hilbert B -module, φ : A → L B ( E ) is an injective ∗ -homomorphismand φ ( A ) = K B ( E ). The C ∗ -algebras A and B are said to be Morita equivalent if thereexists an A - B imprimitivity bimodule.Next we recall Rieffel’s induction procedure. Let A and B be Morita equivalent C ∗ -algebras with E being an A - B imprimitivity bimodule. Suppose π is a representationof B on a Hilbert space H π . Consider the internal tensor product E ⊗ π H π which is.a30ilbert space. The C ∗ -algebra A acts on the Hilbert space E ⊗ π H π as follows: for a ∈ A ,let Ind ( π )( a ) := φ ( a ) ⊗
1. Then
Ind ( π ) is a representation of A . Rieffel’s fundamentaltheorem, Theorem 6.23 of [19], asserts that π → Ind ( π )is a functor which identifies the category of representations of B and the category ofrepresentations of A . Let us isolate two consequences of the above fact in the followingremark. Remark 5.1
With the foregoing notation, we have the following.(1) Let π and π be representations of B . Then π and π are disjoint if and only if Ind ( π ) and Ind ( π ) are disjoint.(2) Let π be a representation of B . Then ( Ind ( π )( A )) ′ = { ⊗ F : F ∈ π ( B ) ′ } . We keep the notation explained in the paragraphs starting from line 8, Page 17 untilthe end of Theorem 3.1. Let us fix a few notation: Let Z := { ( y, s ) : y + s ∈ X } and let ρ : Z → Y and σ : Z → X be defined by ρ ( y, s ) = y and σ ( y, s ) = y + s . Then Z is a( Y ⋊ R d , X ⋊ P )-equivalence where the actions of Y ⋊ R d and X ⋊ P on Z are given bythe formulas: ( y, s )( z, t ) = ( y, s + t ) if y + s = z ( z, t )( x, r ) = ( z, t + r ) if z + t = x for ( x, r ) ∈ X ⋊ P , ( y, s ) ∈ Y ⋊ R d and ( z, t ) ∈ Z . For the definition of an action of agroupoid on a space and for the notion of groupoid equivalence, we refer the reader to[12]. Let A := C ∗ ( Y ⋊ R d ) and B := C ∗ ( X ⋊ P ). Denote C c ( Y ⋊ R d ) and C c ( X ⋊ P )by A and B respectively. Note that A and B are dense in A and B respectively. Denote C c ( Z ) by E . For ξ ∈ A , χ ∈ E , and η ∈ B , let( ξ.χ )( y, r ) = Z ξ ( y, s ) χ ( y + s, r − s ) ds, ( χ.η )( y, r ) = Z χ ( y, s ) η ( y + s, r − s )1 X ( y + s ) ds, and h χ , χ i B ( x, r ) = Z χ ( x + s, − s ) χ ( x + s, r − s ) ds. The above formulas make E into a pre-Hilbert A - B bimodule. On completion, we ob-tain a genuine Hilbert A - B bimodule which we denote by E . Moreover E is an A - B imprimitivity bimodule. For details, we refer the reader to [12].31or a point x ∈ X , we denote the representation of C ∗ ( X ⋊ P ) on L ( Q x ) induced atthe point x by π x and the representation of C ∗ ( Y ⋊ R d ) on L ( R d ) induced at the point x by e π x . We claim that Ind ( π x ) = e π x .Fix x ∈ X . For χ ∈ C c ( Z ), η ∈ C c ( X ⋊ P ) and r ∈ R d , let ^ χ ⊗ η ( r ) = Z χ ( x + r, s − r ) η ( x , s )1 X ( x + s ) ds. (1) It is routine to see that the map C c ( Z ) ⊗ B C c ( X ⋊ P ) ∋ χ ⊗ η → ^ χ ⊗ η ∈ C c ( R d )is well-defined and extends to an isometry from E ⊗ B L ( Q x ) to the Hilbert space L ( R d ).(2) The set { ^ χ ⊗ η : χ ∈ C c ( Z ) , η ∈ C c ( X ⋊ P ) } is total in L ( R d ). Thus we can identify E ⊗ B L ( Q x ) with L ( R d ) via the unitary E ⊗ B L ( Q x ) ∋ χ ⊗ η → ^ χ ⊗ η ∈ L ( R d ).Once this identification is made, a direct calculation shows that Ind ( π x ) = e π x .Thus, in view of Theorem 6.23 of [19] and Remark 5.1, it suffices to prove Theorem 3.1for a transformation groupoid Y ⋊ R d . We do not claim any originality of what followsas it is well known. We include the details for completeness.Let us fix notation. Let Y be a second countable, locally compact Hausdorff spaceon which R d acts. Let y ∈ Y be given. We denote the representation of C ∗ ( Y ⋊ R d )induced at y by π y . Let B ( Y ) be the algebra of bounded measurable functions on Y .For f ∈ B ( Y ), let M y ( f ) ∈ B ( L ( R d )) be defined by the equation M y ( f ) ξ ( t ) = f ( y + t ) ξ ( t )for ξ ∈ L ( R d ). For s ∈ R d , let L s be the unitary on L ( R d ) defined by the equation L s ξ ( t ) = ξ ( t + s ) . Then ( M y , L ) is a covariant representation of the dynamical system ( C ( Y ) , R d ). If weidentify C ∗ ( Y ⋊ R d ) with C ( Y ) ⋊ R d , the covariant representation that corresponds tothe non-degenerate representation π y is ( M y , L ). Proposition 5.2
With the foregoing notation, we have the following.(1) Let y , y ∈ Y be given. The representations π y and π y are non-disjoint if andonly if there exists s ∈ R d such that y + s = y .(2) For y ∈ Y , the commutant of the von Neumann algebra generated by the set { π y ( ξ ) : ξ ∈ C ∗ ( Y ⋊ R d ) } is the von Neumann algebra generated by { L s : s ∈ H } where H is the stabiliser group of y , i.e. H := { s ∈ R d : y + s = y } . roof. Suppose there exists s ∈ R d such that y + s = y . Then L s intertwines π y and π y . Conversely, suppose π y and π y are non-disjoint. Let T be a non-zero intertwiner.Then T intetwines M y and M y . As a consequence, we have T M y ( f ) = M y ( f ) T forevery f ∈ B ( Y ). For i = 1 ,
2, let H i be the stabiliser group of y i . Note that the map R d /H i ∋ s + H i → y i + s ∈ Y is 1-1 and continuous. We denote its image by E i . Thanksto Theorem 3.3.2 of [3], it follows that E i is a Borel set. Note that M y (1 E ) = 1. Theequation T = T M y (1 E ) = M y (1 E ) T implies that M y (1 E ) = 0. This implies that theorbit of y meets the orbit of y . This proves (1).It is clear that { L s : s ∈ H } lies in the commutant of π y ( C ( Y ) ⋊ R d ). Conversely,suppose T lies in the commutant of π y ( C ( Y ) ⋊ R d ). Then T commutes with the algebra { M y ( f ) : f ∈ B ( Y ) } and { L s : s ∈ R d } .Note that R d H ∋ s + H → y + s ∈ Y is continuous and 1-1. Since R d /H and Y are Polish spaces, it follows from Theorem 3.3.2 of [3] that via the embedding R d /H ∋ s + H → y + s ∈ Y , we can identify the Borel space R d /H with a subspace of Y .Thus bounded measurable functions on R d /H can be considered as bounded measurablefunctions on Y . More precisely, suppose f is a bounded measurable function on R d /H ,then we consider f as a function on Y simply by declaring the values of f on thecomplement of R d /H to be zero. This way we embedd C ( R d /H ) inside B ( Y ).Hence we get a covariant representation ( M y , L ) of ( C ( R d /H ) , R d ). By Mackey’simprimitivity theorem, it follows that C ( R d /H ) ⋊ R d is Morita equivalent to C ∗ ( H ).Then the representation M y ⋊ L of C ( R d /H ) ⋊ R d is Ind ( ρ ) where ρ is the regularrepresentation of H on L ( H ). Note that T lies in the commutant of Ind ( ρ ). Sincethe group H is abelian, it follows that the commutant of ρ is the von Neumann algebragenerated by { ρ ( s ) : s ∈ H } . The statement now follows by appealing to Remark 5.1. ✷ References [1] C. Anantharaman-Delaroche and J. Renault,
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