The Fourier-Stieltjes algebra of a C*-dynamical system II
aa r X i v : . [ m a t h . OA ] M a r The Fourier-Stieltjes algebra of a C ∗ -dynamical system II Erik B´edos, Roberto ContiMarch 24, 2020
Abstract
We continue our study of the Fourier-Stieltjes algebra associated to a twisted (uni-tal, discrete) C*-dynamical system and discuss how the various notions of equivalenceof such systems are reflected at the algebra-level. As an application, we show that theamenability of a system, as defined in our previous work, is preserved under Moritaequivalence.2020
Mathematics Subject Classification : Primary 46L55; Secondary 37A55, 43A35,46H25.
Keywords : Fourier-Stieltjes algebra, C ∗ -dynamical system, equivariant representation,cocycle conjugacy, Hilbert bimodule, Morita equivalence, amenability. The classical notion of Fourier-Stieltjes algebra of a locally compact group G [24] wasextended in [9] to a (unital, discrete) twisted C*-dynamical system Σ = ( A, G, α, σ ).In short, the outcome is a Banach algebra B (Σ) attached to Σ with a rich analyticalstructure that can be better described in terms of coefficients of the so-called equivariantrepresentations of Σ. In the case where A is trivial, any such a representation is nothing buta unitary representation of G on a Hilbert space, and one therefore recovers the Fourier-Stieltjes algebra B ( G ). Some aspects of the classical theory survive to the new setting,notably the inclusion of B (Σ) in the completely bounded full/reduced multipliers of Σ, aswell as the fact that B (Σ) is spanned by the Σ-positive definite functions, which themselvesgive rise to completely positive maps of the full and reduced twisted crossed product C ∗ -algebras associated to Σ. We note here that in the case of an untwisted system our conceptof Σ-positive definiteness can be reformulated using the notion of completely positive Herz-Schur Σ-multiplier (cf. [29]). We also recall that B ( G ) continuously embeds into B (Σ),although these two algebras differ significantly from each other for it can be shown that,under mild assumptions, B (Σ) is always noncommutative (actually, B ( G ) is contained inthe center of B (Σ)). Finally, we mention that one can use the aforementioned coefficientsof equivariant representations of Σ to introduce suitable approximation properties forΣ, such as amenability (cf. [9]) and the Haagerup property (cf. [29]), that parallel theanalogous notions for G and provide intrinsic features of the dynamical system Σ.The main motivation for this paper was to explore to which extent the Fourier-Stieltjesalgebra B (Σ) depends on Σ. We recall that if G and G are locally compact groups, then1alter showed in [32] that B ( G ) and B ( G ) are isometrically isomorphic as Banachalgebras if and only if G and G are topologically isomorphic. Hence one may hope that B (Σ) is better suited to characterize Σ than other algebras associated to it. Now thereare several natural notions of equivalence between two dynamical systems Σ = ( A, G, α, σ )and Θ = (
B, H, β, θ ), most notably exterior equivalence, conjugacy, and cocycle conjugacy,but also Morita equivalence (in the case where G = H ). It is immediate that the firsttwo notions are stronger than the third one, which is itself stronger than the last one. Weshow in Theorem 3.8 that B (Σ) and B (Θ) are isometrically isomorphic whenever Σ andΘ are cocycle conjugate (up to a group isomorphism), in a way that preserves the classicalFourier-Stieltjes algebras of the corresponding groups, and also the canonical copies ofthe corresponding algebras. In connection with this result, we also note that the Fourier-Stieltjes algebra of a system does not detect a perturbation of the system by a T -valuedgroup 2-cocycle, cf. Remark 3.10. In the case of Morita equivalent systems, the connectionbetween the Fourier-Stieltjes algebras remains somewhat more elusive, but we are at leastable to show that these algebras can be determined from each other, see Corollary 4.6.However, as a byproduct of this study, we obtain an interesting consequence for Moritaequivalent systems, namely we show in Theorem 5.1 that the amenability of a system (asdefined in [9]) is preserved under such an equivalence.The paper is organized as follows. After some preliminaries in Section 2, we reviewin Section 3 some of the natural notions of equivalence for twisted C ∗ -dynamical systems(exterior equivalence, (group) conjugacy, and cocycle (group) conjugacy) and prove thatthe Fourier-Stieltjes algebra is invariant, up to isometric isomorphism, under cocycle groupconjugacy (which is the most general among these notions). In Section 4 we consider twoMorita equivalent systems and point out that there is, up to isomorphism, a one-to-onecorrespondence between the equivariant representations of the respective systems. We usethis to show that the corresponding Fourier-Stieltjes algebras can then be recovered fromeach other. Finally, in Section 5, we recall our definition of amenability for a system andshow that this property is Morita invariant. We only consider unital C ∗ -algebras in this paper, and a homomorphism between twosuch algebras will always mean a unit preserving ∗ -homomorphism. Isomorphisms andautomorphisms between C ∗ -algebras are therefore also assumed to be ∗ -preserving. Thegroup of unitary elements in a C ∗ -algebra A will be denoted by U ( A ), the center of A by Z ( A ), and the group of automorphisms of A by Aut( A ). The identity map on A will bedenoted by id (or id A ). If B is another C ∗ -algebra, A ⊗ B will denote their minimal tensorproduct.By a Hilbert C ∗ -module, we will always mean a right Hilbert C ∗ -module, unless other-wise specified, and follow the notation introduced in [28]. In particular, all inner productswill be assumed to be linear in the second variable, L B ( X, Y ) will denote the space of alladjointable operators between two Hilbert C ∗ -modules X and Y over a C ∗ -algebra B , and L B ( X ) = L B ( X, X ). A representation of a C ∗ -algebra A on a Hilbert B -module Y is thena homomorphism from A into the C ∗ -algebra L B ( Y ). If Z is another Hilbert C ∗ -module(over C ), we will let π ⊗ ι : A → L B ⊗ C ( Y ⊗ Z ) denote the amplified representation of A on Y ⊗ Z given by ( π ⊗ ι )( a ) = π ( a ) ⊗ I Z , where the Hilbert B ⊗ C -module Y ⊗ Z is the2xternal tensor product of Y and Z (cf. [28]), and I Z denotes the identity operator on Z .If Z is a Hilbert space, then we consider Y ⊗ Z as a Hilbert B -module.The quadruple Σ = ( A, G, α, σ ) will always denote a twisted unital discrete C ∗ -dynamicalsystem . This means that A is a C ∗ -algebra with unit 1 A , G is a discrete group with iden-tity e and ( α, σ ) is a twisted action of G on A (sometimes called a cocycle G -action on A ),that is, α is a map from G into Aut( A ) and σ : G × G → U ( A ) is a normalized 2-cocyclefor α , such that α g α h = Ad( σ ( g, h )) α gh ,σ ( g, h ) σ ( gh, k ) = α g ( σ ( h, k )) σ ( g, hk ) ,σ ( g, e ) = σ ( e, g ) = 1 A for all g, h, k ∈ G . Of course, Ad( u ) denotes here the (inner) automorphism of A imple-mented by the unitary u in U ( A ). If σ = 1 is the trivial 2-cocycle, that is, σ ( g, h ) = 1 A for all g, h ∈ G , then α is a genuine action and Σ is an ordinary C ∗ -dynamical system (seee.g. [33, 12]), usually denoted by Σ = ( A, G, α ). If σ is central , that is, it takes values in U ( Z ( A )), then α is also a genuine action of G on A , and this is the case studied in [34].In the sequel we will often just use the word system to mean a discrete unital twisted C ∗ -dynamical system.An equivariant representation of Σ on a Hilbert A -module X (see e.g. [7, 8]) is a pair( ρ, v ) where ρ : A → L A ( X ) is a representation of A on X and v is a map from G into thegroup I ( X ) of all C -linear, invertible, bounded maps from X into itself, which satisfy:(i) ρ ( α g ( a )) = v ( g ) ρ ( a ) v ( g ) − , g ∈ G, a ∈ A, (ii) v ( g ) v ( h ) = ad ρ ( σ ( g, h )) v ( gh ) , g, h ∈ G, (iii) α g (cid:0) h x, x ′ i (cid:1) = h v ( g ) x, v ( g ) x ′ i , g ∈ G, x, x ′ ∈ X, (iv) v ( g )( x · a ) = ( v ( g ) x ) · α g ( a ) , g ∈ G, x ∈ X, a ∈ A .In (ii) above, ad ρ ( σ ( g, h )) ∈ I ( X ) is defined byad ρ ( σ ( g, h )) x = (cid:0) ρ ( σ ( g, h )) x (cid:1) · σ ( g, h ) ∗ , g, h ∈ G, x ∈ X. Note that the equivariant representations of Σ may instead be presented in terms of (Σ,Σ)-compatible actions, as in [18, 19], cf. Remark 4.1. Note also that condition (iii) impliesthat each v ( g ) is isometric.For completeness, we mention some examples of equivariant representations. First,the trivial equivariant representation of Σ, which is the pair ( ℓ, α ) acting on A , consideredas a right A -module over itself in the canonical way, where ℓ : A → L A ( A ) is given byleft-multiplication. Next, let A G := ℓ ( G, A ) denote the right A -module given by A G = n ξ : G → A | X g ∈ G ξ ( g ) ∗ ξ ( g ) is norm-convergent in A o , with the obvious right A -module structure, and inner product given by h ξ, η i = X g ∈ G ξ ( g ) ∗ η ( g ) . regular equivariant representation of Σ on A G is the pair (ˇ ℓ, ˇ α ) acting on A G defined by (ˇ ℓ ( a ) ξ )( h ) = aξ ( h ) , ( ˇ α ( g ) ξ )( h ) = α g ( ξ ( g − h ))for a ∈ A, ξ ∈ A G and g, h ∈ G .More generally, if ( ρ, v ) is an equivariant representation of A on a right Hilbert A -module X and w is a unitary representation of G on some Hilbert space H , then ( ρ ⊗ ι, v ⊗ w )is an equivariant representation of Σ on X ⊗ H .One can also form the tensor product of equivariant representations. Assume that( ρ , v ) and ( ρ , v ) are equivariant representations of Σ on some Hilbert A -modules X and X , respectively. We can then form the internal tensor product X ⊗ ρ Y , whichis a right Hilbert A -module (cf. [28]); we will suppress ρ in our notation and denote X ⊗ ρ X by X ⊗ A X , as it is quite common in the literature. Then the tensor product( ρ , v ) ⊗ ( ρ , v ) acts on X ⊗ A X as follows. For a ∈ A , let ( ρ ⊗ ρ )( a ) ∈ L A ( X ⊗ A X )be the map determined on simple tensors by( ρ ⊗ ρ )( a )( x ˙ ⊗ x ) = ρ ( a ) x ˙ ⊗ x for x ∈ X and x ∈ X . Moreover, for every g ∈ G , let ( v ⊗ v )( g ) in I ( X ⊗ A X ) be the map determined onsimple tensors by( v ⊗ v )( g )( x ˙ ⊗ x ) = v ( g ) x ˙ ⊗ v ( g ) x for x ∈ X and x ∈ X . Then ( ρ , v ) ⊗ ( ρ , v ) := ( ρ ⊗ ρ , v ⊗ v ) is an equivariant representation of Σ on theright Hilbert A -module X ⊗ A X (cf. [19, 7]).Let ( ρ, v ) be an equivariant representation of Σ on a Hilbert A -module X and let x, y ∈ X . Then we define T ρ,v,x,y : G × A → A by T ρ,v,x,y ( g, a ) = (cid:10) x, ρ ( a ) v ( g ) y (cid:11) for a ∈ A, g ∈ G, and think of T ρ,v,x,y as an A -valued coefficient function associated with ( ρ, v ).The Fourier-Stieltjes algebra B (Σ) is defined in [9] as the collection of all the mapsfrom G × A into A of the form T ρ,v,x,y for some equivariant representation ( ρ, v ) of Σ ona Hilbert A -module X and x, y ∈ X . Then B (Σ) becomes a unital subalgebra of L (Σ),where L (Σ) = { T : G × A → A | T is linear in the second variable } is equipped with its natural algebra structure: for T, T ′ ∈ L (Σ) and λ ∈ C , we let T + T ′ , λT , T · T ′ and I Σ be the maps in L (Σ) defined by( T + T ′ )( g, a ) := T ( g, a ) + T ′ ( g, a )( λT )( g, a ) := λT ( g, a )( T · T ′ )( g, a ) := T ( g, T ′ ( g, a )) I Σ ( g, a ) := a for g ∈ G and a ∈ A . Given T ∈ L (Σ) and g ∈ G , we will sometimes write T g for thelinear map from A into itself given by T g ( a ) = T ( g, a ) for all a ∈ A .If T ∈ B (Σ), letting k T k denote the infimum of the set of values k x kk y k associatedwith the possible decompositions of T of the form T = T ρ,v,x,y , one gets a norm on B (Σ)such that B (Σ) is a unital Banach algebra w.r.t. k · k .4e also recall that there is a canonical way of embedding B ( G ) into B (Σ) (cf. [9,Proposition 3.2]): For f ∈ B ( G ), define T f ∈ L (Σ) by T f ( g, a ) = f ( g ) a for g ∈ G and a ∈ A . Then T f ∈ B (Σ), and the map f → T f gives an injective, contractive, algebra-homomorphism of B ( G ) into B (Σ).The Fourier-Stieltjes algebra B (Σ) also contains a copy of A . Indeed, for b ∈ A , let T b ∈ L (Σ) be given by T b ( g, a ) = ba for all g ∈ G and a ∈ A . Then we have that T b = T ℓ,α,b ∗ , A ∈ B (Σ) and k T b k ≤ k b k . From this, one readily deduces that the map b → T b gives an isometric algebra-homomorphism from A into B (Σ).Finally, we recall that, as in the classical case, B (Σ) is spanned by its positive defi-nite elements (cf. [9, Corollary 4.5]). For the ease of the reader, we review how positivedefiniteness is defined in our setting. Let T ∈ L (Σ). Then T is called positive definite (w.r.t. Σ), or Σ -positive definite , when for any n ∈ N , g , . . . , g n ∈ G and a , . . . , a n ∈ A ,the matrix h α g i (cid:16) T g − i g j (cid:0) α − g i (cid:0) a ∗ i a j σ ( g i , g − i g j ) ∗ (cid:1)(cid:1)(cid:17) σ ( g i , g − i g j ) i is positive in M n ( A ) (the n × n matrices over A ). As shown in [9, Corollary 4.4], whichis an analogue of the Gelfand-Raikov theorem, this is equivalent to requiring that T maybe written as T = T ρ,v,x,x for some equivariant representation ( ρ, v ) of Σ on some Hilbert A -module X and some x ∈ X . It then follows that k T k ∞ := sup {k T g k | g ∈ G } = k T e (1 A ) k = kh x, x i A k (cf. [9, Corollary 4.3]). We set P (Σ) = (cid:8) T ∈ L (Σ) | T is positive definite (w.r.t. Σ) (cid:9) . There are various notions of equivalence for C ∗ -dynamical systems in the literature. Inthis section we will study how the notions of exterior equivalence, (group) conjugacy andcocycle (group) conjugacy are reflected at the level of the Fourier-Stieltjes algebras. Definition 3.1.
Consider a system Σ = (
A, G, α, σ ), and let w : G → U ( A ) be a normal-ized map, that is, such that w ( e ) = 1 A . Then it is well known (cf. [30, Section 3]) that weget another twisted action ( α w , σ w ) of G on A by setting α wg = Ad( w ( g )) ◦ α g and σ w ( g, g ′ ) = w ( g ) α g ( w ( g ′ )) σ ( g, g ′ ) w ( gg ′ ) ∗ for all g, g ′ ∈ G . We then set Σ w := ( A, G, α w , σ w ) and call Σ w a perturbation of Σ by w . Remark 3.2.
Another way to perturb a system Σ = (
A, G, α, σ ) is as follows. Let α ′ denote the restriction of α to a (genuine) action of G on Z ( A ), and let η : G × G → U ( Z ( A ))be a normalized 2-cocycle for α ′ . (For example, we can let η : G × G → T be any normalized2-cocycle for the group G and consider η as a 2-cocycle for α ′ .) Then we get a twistedaction ( α, σ η ) of G on A by setting σ η ( g, g ′ ) := σ ( g, g ′ ) η ( g, g ′ )for all g, g ′ ∈ G . The system Σ( η ) := ( A, G, α, σ η ) is called a perturbation of Σ by η .5 efinition 3.3. Two systems Σ = (
A, G, α, σ ) and Θ = (
A, G, β, θ ) are called exteriorequivalent , and we write Σ ∼ e Θ, when Θ = Σ w for some map w : G → U ( A ) (which isthen necessarily normalized). Example 3.4.
Let α and β be two genuine actions of G on A and set Σ = ( A, G, α,
A, G, β, w : G → U ( A ) is called a 1 -cocycle for α whenit satisfies that w ( gg ′ ) = w ( g ) α g ( w ( g ′ )) for all g, g ′ ∈ G . Then we have that Σ ∼ e Θ ifand only if there exists some 1-cocycle w : G → U ( A ) for α such that β g = Ad( w ( g )) ◦ α g for all g ∈ G . One usually says that β is a perturbation of α by w in this case.Assume now that α and β agree up to inner automorphisms, that is, they satisfy that β g = Ad( u ( g )) ◦ α g for some map u : G → U ( A ), which may be assumed to be normalized.Set ∂u ( g, h ) := u ( g ) α g ( u ( h )) u ( gh ) ∗ for all g, h ∈ G . Then it can easily be checked that ∂u is a 2-cocycle for β taking itsvalues in U ( Z ( A )). If ∂u = 1, i.e., u is not a 1-cocycle for α , then we get that ( β, ∂u )is a twisted action of G on A satisfying that Σ = ( A, G, α, ∼ e ( A, G, β, ∂u ). Similarly,Θ ∼ e ( A, G, α, ∂u ∗ ), where u ∗ ( g ) := u ( g ) ∗ for all g ∈ G .We note that if the map u above takes its values in U ( Z ( A )) (so we have β = α ),and α ′ denotes the restriction of α to an action of G on Z ( A ), then ∂u is a normalized2-cocycle for α ′ (called a coboundary for α ′ ). A perturbation of Σ by u is then clearly thesame as a perturbation of Σ by ∂u (in the sense of Remark 3.2), i.e., we have Σ u = Σ( ∂u ),and we get that Σ ∼ e Σ( ∂u ) in this case.Next, consider Σ = ( A, G, α, σ ) and note that if φ : A → B is an isomorphism of C ∗ -algebras and ϕ : G → H is an isomorphism of groups, then we get a new systemΘ = ( B, H, β, θ ) by setting β h = φ ◦ α ϕ − ( h ) ◦ φ − and θ ( h, h ′ ) = φ (cid:0) σ ( ϕ − ( h ) , ϕ − ( h ′ )) (cid:1) for all h, h ′ ∈ H . This motivates the following notion. Definition 3.5.
Two systems Σ = (
A, G, α, σ ) and Θ = (
B, H, β, θ ) are said to be groupconjugate if there exist an isomorphism φ : A → B and an isomorphism ϕ : G → H suchthat(i) β ϕ ( g ) = φ ◦ α g ◦ φ − ,(ii) θ (cid:0) ϕ ( g ) , ϕ ( g ′ ) (cid:1) = φ (cid:0) σ ( g, g ′ ) (cid:1) for all g, g ′ ∈ G , in which case we write Σ ∼ gc Θ. In the case where H = G , we will saythat Σ and Θ are conjugate , and write Σ ∼ c Θ, if ϕ can be chosen to be the identity map. Definition 3.6.
Two systems Σ = (
A, G, α, σ ) and Θ = (
B, H, β, θ ) are said to be cocyclegroup conjugate if Σ w ∼ gc Θ for some normalized w : G → U ( A ), in which case we writeΣ ∼ cgc Θ. Equivalently, as one readily checks, Σ ∼ cgc Θ if and only if Θ is exteriorequivalent to some group conjugate of Σ. In the case where H = G , we will say thatΣ and Θ are cocycle conjugate , and write Σ ∼ cc Θ, if Σ w is conjugate to Θ for somenormalized w : G → U ( A ). 6iscarding set-theoretical problems, one may show without much trouble that ∼ cgc (resp. ∼ cc ) satisfies the properties of an equivalence relation. Moreover, it is evident fromthe definitions that (group) conjugacy and exterior equivalence are stronger notions thancocycle (group) conjugacy. Example 3.7.
Assume again α and β are genuine actions of G on A . Then we have( A, G, α, ∼ cc ( A, G, β,
1) if and only if (
A, G, α w , w ) ∼ c ( A, G, β,
1) for some normalized w : G → U ( A ), in which case we get 1 = 1 w ( g, g ′ ) = w ( g ) α g ( w ( g ′ )) w ( gg ′ ) ∗ for all g ∈ G ,so that w is a 1-cocycle for α . Hence ( A, G, α, ∼ cc ( A, G, β,
1) if and only if there is aperturbation of α by a 1-cocycle for α which is conjugate to β , i.e., α is cocycle conjugateto β (as defined for example in [11, II.10.3.18]).It is part of the folklore that the C ∗ -crossed products associated to cocycle conjugatesystems are isomorphic, both in the full and in the reduced case, via an isomorphism thatpreserves the “diagonal” algebra (for partial results in this direction, see e.g. [30, Lemma3.2] and [33, Lemma 2.68]). In our setting, we have: Theorem 3.8.
Assume
Σ = (
A, G, α, σ ) and Θ = (
B, H, β, θ ) are cocycle group conjugate.Then B (Σ) and B (Θ) are isometrically isomorphic.More precisely, there exists an algebra-isomorphism Ψ : B (Θ) → B (Σ) such that1 ) Ψ is isometric ; ) Ψ maps the copy of B ( H ) inside B (Θ) isometrically onto the copy of B ( G ) inside B (Σ) ( w.r.t. the norms of B ( G ) and B ( H )); ) Ψ restricts to an isomorphism from the copy of B inside B (Θ) onto the copy of A inside B (Σ) , and the associated map from B to A is ∗ -preserving ( hence isometric ) .Proof. It clearly suffices to prove the result in the two separate cases where Σ and Θ aregroup conjugate or exterior equivalent.Assume first that Σ ∼ gc Θ via isomorphisms φ : A → B and ϕ : G → H . Then thereader should have no trouble in verifying that the map Ψ : B (Θ) → B (Σ) given by[Ψ( S )]( g, a ) = φ − (cid:0) S ( ϕ ( g ) , φ ( a )) (cid:1) for S ∈ B (Θ), g ∈ G and a ∈ A , is a well-defined algebra-isomorphism satisfying 1), 2)and 3).Next, assume that Σ and Θ are exterior equivalent, so we have Θ = Σ w for somenormalized map w : G → U ( A ), where Σ w = ( A, G, α w , σ w ). Noting that L (Σ w ) = L (Σ),it is straightforward to check that the map Π : L (Σ) → L (Σ w ) given by[Π( T )]( g, a ) = T (cid:0) g, aw ( g ) (cid:1) w ( g ) ∗ for T ∈ L (Σ), g ∈ G and a ∈ A , is an algebra-isomorphism.Now, let T ∈ B (Σ), so T = T ρ,v,x,y for some equivariant representation ( ρ, v ) of Σon a Hilbert A -module X and x, y ∈ X . Then set e ρ = ρ and define e v : G → I ( X ) by e v ( g ) = ad ρ ( w ( g )) v ( g ), i.e., for each g ∈ G , e v ( g ) x = (cid:0) ρ ( w ( g )) v ( g ) x (cid:1) · w ( g ) ∗ x ∈ X . We claim that ( e ρ, e v ) is an equivariant representation of Σ w on X .Indeed, let g, h ∈ G, a ∈ A and x, y ∈ X . Then, using the properties of ( ρ, v ) repeat-edly, we get:(i) e ρ (cid:0) α wg ( a ) (cid:1)e v ( g ) x = ρ (cid:0) w ( g ) α g ( a ) w ( g ) ∗ (cid:1)(cid:16)(cid:0) ρ ( w ( g )) v ( g ) x (cid:1) · w ( g ) ∗ (cid:17) = (cid:0) ρ ( w ( g )) ρ ( α g ( a )) v ( g ) x (cid:1) · w ( g ) ∗ = (cid:0) ρ ( w ( g )) v ( g ) ρ ( a ) x (cid:1) · w ( g ) ∗ = e v ( g ) e ρ ( a ) x, (ii) e v ( g ) e v ( h ) x = (cid:0) ρ ( w ( g )) v ( g ) e v ( h ) x (cid:1) · w ( g ) ∗ = (cid:16) ρ ( w ( g )) v ( g ) (cid:0) ( ρ ( w ( h )) v ( h ) x ) · w ( h ) ∗ (cid:1)(cid:17) · w ( g ) ∗ = (cid:16) ρ ( w ( g )) (cid:16)(cid:0) ( v ( g ) ρ ( w ( h )) v ( h ) x ) · α g ( w ( h )) ∗ (cid:1)(cid:17)(cid:17) · w ( g ) ∗ = (cid:16) ρ ( w ( g )) (cid:0) ( ρ ( α g (( w ( h ))) v ( g ) v ( h ) x ) · α g ( w ( h )) ∗ (cid:1)(cid:17) · w ( g ) ∗ = (cid:16)(cid:0) ρ ( w ( g )) ρ ( α g ( w ( h ))) v ( g ) v ( h ) x (cid:1) · α g ( w ( h )) ∗ (cid:17) · w ( g ) ∗ = (cid:16) ρ ( w ( g )) ρ ( α g ( w ( h ))) (cid:0) ( ρ ( σ ( g, h )) v ( gh ) x ) · σ ( g, h ) ∗ ) (cid:1)(cid:17) · α g ( w ( h )) ∗ w ( g ) ∗ = (cid:16) ρ ( σ w ( g, h )) ρ ( w ( gh )) v ( gh ) x (cid:17) · σ ( g, h ) ∗ α g ( w ( h )) ∗ w ( g ) ∗ = (cid:16) ρ ( σ w ( g, h )) ρ ( w ( gh )) v ( gh ) x (cid:17) · w ( gh ) ∗ w ( gh ) σ ( g, h ) ∗ α g ( w ( h )) ∗ w ( g ) ∗ = (cid:16) ρ ( σ w ( g, h )) (cid:0) ( ρ ( w ( gh )) v ( gh ) x ) · w ( gh ) ∗ (cid:1)(cid:17) · σ w ( g, h ) ∗ = (cid:16) ρ ( σ w ( g, h )) (cid:0)e v ( gh ) x (cid:1)(cid:17) · σ w ( g, h ) ∗ = ad e ρ ( σ w ( g, h )) e v ( gh ) x, (iii) α wg (cid:0) h x, y i (cid:1) = w ( g ) α g (cid:0) h x, y i (cid:1) w ( g ) ∗ = w ( g ) (cid:10) v ( g ) x, v ( g ) y (cid:11) w ( g ) ∗ = w ( g ) (cid:10) ρ ( w ( g )) v ( g ) x, ρ ( w ( g )) v ( g ) y (cid:11) w ( g ) ∗ = (cid:10)(cid:0) ρ ( w ( g )) v ( g ) x (cid:1) · w ( g ) ∗ , (cid:0) ρ ( w ( g )) v ( g ) y (cid:1) · w ( g ) ∗ (cid:11) = (cid:10)e v ( g ) x, e v ( g ) y (cid:11) , e v ( g )( x · a ) = (cid:0) ρ ( w ( g )) v ( g )( x · a ) (cid:1) · w ( g ) ∗ = ρ ( w ( g )) (cid:0) ( v ( g ) x ) · ( α g ( a ) w ( g ) ∗ ) (cid:1) = ρ ( w ( g )) (cid:0) ( v ( g ) x ) · ( w ( g ) ∗ α wg ( a )) (cid:1) = (cid:16) ρ ( w ( g )) (cid:0) ( v ( g ) x ) · w ( g ) ∗ (cid:1)(cid:17) · α wg ( a )= ( e v ( g ) x ) · α wg ( a ) , as claimed. Now for all g ∈ G and a ∈ A we have[Π( T )]( g, a ) = T ρ,v,x,y (cid:0) g, aw ( g ) (cid:1) w ( g ) ∗ = (cid:10) x, ρ ( aw ( g )) v ( g ) y (cid:11) w ( g ) ∗ = (cid:10) x, (cid:0) ρ ( a ) ρ ( w ( g )) v ( g ) y (cid:1) · w ( g ) ∗ (cid:11) = (cid:10) x, e ρ ( a ) e v ( g ) y (cid:11) = T e ρ, e v,x,y ( g, a ) , so we get that Π maps B (Σ) into B (Σ w ) and that k Π( T ) k ≤ k x kk y k . Since this inequalityholds for any ρ, v, x, y such that T = T ρ,v,x,y , it follows that k Π( T ) k ≤ k T k . By symmetry,we then see that Π restricts to an isometric algebra-isomorphism between B (Σ) and B (Σ w ).It follows that Ψ := Π − is an algebra-isomorphism from B (Θ) = B (Σ w ) onto B (Σ) suchthat 1) holds. In passing, we note that one can also easily deduce that Π( T ) is Σ w -positivedefinite whenever T is Σ-positive definite, either by a direct computation, or using whatwe just have done in combination with the Gelfand-Raikov characterization of positivedefiniteness (cf. [9, Corollary 4.4]).Let now f ∈ B ( G ) and consider T f ∈ B (Σ). Then we have thatΠ( T f )( g, a ) = T f (cid:0) g, aw ( g ) (cid:1) w ( g ) ∗ = (cid:0) f ( g ) aw ( g ) (cid:1) w ( g ) ∗ = f ( g ) a = T f ( g, a )for all g ∈ G and a ∈ A , which shows that Π( T f ) = T f ∈ B (Σ w ). Thus it is clear that Πrestricts to the identity map from B ( G ) (inside B (Σ)) into B ( G ) (inside B (Σ w )), hencethat Ψ = Π − satisfies 2).Finally, let b ∈ A and consider T b ∈ B (Σ). Then we have thatΠ( T b )( g, a ) = T b (cid:0) g, aw ( g ) (cid:1) w ( g ) ∗ = baw ( g ) w ( g ) ∗ = ba = T b ( g, a )for all g ∈ G and a ∈ A . Thus it is clear that Π restricts to the identity map from A (inside B (Σ)) into A (inside B (Σ w )), hence that Ψ = Π − satisfies 3). Remark 3.9.
The converse of Theorem 3.8 is not true in general. Indeed, set Z ( G, T ) = { ω : G × G → T | ω is a normalized 2-cocycle on G } . Then let ω ∈ Z ( G, T ) and consider the systems Σ = ( C , G, triv ,
1) and Θ = ( C , G, triv , ω ),where triv denotes the obvious action of G on C . Then we have that B (Σ) = B ( G ) = B (Θ),but Σ is not cocycle group conjugate to Θ if ω is not a coboundary. Remark 3.10.
In order to look for a converse of Theorem 3.8 one option is to weakencocycle group conjugacy as follows. If Σ = (
A, G, α, σ ) is a system and ω ∈ Z ( G, T ), thenwe may regard ω as a normalized 2-cocycle for the restriction of α to Z ( A ) and perturb9 by ω (cf. Remark 3.2). Obviously, Σ and Σ( ω ) = ( A, G, α, σ ω ) have then the sameequivariant representations, so we have that B (Σ) = B (Σ( ω )).If Θ = ( B, H, β, θ ) is another system, let us say that Σ and Θ are weakly cocycle groupconjugate if Σ( ω ) is cocycle group conjugate to Θ for some ω ∈ Z ( G, T ). Using Theorem3.8 we get that B (Θ) is then isomorphic to B (Σ( ω )) = B (Σ) via an algebra-isomorphismsatisfying 1), 2) and 3).Let us now assume that the conclusion of Theorem 3.8 holds. One may then wonderunder which additional requirements it would be possible to conclude that Σ and Θ areweakly cocycle group conjugate. A result in this direction goes as follows.By invoking Walter’s theorem recalled in the introduction we get from 2) that Ψdetermines an isomorphism ϕ : G → H , while 3) gives that there is a ∗ -isomorphism φ : A → B . For each g ∈ G , set γ g := φ − β ϕ ( g ) φ ∈ Aut( A ). Then one may checkwhether γ g and α g agree up to inner automorphisms for every g ∈ G . Assume that thishappens to be the case, i.e., there exists some normalized map w : G → U ( A ) such that γ g = Ad( w ( g )) α g for all g ∈ G . Then, letting u : G × G → U ( A ) be defined by u ( g, g ′ ) = φ − (cid:0) θ ( ϕ ( g ) , ϕ ( g ′ )) (cid:1) for all g, g ′ ∈ G, we get a twisted action ( γ, u ) of G on A . Define then a map ω : G × G → U ( A ) by ω ( g, g ′ ) := u ( g, g ′ ) σ w ( g, g ′ ) ∗ for all g, g ′ ∈ G . Then, using the two expressions for γ and making use of some cocycleidentities, one verifies that ω takes its values in Z ( A ), and that it is a 2-cocycle for α ′ (therestriction of α to Z ( A )). Since u = ( σ w ) ω , it follows thatΘ = ( B, H, β, θ ) ∼ gc ( A, G, γ, u ) = (
A, G, α w , ( σ w ) ω ) = Σ w ( ω )(using notation as in Remark 3.2). Hence, if A (and therefore B ) has trivial center, weget that ω ∈ Z ( G, T ) and Θ is group conjugate to Σ w ( ω ), which is exterior equivalent toΣ( ω ). Thus, Σ and Θ are weakly group cocycle conjugate in this case.As a consequence, we obtain the following. Theorem 3.11.
Consider two systems
Σ = (
A, G, α, σ ) and Θ = (
B, H, β, θ ) . Assumethat there exists an algebraic isomorphism Π : B (Σ) → B (Θ) satisfying that Π( T ) (cid:0) ϕ ( g ) , φ ( a ) (cid:1) = φ (cid:0) T ( g, aw ( g )) w ( g ) ∗ (cid:1) for all g ∈ G, a ∈ A, (3.1) for some isomorphism ϕ : G → H , some ∗ -isomorphism φ : A → B and some map w : G → U ( A ) , which also satisfies Π( T ℓ A ,α,x,y ) = T ℓ B ,β,φ ( x ) ,φ ( y ) (3.2) for all x, y ∈ A . If the center of A is trivial, then Σ and Θ are weakly cocycle groupconjugate.Proof. Using (3.1) and (3.2), one deduces that φ − β ϕ ( g ) φ = Ad( w ( g )) α g for all g ∈ G . Weare then in the position to proceed as we did above, and the desired assertion follows atonce. 10 On Morita equivalent systems
Let us consider two twisted unital discrete C ∗ -dynamical systems Σ = ( A, G, α, σ ) andΘ = (
B, G, β, θ ) over the same group G . (We will briefly discuss the more general situationin Remark 4.8.) Our main aim in this section is to show that if Σ and Θ are Moritaequivalent in the sense of [13, 27], then the Fourier-Stieltjes algebras B (Σ) and B (Θ) canbe determined from each other. Morita equivalence for (untwisted) C ∗ -dynamical systemsgoes at least back to [16]. For the ease of the reader, we review the definitions of theconcepts that we will use.Following [19], we say that a right Hilbert B -module Z is a right Hilbert A - B bimodule if there is a homomorphism κ : A → L B ( Z ). We set a · z = κ ( a ) z for a ∈ A and z ∈ Z , andfrequently write A Z B for Z . A right Hilbert A - B bimodule isomorphism Φ : A Z B → A W B between two right A - B Hilbert bimodules Z and W (or simply an isomorphism, for short)is a bimodule isomorphism such that h Φ( z ) , Φ( z ′ ) i B = h z, z ′ i B for z, z ′ ∈ A Z B . Left Hilbert A - B bimodules and their isomorphisms are defined in a similar way.Let A Z B be a right Hilbert A - B -bimodule. A map δ from G into I ( Z ) (the group ofinvertible C -linear bounded maps from Z into itself) is called a (Σ , Θ)- compatible actionof G on A Z B when the following conditions are satisfied for g ∈ G, a ∈ A , z, ζ ∈ Z and b ∈ B : • δ ( g )( a · z ) = α g ( a ) · ( δ ( g ) z ), • δ ( g )( z · b ) = ( δ ( g ) z ) · β g ( b ), • δ ( g ) δ ( h ) z = σ ( g, h ) · ( δ ( gh ) z ) · θ ( g, h ) ∗ , • (cid:10) δ ( g ) z, δ ( g ) ζ (cid:11) B = β g ( h z, ζ i B ).We will let S δ,z,ζ : G × A → B be the map defined by S δ,z,ζ ( g, a ) = (cid:10) z, a · ( δ ( g ) ζ ) (cid:11) B for all g ∈ G and a ∈ A . Clearly, if g ∈ G is fixed, the map a → S δ,z,ζ ( g, a ) from A into B is linear; moreover, it is bounded, since one easily shows that k S δ,z,ζ ( g, a ) k ≤ k z kk ζ kk a k for all a ∈ A .Two (Σ , Θ)-compatible actions δ and δ ′ of G , acting respectively on A Z B and A Z ′ B ,are called equivariantly isomorphic if there exists an isomorphism of right Hilbert A - B -bimodules between A Z B and A Z ′ B which intertwines δ and δ ′ . Remark 4.1.
If ( ρ, v ) is an equivariant representation of Σ on a right Hilbert A -module X , then X is a right Hilbert A - A -bimodule (using ρ as the left action of A on X ) and v isa (Σ , Σ)-compatible action of G on A X A . Conversely, if v is a (Σ , Σ)-compatible action of G on a right Hilbert A - A -bimodule X , where the left action of A on X is given by somehomomorphism ρ : A → L A ( X ), then ( ρ, v ) is an equivariant representation of Σ on X . We recall that by our standing assumptions, κ is then unit preserving, hence nondegenerate, as requiredin [19]. A as a right Hilbert A - A -bimodule in the obvious way, thenthe map α : G → I ( A ) is a (Σ , Σ)-compatible action of G on A A A , corresponding to thetrivial equivariant representation ( α, ℓ ) of Σ on A .We recall that a right Hilbert B -module X is called full when h X, X i = B . Fullnessof a left Hilbert C ∗ -module is defined in a similar way. An A - B imprimitivity bimodule Z = A Z B (sometimes called an equivalence A - B -bimodule) is a full right Hilbert A - B -bimodule w.r.t. a B -valued inner product h· , ·i B , which is also a full left Hilbert A - B -bimodule w.r.t. to an A -valued inner product A h· , ·i , in such a way that A h z, z ′ i · z ′′ = z · h z ′ , z ′′ i B for all z, z ′ , z ′′ ∈ Z . It then follows that k A h z, z ik = kh z, z i B k for all z ∈ Z , hence that thetwo norms on Z associated to the left and the right inner products coincide.Following [13, 27], we say that the two systems Σ and Θ are Morita equivalent whenthere exist an A - B imprimitivity bimodule Z together with a (Σ , Θ)-compatible action δ of G on Z ; we then write Σ ∼ ( Z,δ ) Θ. We note that δ automatically satisfies • A (cid:10) δ ( g ) z, δ ( g ) ζ (cid:11) = α g ( A h z, ζ i ) , see e.g. the argument given in [19, Remark 2.6 (2)].It is easy to check that Σ and Θ are Morita equivalent whenever they are cocycle conju-gate (see e.g. [16, Section 9] for the untwisted case). Moreover, Morita equivalent twisted C ∗ -dynamical systems have Morita equivalent C ∗ -crossed products (see [13, Theorem 2.3]for the full case, and [17, Sections 2.5.4 and 2.8.6] for the reduced case). We also mentionthe following result, which is probably a part of the folklore on this topic. Proposition 4.2.
Assume that
Σ = (
A, G, α, σ ) and Θ = (
B, G, β, θ ) are Morita equiva-lent, and that A and B are commutative. Then the action α of G on A is conjugate to theaction β of G on B , i.e., there exists an isomorphism φ from A onto B which intertwinesthese actions. Moreover, Σ is conjugate to the system ( B, G, β, σ φ ) , while Θ is conjugateto the system ( A, G, α, θ φ − ) , where σ φ ( g, h ) := φ ( σ ( g, h )) and θ φ − ( g, h ) := φ − ( θ ( g, h )) for all g, h ∈ G .Proof. The assumption says that Σ ∼ ( X,δ ) Θ for some A - B imprimitivity bimodule Z andsome (Σ , Θ)-compatible action δ of G on Z . In particular, A and B are Morita equivalent.As A, B are both commutative, we can then apply [10, Theorem 2.24] to conclude thatthere is a unique isomorphism φ : A → B satisfying that φ ( A h z, z ′ i ) = h z ′ , z i B for all z, z ′ ∈ Z, (4.1)and we also have that a · z = z · φ ( a ) for all a ∈ A and z ∈ Z . Using properties of δ incombination with (4.1) we get φ (cid:0) α g ( A h z, z ′ i ) (cid:1) = φ (cid:0) A h δ ( g ) z, δ ( g ) z ′ i (cid:1) = h δ ( g ) z ′ , δ ( g ) z i B = β g ( h z ′ , z i B ) = β g (cid:0) φ ( A h z, z ′ i ) (cid:1) for all g ∈ G and z, z ′ ∈ Z . Since Z is full as a left Hilbert A -module, it follows that φα g = β g φ for every g ∈ G , hence that α and β are conjugate. This shows the first partof the proposition. The second part follows immediately.12n the setting of Proposition 4.2, it is not clear that Σ and Θ are conjugate. However,this is certainly the case when σ and θ are both trivial: Corollary 4.3.
Suppose that ( A, G, α ) and ( B, G, β ) are ( untwisted discrete unital ) C ∗ -dynamical systems with both A and B commutative. Then these systems are Morita equiva-lent if and only if they are conjugate, in which case the associated Fourier-Stieltjes algebrasare isometrically isomorphic.Proof. This follows from Proposition 4.2 and Theorem 3.8.Assume now that Ω = (
C, G, γ, ω ) is another twisted discrete unital C ∗ -dynamicalsystem, δ is a (Σ , Θ)-compatible action of G on A X B and η is a (Θ , Ω)-compatible actionof G on B Y C . If π : B → L C ( Y ) denotes the left action of B on Y , we can form the internaltensor product X ⊗ π Y , which is a right Hilbert C -module (cf. [28]); we will suppress π inour notation and denote X ⊗ π Y by X ⊗ B Y in the sequel, as is common in the literature.Moreover, X ⊗ B Y can be turned into a right Hilbert A - C bimodule, the left action of A on X ⊗ B Y being given on simple tensors by a · ( x ˙ ⊗ y ) = ( a · x ) ˙ ⊗ y , and we can define a(Σ , Ω)-compatible product action δ ⊗ B η of G on A ( X ⊗ B Y ) C , which is given on simpletensors by ( δ ⊗ B η )( g )( x ˙ ⊗ y ) = δ ( g ) x ˙ ⊗ η ( g ) y . Indeed, as a sample, consider g, h ∈ G , x ∈ X and y ∈ Y . Then we have (cid:0) ( δ ⊗ B η )( g )( δ ⊗ B η )( h ) (cid:1) ( x ˙ ⊗ y ) = ( δ ⊗ B η )( g ) (cid:0) δ ( h ) x ˙ ⊗ η ( h ) y (cid:1) = δ ( g ) δ ( h ) x ˙ ⊗ η ( g ) η ( h ) y = (cid:0) σ ( g, h ) · ( δ ( gh ) x ) · θ ( g, h ) ∗ (cid:1) ˙ ⊗ (cid:0) θ ( g, h ) · ( η ( gh ) x ) · ω ( g, h ) ∗ (cid:1) = (cid:0)(cid:0) σ ( g, h ) · ( δ ( gh ) x ) · θ ( g, h ) ∗ (cid:1) · θ ( g, h ) (cid:1) ˙ ⊗ (cid:0)(cid:0) η ( gh ) y (cid:1) · ω ( g, h ) ∗ (cid:1) = σ ( g, h ) · (cid:0) δ ( gh ) x ˙ ⊗ η ( gh ) y (cid:1) · ω ( g, h ) ∗ = σ ( g, h ) · (cid:0) ( δ ⊗ B η )( gh )( x ˙ ⊗ y ) (cid:1) · ω ( g, h ) ∗ Thus, by continuity, it follows that δ ⊗ B η satisfies the third property required for beinga (Σ , Ω)-compatible action. The reader will find more details about this construction andits properties in [18, 19]. These articles deal with the untwisted case, but it is easy toadapt the proofs to our setting. In particular, arguing as in the proof of [19, Theorem 2.8and Remark 2.9], we obtain that the following facts hold: • Up to equivariant isomorphism, the product of compatible actions is associative. • Recalling that α is a (Σ , Σ)-compatible action of G on A A A , the (Σ , Θ)-compatibleproduct action α ⊗ A δ of G on A ( A ⊗ A X ) B is equivariantly isomorphic to δ . Ina similar way, the product action δ ⊗ B β of G on A ( X ⊗ B B ) B is equivariantlyisomorphic to δ . • Assume that Σ and Θ are Morita equivalent with Σ ∼ ( Z,δ ) Θ. Then we have: – Θ ∼ ( e Z, e δ ) Σ, where e Z is the right Hilbert B - A bimodule conjugate (or reverse)to Z and e δ is the (Θ , Σ)-compatible action of G on e Z given by e δ ( g ) e z = ] δ ( g ) z . – The product action δ ⊗ B e δ of G on A ( Z ⊗ B e Z ) A is equivariantly isomorphic, asa (Σ , Σ)-compatible action, to α .13 The product action e δ ⊗ A δ of G on B ( e Z ⊗ A Z ) B is equivariantly isomorphic, asa (Θ , Θ)-compatible action, to β .Next, consider a (Σ , Σ)-compatible action v of G on a right Hilbert A - A bimodule X . Wewill use the same notation as in [19] and let [ X, v ] denote the class of all pairs ( X ′ , v ′ )where v ′ is a (Σ , Σ)-compatible action of G on a right Hilbert A - A -module X ′ such that v ′ is equivariantly isomorphic to v . Further, we will let A (Σ) denote the collection of theseequivalence classes. Using the above properties, one sees that A (Σ) can be equipped withan associative product given by[ X , v ][ X , v ] := [ X ⊗ A X , v ⊗ A v ] , and that [ A, α ] acts as a unit in A (Σ). Moreover, one readily gets the following result. Proposition 4.4.
Assume that the systems Σ and Θ are Morita equivalent with Σ ∼ ( Z,δ ) Θ , and let v be a (Σ , Σ) -compatible action on a right Hilbert A - A bimodule X .Then w := ( e δ ⊗ A v ) ⊗ A δ is a (Θ , Θ) -compatible action on the right Hilbert B - B -bimodule Y := ( e Z ⊗ A X ) ⊗ A Z .Moreover, the action δ ⊗ B ( w ⊗ B e δ ) on the right Hilbert A - A -bimodule e Z ⊗ B ( Y ⊗ B Z ) is equivariantly isomorphic to v .Hence, the map [ X, v ] [ Y, w ] gives a one-to-one correspondence between A (Σ) and A (Θ) which preserves products. Taking into account Remark 4.1 this result says that, up to isomorphism, the equivari-ant representations of two Morita equivalent systems are in a one-to-one correspondence.As we will soon see, this has some relevance for the associated Fourier-Stieltjes algebras.By isomorphism of equivariant representations of a system, we mean the following.Let ( ρ, v ) , ( ρ ′ , v ′ ) be equivariant representations of Σ on right Hilbert A -modules X and X ′ , respectively. Then ( ρ, v ) and ( ρ ′ , v ′ ) are said to be isomorphic if v and v ′ areequivariantly isomorphic as (Σ , Σ)-compatible actions of G , i.e., there exists an isomor-phism of right Hilbert A -modules φ : X → X ′ which intertwines v and v ′ , as well as ρ and ρ ′ . We note that in this case we have T ρ,v,x,y = T ρ ′ ,v ′ ,φ ( x ) ,φ ( y ) (4.2)for all x, y ∈ X . Indeed, for each a ∈ A and g ∈ G , we have T ρ,v,x,y ( g, a ) = (cid:10) x, ρ ( a ) v ( g ) y (cid:11) = (cid:10) φ ( x ) , φ (cid:0) ρ ( a ) v ( g ) y (cid:1)(cid:11) ′ = (cid:10) φ ( x ) , ρ ′ ( a ) v ′ ( g ) φ ( y ) (cid:11) ′ = T ρ ′ ,v ′ ,φ ( x ) ,φ ( y ) ( g, a ) . The following notation will be useful. If S : G × A → B, T : G × A → A and R : G × B → A are maps, then we let S · T : G × A → B and T · R : G × B → A be themaps given by ( S · T )( g, a ) = S ( g, T ( g, a )) , ( T · R )( g, b ) = T ( g, R ( g, b ))for all g ∈ G, a ∈ A and b ∈ B . Moreover, we let S · T · R : G × B → B be given by S · T · R := ( S · T ) · R = S · ( T · R ) . roposition 4.5. Assume that the systems Σ and Θ are Morita equivalent with Σ ∼ ( Z,δ ) Θ , and let ( ρ, v ) be an equivariant representation of Σ on a right Hilbert A -module X . Let x, x ′ ∈ X and z, z ′ , ζ, ζ ′ ∈ Z . Then the map S δ,z ′ ,ζ ′ · T ρ,v,x,x ′ · S ˜ δ, ˜ z, ˜ ζ : G × B → B belongs to B (Θ) . Thus we get a linear map F z,z ′ ,ζ,ζ ′ : B (Σ) → B (Θ) given by F z,z ′ ,ζ,ζ ′ ( T ) = S δ,z ′ ,ζ ′ · T · S ˜ δ, ˜ z, ˜ ζ for every T ∈ B (Σ) . Similarly, the assignment T ′ S ˜ δ, ˜ z, ˜ ζ · T ′ · S δ,z ′ ,ζ ′ gives a linear mapfrom B (Θ) into B (Σ) .Proof. Let Y = ( e Z ⊗ A X ) ⊗ A Z and w = ( e δ ⊗ v ) ⊗ δ : G → I ( Y ) be as in Proposition 4.4,and let τ : B → L B ( Y ) denote the homomorphism coming from the left action of B on Y , so ( τ, w ) is an equivariant representation of Θ on the right Hilbert B -module Y .Let g ∈ G and b ∈ B . Then we have T τ,w, (˜ z ˙ ⊗ x ) ˙ ⊗ z ′ , (˜ ζ ˙ ⊗ x ′ ) ˙ ⊗ ζ ′ ( g, b ) = D z ′ , T ρ,v,x,x ′ (cid:0) g, A (cid:10) z · b, δ ( g ) ζ (cid:11)(cid:1) · δ ( g ) ζ ′ E B . Indeed, T τ,w, (˜ z ˙ ⊗ x ) ˙ ⊗ z ′ , (˜ ζ ˙ ⊗ x ′ ) ˙ ⊗ ζ ′ ( g, b ) = D (˜ z ˙ ⊗ x ) ˙ ⊗ z ′ , τ ( b ) w ( g )(˜ ζ ˙ ⊗ x ′ ) ˙ ⊗ ζ ′ E B = D (˜ z ˙ ⊗ x ) ˙ ⊗ z ′ , (cid:0) (( δ ( g ) ζ ) · b ∗ ) e ˙ ⊗ v ( g ) x ′ (cid:1) ˙ ⊗ δ ( g ) ζ ′ E B = D z ′ , (cid:10) ˜ z ˙ ⊗ x, (( δ ( g ) ζ ) · b ∗ ) e ˙ ⊗ v ( g ) x ′ (cid:11) A · δ ( g ) ζ ′ E B = D z ′ , (cid:10) x, h ˜ z, (( δ ( g ) ζ ) · b ∗ ) e i A · v ( g ) x ′ (cid:11) A · δ ( g ) ζ ′ E B = D z ′ , (cid:10) x, A h z, ( δ ( g ) ζ ) · b ∗ i · v ( g ) x ′ (cid:11) A · δ ( g ) ζ ′ E B = D z ′ , (cid:10) x, ρ (cid:0) A h z · b, δ ( g ) ζ i (cid:1) v ( g ) x ′ (cid:11) A · δ ( g ) ζ ′ E B = D z ′ , T ρ,v,x,x ′ (cid:0) g, A (cid:10) z · b, δ ( g ) ζ (cid:11)(cid:1) · δ ( g ) ζ ′ E B , as asserted. Since S ˜ δ, ˜ z, ˜ ζ ( g, b ) = (cid:10) ˜ z, b · (˜ δ ( g )˜ ζ ) (cid:11) A = (cid:10) ˜ z, (cid:0) ( δ ( g ) ζ ) · b ∗ (cid:1)e(cid:11) A = A (cid:10) z, ( δ ( g ) ζ ) · b ∗ (cid:11) = A (cid:10) z · b, δ ( g ) ζ (cid:11) , we get that (cid:0) S δ,z ′ ,ζ ′ · T ρ,v,x,x ′ · S ˜ δ, ˜ z, ˜ ζ (cid:1) ( g, b ) = S δ,z ′ ,ζ ′ (cid:0) g, T ρ,v,x,x ′ (cid:0) g, A (cid:10) z · b, δ ( g ) ζ (cid:11)(cid:1)(cid:1) = (cid:10) z ′ , T ρ,v,x,x ′ (cid:0) g, A (cid:10) z · b, δ ( g ) ζ (cid:11)(cid:1) · δ ( g ) ζ ′ (cid:11) B . This shows that S δ,z ′ ,ζ ′ · T ρ,v,x,x ′ · S ˜ δ, ˜ z, ˜ ζ = T τ,w, (˜ z ˙ ⊗ x ) ˙ ⊗ z ′ , (˜ ζ ˙ ⊗ x ′ ) ˙ ⊗ ζ ′ ∈ B (Θ)and the first claim follows. The remaining claims are then easily obtained.15 orollary 4.6. Assume Σ and Θ are Morita equivalent with Σ ∼ ( Z,δ ) Θ . Then B (Θ) can be determined from B (Σ) and Z ( and similarly for the other way around ) . Indeed, wehave B (Θ) = Span n F z,z ′ ,ζ,ζ ′ ( T ) | T ∈ B (Σ) , z, z ′ , ζ, ζ ′ ∈ Z o . (4.3) Proof.
Using Proposition 4.5 we get that the right-hand side of (4.3) is contained in B (Θ).To show the reverse inclusion, we first observe that for z, z ′ , ζ, ζ ′ ∈ Z , g ∈ G and b ∈ B we have (cid:0) S δ,z ′ ,ζ ′ · S ˜ δ, ˜ z, ˜ ζ (cid:1) ( g, b ) = S δ,z ′ ,ζ ′ (cid:0) g, S ˜ δ, ˜ z, ˜ ζ ( g, b ) (cid:1) = S δ,z ′ ,ζ ′ (cid:0) g, A (cid:10) z · b, δ ( g ) ζ (cid:11)(cid:1) = (cid:10) z ′ , A (cid:10) z · b, δ ( g ) ζ (cid:11) · ( δ ( g ) ζ ′ ) (cid:11) B = (cid:10) z ′ , z · b · (cid:10) δ ( g ) ζ, δ ( g ) ζ ′ (cid:11) B (cid:11) B = (cid:10) z ′ , z (cid:11) B b (cid:10) δ ( g ) ζ, δ ( g ) ζ ′ (cid:11) B = (cid:10) z, z ′ (cid:11) ∗ B bβ g (cid:0) h ζ, ζ ′ i B (cid:1) Now, since Z is full as a right Hilbert B -module, we can use Lemma 2.5 in [10] to find z , z ′ , . . . , z n , z ′ n ∈ Z such that n X i =1 h z i , z ′ i i B = 1 B (the unit of B ) . (4.4)(In fact, proceeding as in [25, p. 90], one may even choose z ′ j = z j for all j = 1 , . . . , n , butwe won’t need this). We note that n X i,j =1 F z i ,z ′ i ,z j ,z ′ j ( I Σ ) = n X i,j =1 S δ,z ′ i ,z ′ j · S ˜ δ, ˜ z i , ˜ z j = I Θ . (4.5)Indeed, for g ∈ G and b ∈ B , using (4.4), we get (cid:16) n X i,j =1 S δ,z ′ i ,z ′ j · S ˜ δ, ˜ z i , ˜ z j (cid:17) ( g, b ) = n X i,j =1 (cid:10) z i , z ′ i (cid:11) ∗ B bβ g (cid:0) h z j , z ′ j i B (cid:1) = (cid:0) n X i =1 (cid:10) z i , z ′ i (cid:11) B (cid:1) ∗ bβ g (cid:0) n X j =1 h z j , z ′ j i B (cid:1) = b. Let T ′ ∈ B (Θ). For each i, j, k, l ∈ { , . . . , n } , set T ′ i,j,k,l := S ˜ δ, ˜ z i , ˜ z j · T ′ · S δ,z ′ k ,z ′ l , which belongs to B (Σ) (by Proposition 4.5). Then, using (4.5), we get that n X i,j,k,l =1 F z k ,z ′ i ,z l ,z ′ j (cid:0) T ′ i,j,k,l (cid:1) = n X i,j,k,l S δ,z ′ i ,z ′ j · S ˜ δ, ˜ z i , ˜ z j · T ′ · S δ,z ′ k ,z ′ l · S ˜ δ, ˜ z k , ˜ z l = (cid:16) n X i,j =1 S δ,z ′ i ,z ′ j · S ˜ δ, ˜ z i , ˜ z j (cid:17) · T ′ · (cid:16) n X k,l =1 S δ,z ′ k ,z ′ l · S ˜ δ, ˜ z k , ˜ z l (cid:17) = I Θ · T ′ · I Θ = T ′ , which shows that T ′ ∈ Span n F z,z ′ ,ζ,ζ ′ ( T ) | T ∈ B (Σ) , z, z ′ , ζ, ζ ′ ∈ Z o , as desired.16n view of the last statement of Corollary 4.3, one might wonder under which assump-tions the Fourier-Stieltjes algebras associated to Morita equivalent systems are actually(isometrically) isomorphic, cf. Theorem 3.8 (see also Remark 4.7). Also, it would be inter-esting to investigate whether in general those Fourier-Stieltjes algebras could be Moritaequivalent as Banach algebras in some suitable sense (see e.g. [26] or [31]). However,elaborating on this topic would require the development of additional machinery, and wewon’t discuss this here. Remark 4.7.
It may be worth to point out that in general Morita equivalence of systemsis not sufficient to ensure that the associated Fourier-Stieltjes algebras are isomorphic.Indeed, consider Σ = ( C , G, triv ,
1) and Θ = ( M ( C ) , G, triv ,
1) for some discrete group G (where triv denotes the trivial action in both cases). It is then easy to see that Σ and Θare Morita equivalent. On the other hand, B (Σ) = B ( G ) is commutative, while B (Θ) isnot as it contains a copy of M ( C ). Remark 4.8.
Consider two systems Σ = (
A, G, α, σ ) and Θ = (
B, H, β, θ ) where H mightbe different from G , as in the previous section. If ϕ : G → H is an isomorphism, we obtain anew system Θ ϕ = ( B, G, β ϕ , θ ϕ ) by setting β ϕg = β ϕ ( g ) and θ ϕ ( g, g ′ ) = θ ( ϕ ( g ) , ϕ ( g ′ )). Oneeasily checks that B (Θ) is isometrically isomorphic to B (Θ ϕ ). Now, let us say that Σ andΘ are weakly Morita equivalent if there exist some ω ∈ Z ( G, T ) and some isomorphism ϕ : G → H such that Σ( ω ) is Morita equivalent to Θ ϕ . Corollary 4.6 gives then that B (Σ) = B (Σ( ω )) can be determined from B (Θ ϕ ), hence from B (Θ), and vice-versa. Finally, wemention that Σ and Θ are weakly Morita equivalent whenever they are cocycle groupconjugate, as the reader will easily verify. Amenability is an important topic within operator algebras, and it has received a gooddeal of attention, also in connection with C ∗ -dynamical systems (see e.g. [4, 5, 20, 23, 12,7, 22, 9, 29, 14, 3, 15] and references therein). Using the technique used in the proof ofCorollary 4.6, we will show that amenability of a system, as defined in [9], is preservedunder Morita equivalence. As before, we let Σ = ( A, G, α, σ ) and Θ = (
B, G, β, θ ) denotetwo twisted unital discrete C ∗ -dynamical systems. We recall that Σ is said to be amenable whenever there exists a net { T ν } in P (Σ) such that • each T ν is finitely supported, i.e., the set { g ∈ G | T νg = 0 } is finite for each ν , • { T ν } is uniformly bounded, i.e., sup ν k T ν k ∞ < ∞ , • lim ν k T νg ( a ) − a k = 0 for every g ∈ G and a ∈ A .Assume for example that Σ has Exel’s ( positive ) approximation property [20, 22, 23],that is, there exists a net { ξ ν } of finitely supported functions from G into A such that(a) sup ν (cid:13)(cid:13) P g ∈ G ξ ν ( g ) ∗ ξ ν ( g ) (cid:13)(cid:13) < ∞ ;(b) lim ν (cid:13)(cid:13) P h ∈ G ξ ν ( h ) ∗ aα g (cid:0) ξ ν ( g − h ) (cid:1) − a (cid:13)(cid:13) = 0 for all g ∈ G and a ∈ A .17hen Σ is amenable because setting T νg ( a ) = P h ∈ G ξ ν ( h ) ∗ aα g (cid:0) ξ ν ( g − h ) (cid:1) for all g ∈ G and a ∈ A gives a net { T ν } satisfying the required properties. Note that if all ξ ν ’s take theirvalues in Z ( A ), then (b) is equivalent tolim ν (cid:13)(cid:13) X h ∈ G ξ ν ( h ) ∗ α g (cid:0) ξ ν ( g − h ) (cid:1) − A (cid:13)(cid:13) = 0for all g ∈ G . Thus it readily follows that if σ = 1, then Σ is amenable whenever theaction α is amenable in the sense of [12], a notion that is stronger than Anantharaman-Delaroche’s original definition of amenability of α in [4]. Notice also that as long as σ isscalar-valued then the amenability of Σ does not depend on σ . As shown in [9, Theorem4.6], amenability of Σ implies that Σ is regular , i.e., the full and the reduced C ∗ -crossedproducts associated to Σ are canonically isomorphic. Several other notions of amenability(for untwisted systems) are discussed in [14, 15]. We note that if A is commutative, G isexact and σ = 1, then it follows readily from [14, Theorem 5.2] that all existing notionsof amenability for Σ (including ours, and regularity) are equivalent.Strong and weak equivalence of Fell bundles over groups are studied in [1, 2, 3]. Havingin mind that Σ gives rise to a Fell bundle over G in a canonical way (cf. [21]), one may forinstance deduce from [2, Corollary 4.5] and [3, Theorem 6.23] that regularity and Exel’sapproximation property are preserved under Morita equivalence of systems. We provebelow that this is also true for amenability in our sense. Theorem 5.1.
Assume that the systems Σ and Θ are Morita equivalent, with Σ ∼ ( Z,δ ) Θ .Then Θ is amenable whenever Σ is amenable.Proof. Assume that Σ is amenable. As in the proof of Proposition 4.5, we can find z , z ′ , . . . , z n , z ′ n ∈ Z such that n X i =1 h z i , z ′ i i B = 1 B . For later use, we set K = (cid:0) P ni =1 k z i kk z ′ i k ) . Let then F : B (Σ) → B (Θ) be the linearmap given by F = n X i,j =1 F z i ,z ′ i ,z j ,z ′ j . We first note that F maps P (Σ) into P (Θ). To show this, we use the notation introducedin the proof of Proposition 4.5. Let T = T ρ,v,x,x ∈ P (Σ) and set y := P ni =1 (˜ z i ˙ ⊗ x ) ˙ ⊗ z ′ i ∈ Y := ( ˜ Z ⊗ A X ) ⊗ A Z . Then we have F ( T ) = n X i,j =1 F z i ,z ′ i ,z j ,z ′ j ( T ρ,v,x,x ) = n X i,j =1 T τ,w, (˜ z i ˙ ⊗ x ) ˙ ⊗ z ′ i , (˜ z j ˙ ⊗ x ) ˙ ⊗ z ′ j = n X j =1 T τ,w,y, (˜ z j ˙ ⊗ x ) ˙ ⊗ z ′ j = T τ,w,y,y ∈ P (Θ) . k F ( T ) k ∞ = kh y, y i B k = k y k ≤ (cid:16) n X i =1 k (˜ z i ˙ ⊗ x ) ˙ ⊗ z ′ i k (cid:17) ≤ (cid:16) n X i =1 k ˜ z i kk x kk z ′ i k (cid:17) = K k x k = K k T k ∞ (since k ˜ z k = k z k for all z ∈ Z ). Further, we note that F ( T ) is easily seen to be finitelysupported whenever T ∈ B (Σ) is finitely supported.Let now { T ν } be a net in P (Σ) witnessing the amenability of Σ. Then { F ( T ν ) } isclearly a net of finitely supported elements in P (Θ). Moreover, we have thatsup ν k F ( T ν ) k ∞ ≤ K sup ν k T ν k ∞ < ∞ , so { F ( T ν ) } is uniformly bounded. Finally, let g ∈ G and b ∈ B . Then F ( T ν )( g, b ) = n X i,j =1 (cid:0) F z i ,z ′ i ,z j ,z ′ j ( T ν ) (cid:1) ( g, b ) = n X i,j =1 (cid:0) S δ,z ′ i ,z ′ j · T ν · S ˜ δ, ˜ z i , ˜ z j (cid:1) ( g, b )= n X i,j =1 S δ,z ′ i ,z ′ j (cid:0) g, T νg (cid:0) S ˜ δ, ˜ z i , ˜ z j ( g, b ) (cid:1)(cid:1) . Consider now i, j ∈ { , . . . , n } . Using that the map a → S δ,z ′ i ,z ′ j (cid:0) g, a (cid:1) from A into B iscontinuous, we get thatlim ν S δ,z ′ i ,z ′ j (cid:0) g, T νg (cid:0) S ˜ δ, ˜ z i , ˜ z j ( g, b ) (cid:1)(cid:1) = S δ,z ′ i ,z ′ j (cid:0) g, lim ν T νg (cid:0) S ˜ δ, ˜ z i , ˜ z j ( g, b ) (cid:1)(cid:1) = S δ,z ′ i ,z ′ j (cid:0) g, (cid:0) S ˜ δ, ˜ z i , ˜ z j ( g, b ) (cid:1)(cid:1) = (cid:0) S δ,z ′ i ,z ′ j · S ˜ δ, ˜ z i , ˜ z j (cid:1) ( g, b ) . Hence, using Equation (4.5), we get thatlim ν F ( T ν )( g, b ) = n X i,j =1 lim ν S δ,z ′ i ,z ′ j (cid:0) g, T νg (cid:0) S ˜ δ, ˜ z i , ˜ z j ( g, b ) (cid:1)(cid:1) = n X i,j =1 (cid:0) S δ,z ′ i ,z ′ j · S ˜ δ, ˜ z i , ˜ z j (cid:1) ( g, b ) = I Θ ( b ) = b. This shows that Θ is amenable, as desired.An immediate consequence of this result is that amenability of a system is also pre-served under weak Morita equivalence (as defined in Remark 4.8).
Remark 5.2.
A result of a nature similar to Theorem 5.1 is Theorem 2.2.17 in [6],which says that topological amenability of locally compact groupoids is invariant undertopological equivalence (whose definition is hinted by Morita equivalence of C ∗ -algebras).19 cknowledgements Most of the present work has been done during the visits made by E.B. at the SapienzaUniversity of Rome and by R.C. at the University of Oslo in the period 2018-2019. Boththe authors are grateful to the hosting institutions for the kind hospitality, and to theTrond Mohn Foundation (TMS) for financial support. We are also indebted to the refereesfor their many helpful comments and suggestions, leading to an improved version of thisarticle.
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