Amenable and inner amenable actions and approximation properties for crossed products by locally compact groups
aa r X i v : . [ m a t h . OA ] J a n AMENABLE AND INNER AMENABLE ACTIONS ANDAPPROXIMATION PROPERTIES FOR CROSSEDPRODUCTS BY LOCALLY COMPACT GROUPS
ANDREW MCKEE AND REYHANEH POURSHAHAMI
Abstract.
Amenable actions of locally compact groups on von Neu-mann algebras are investigated by exploiting the natural module struc-ture of the crossed product over the Fourier algebra of the acting group.The resulting characterisation of injectivity for crossed products gener-alises a result of Anantharaman-Delaroche on discrete groups. Amenableactions of locally compact groups on C ∗ -algebras are investigated in thesame way, and amenability of the action is related to nuclearity of thecorresponding crossed product. A survey is given to show that this no-tion of amenable action for C ∗ -algebras satisfies a number of expectedproperties. A notion of inner amenability for actions of locally com-pact groups is introduced, and a number of applications are given inthe form of averaging arguments, relating approximation properties ofcrossed product von Neumann algebras to properties of the componentsof the underlying w ∗ -dynamical system. We use these results to answera recent question of Buss–Echterhoff–Willett. Introduction
In the setting of group actions on operator algebras there are a numberof questions which have satisfactory answers for actions of discrete groups,but which are still open for the case of actions of locally compact groups.(1) (a) How is injectivity of the crossed product corresponding to a w ∗ -dynamical system ( M, G, α ) related to amenability of the action α and injectivity of M ? (See [1, Corollaire 4.2] for the answerin the discrete case.)(b) How is nuclearity of the (full or reduced) crossed product ofa C ∗ -dynamical system ( A, G, α ) related to amenability of theaction α and nuclearity of A ? (See [3, Th´eor`eme 4.5] for theanswer in the discrete case.)(2) How can one carry out Haagerup’s averaging arguments, in orderto pass from approximation properties of a crossed product to ap-proximation properties of the components of a ( C ∗ or w ∗ )-dynamicalsystem? (See e.g. [27, Proposition 3.4] for the answer in the discretecase.) In this paper we show that these problems can be solved by accounting forthe module structure over the Fourier algebra of the acting group, whichcrossed products naturally carry.Restricting for a moment to group C ∗ -algebras and group von Neumannalgebras (the case of trivial actions) question (1) goes back to Lance [24, The-orem 4.3], who showed that a discrete group G is amenable if and only if C ∗ λ ( G ) is a nuclear C ∗ -algebra (and a similar technique shows vN( G ) is aninjective von Neumann algebra). Connes [12] showed that this does not gen-eralise directly to locally compact groups; indeed, though amenable locallycompact groups have injective group von Neumann algebras, there exist non-amenable locally compact groups G for which vN( G ) is injective (any non-amenable connected Lie group is an example). Lau and Paterson [30, page85] recognised that it is only in the class of inner amenable groups thatinjectivity of vN( G ) implies amenability of G . Since every amenable groupis inner amenable this provides a complete answer to the problem of un-derstanding amenability of locally compact groups in terms of their groupvon Neumann algebras. Recently Crann and Tanko [17, Theorem 3.4] gavea reinterpretation of this result in terms of module maps: it is injectivityof vN( G ) as an A ( G ) -module which implies that G is amenable, and in-ner amenable groups are the locally compact groups for which injectivity ofvN( G ) implies injectivity of vN( G ) as an A ( G )-module — see Theorem 2.6.All discrete groups are inner amenable, so in the discrete case one does notneed to account for the A ( G )-module structure.For trivial actions the averaging arguments mentioned in (2) were intro-duced by Haagerup [21] (written in 1986 and circulated as an unpublishedmanuscript), where such arguments were used to study discrete groups.Later work of Lau–Paterson [30, page 85] indicated that one can use in-ner amenability, rather than discreteness, to give such averaging arguments,extending these techniques beyond discrete groups. Crann [14] (see also[17, Section 3]) rephrases inner amenability in terms of relative module in-jectivity, clarifying the exact relationship between amenability of a locallycompact group and injectivity of its group von Neumann algebra.Moving on to group actions, Anantharaman-Delaroche [1] introduced adefinition of amenable action of a locally compact group on a von Neu-mann algebra, building on Zimmer’s work [35–37] on amenable actions oncommutative von Neumann algebras. For actions of discrete groups thisdefinition is used to solve question (1) part (a) [1, Corollaire 4.2], and wassubsequently adapted to solve question (1) part (b) [3, Th´eor`eme 4.5]. Wecaution the reader that there is another definition of amenable action on a setin the literature which is different from Zimmer’s; in this paper we consideronly Zimmer’s notion and its generalisation by Anantharaman-Delaroche.The question of how to define amenable actions of locally compact groupson C ∗ -algebras remained open (see [5, Section 9]). This question has at-tracted significant recent attention — we note work of Bearden–Crann [6],Buss–Echterhoff–Willett [10, 11], Ozawa–Suzuki [29] and Suzuki [32], all of MENABLE AND INNER AMENABLE ACTIONS 3 which are concerned with amenable actions of locally compact groups on C ∗ -algebras.Since we began this work there has been a flurry of papers on the topicof amenable actions of locally compact groups. As well as the papers [6,10, 11, 29, 32] mentioned above, which focus on amenable actions on C ∗ -algebras, we have recent work of Crann [15] and Bearden–Crann [7]. We haverewritten this paper several times in attempts to account for these works. Inparticular, Crann and Bearden–Crann [7, 15] also discovered the relationshipbetween amenable actions and module injectivity, and gave another relatedmodule property. Our results were obtained independently, but we notethat we have been heavily influenced by Crann’s earlier work on moduleinjectivity for quantum groups [13, 14].The organisation of this paper is as follows. In Section 2 we review thedefinitions and results surronding the notion of group operator algebrasand crossed products. Then we review the module versions of injectivity,and their link to amenability, which we aim to generalise. In Section 3 wegeneralise amenable actions on von Neumann algebras by using the nat-ural module structure of a crossed product over the Fourier algebra. InSection 4 we introduce inner amenable actions on von Neumann algebras.We give some basic properties of this notion and extend Lau–Paterson’sresult [25, Theorem 3.1] to inner amenable actions on non-commutativeinjective von Neumann algebras. In Section 5 we consider the weak* com-pletely bounded approximation property for crossed product von Neumannalgebras, giving an example of how inner amenable actions can be used foraveraging arguments in Proposition 5.2. Section 6 is devoted to amenableactions on C ∗ -algebras. We use the natural extension of Anantharaman-Delaroche’s definition of amenable action of a discrete group on a C ∗ -algebrato locally compact groups, and relate this to nuclearity of the correspond-ing crossed products. We also give a survey of known results, mainly fromwork of Buss–Echterhoff–Wilett [10], Bearden–Crann [6] and Suzuki [32],to demonstrate how close our notion of amenable action comes to satisfy-ing the requirements for an amenable action on a C ∗ -algebra suggested byAnantharaman-Delaroche [5, Section 9.2].2. Preliminaries
For von Neumann algebras M and N we will write their normal spatialtensor product as M ⊗ N . We will use throughout the theory of operatorspaces and completely bounded maps; the reference [19] is suggested for thisbackground material.2.1. Group operator algebras and crossed products.
Let G be a lo-cally compact group. We write λ for the left regular representation λ : G → B ( L ( G )); λ r ξ ( s ) := ξ ( r − s ) , r, s ∈ G, ξ ∈ L ( G ) , ANDREW MCKEE AND REYHANEH POURSHAHAMI which is a unitary representation of G , so extends to a ∗ -representation of C c ( G ) on L ( G ) by λ : C c ( G ) → B ( L ( G )); λ ( f ) := Z G f ( r ) λ r dr, f ∈ C c ( G ) . The reduced group C ∗ -algebra C ∗ λ ( G ) is the C ∗ -algebra obtained by complet-ing C c ( G ) in the norm induced by λ , while the group von Neumann algebrais the double commutant vN( G ) := C ∗ λ ( G ) ′′ = { λ r : r ∈ G } ′′ .The Fourier algebra of G , denoted by A ( G ), is the set A ( G ) = { u : G → C : u ( r ) = h λ r ξ, η i , ξ, η ∈ L ( G ) } , with pointwise operations and the norm k u k = inf k ξ kk η k , where the infi-mum is taken over all possible representations u ( · ) = h λ · ξ, η i . The Fourieralgebra is a completely contractive Banach algebra, and is naturally iden-tified with the predual of vN( G ) by the pairing h λ r , u i = u ( r ) ( u ∈ A ( G )).We refer to [20] for further background.Now let A be a C ∗ -algebra, and α : G → Aut( A ) a homomorphismwhich is continuous in the point-norm topology, i.e. let ( A, G, α ) be a C ∗ -dynamical system. A pair ( φ, ρ ), where φ : A → B ( H ) is a ∗ -representationand ρ : G → B ( H ) is a unitary representation, is called a covariant pairfor ( A, G, α ) if ρ r φ ( a ) ρ ∗ r = φ ( α r ( a )) for all r ∈ G, a ∈ A . In particular, a regular covariant pair is constructed as follows: suppose that π : A → B ( H )is a faithful ∗ -representation and define(1) π α : A → B ( L ( G ) ⊗ H ); π α ( a ) ξ ( s ) := α s − ( a ) ξ ( s ) ,λ : G → B ( L ( G ) ⊗ H ); λ r ξ ( s ) := ξ ( r − s ) . A covariant pair ( φ, ρ ) as above defines a ∗ -representation of C c ( G, A ) on H by φ ⋊ ρ ( f ) := Z G φ (cid:0) f ( r ) (cid:1) ρ r dr, f ∈ C c ( G, A ) . The reduced crossed product A ⋊ α,r G is the C ∗ -algebra obtained by com-pleting C c ( G, A ) in the norm induced by π α ⋊ λ ; it can be shown that thisdefinition is independent of the original faithful ∗ -representation π . The fullcrossed product A ⋊ α G is the completion of C c ( G, A ) in the universal norm k f k := sup {k φ ⋊ ρ ( f ) k : ( φ, ρ ) is a covariant pair for ( A, G, α ) } . We refer to Williams [34, Chapter 2] for the details of this construction,and the fact that the full crossed product is universal in the following sense:every ∗ -representation of the full crossed product A ⋊ α G is of the form φ ⋊ ρ for some covariant pair ( φ, ρ ) for ( A, G, α ). In particular we will use thecovariant pair ( i A , i G ) for which i A ⋊ i G is the universal ∗ -representation of A ⋊ α G in Section 6.If M ⊂ B ( H ) is a von Neumann algebra and α is a point-weak* continuousaction of G on M then we say that ( M, G, α ) is a w ∗ -dynamical system . MENABLE AND INNER AMENABLE ACTIONS 5
In this case the definition of π α in (1) is an injective, weak*-continuoushomomorphism π α : M → L ∞ ( G ) ⊗ M satisfying(id ⊗ π α ) ◦ π α = (∆ L ∞ ( G ) ⊗ id) ◦ π α . Here ∆ L ∞ ( G ) is the natural coproduct on L ∞ ( G ), given by ∆ L ∞ ( G ) ( φ )( s, t ) := φ ( st ) ( φ ∈ L ∞ ( G ) , s, t ∈ G ). The crossed product associated to ( M, G, α ) isthe von Neumann algebra M ⋊ α G := { π α ( M ) , vN( G ) ⊗ C } ′′ ⊂ B ( L ( G ) ⊗H ).We write π ˆ α : M ⋊ α G → vN( G ) ⊗ M ⋊ α G for the natural coaction of G on M ⋊ α G (see [28, Proposition 2.4]), given by π ˆ α (cid:0) π α ( a ) (cid:1) := 1 ⊗ π α ( a ) , π ˆ α ( λ r ⊗
1) := λ r ⊗ λ r ⊗ , for all r ∈ G and all a ∈ M . This coaction induces a module action of A ( G )on M ⋊ α G , with the module structure given by u ∗ x := ( u ⊗ id) π ˆ α ( x ) , u ∈ A ( G ) , x ∈ M ⋊ α G. In fact, with this definition M ⋊ α G is a faithful A ( G )-module. This modulestructure induces an inclusion∆ : M ⋊ α G ֒ → CB ( A ( G ) , M ⋊ α G ); ∆( x )( u ) := u ∗ x, u ∈ A ( G ) , x ∈ M ⋊ α G. There is a natural A ( G )-module structure on CB ( A ( G ) , M ⋊ α G ) given by( u · Ψ)( v ) := Ψ( vu ) , u, v ∈ A ( G ) , and ∆ is an A ( G )-module map under this definition. Remark 2.1. By e.g. [19, Corollary 7.1.5] there is a natural isomorphism CB ( A ( G ) , M ⋊ α G ) ∼ = vN( G ) ⊗ ( M ⋊ α G ) , and under this isomorphism the inclusion ∆ is induced by the coaction π ˆ α .Indeed, the image of ∆ is identified by this isomorphism with the image of π ˆ α in vN( G ) ⊗ ( M ⋊ α G ).The coproduct on vN( G ) is implemented by the multiplicative unitary U ∈ vN( G ) ⊗ L ∞ ( G ), given by U ξ ( s, t ) := ξ ( ts, t ) , ξ ∈ L ( G ) ⊗ L ( G ) , s, t ∈ G, so that ∆ vN( G ) ( x ) = U ∗ (1 ⊗ x ) U ( x ∈ vN( G )). This extends to a coactionof G on B ( L ( G )) by the same formula. Thus the module action ∗ of A ( G )on vN( G ) extends to a module action on B ( L ( G )) by u ∗ x := ( u ⊗ id) U ∗ (1 ⊗ x ) U, u ∈ A ( G ) , x ∈ B ( L ( G )) . We include the following easy lemma because it is used several times inour arguments.
Lemma 2.2.
Let ( M, G, α ) and ( N, G, β ) be w*-dynamical systems and sup-pose that Φ : M → N is a G -equivariant map, i.e. Φ ◦ α r = β r ◦ Φ for all r ∈ G . Then there is an A ( G ) -module map ˜Φ : B ( L ( G )) ⊗ M → B ( L ( G )) ⊗ N which restricts to an A ( G ) -module map ˜Φ | M ⋊ α G : M ⋊ α G → N ⋊ β G . More-over, if Φ is completely bounded then so is ˜Φ , and k ˜Φ k cb ≤ k Φ k cb . ANDREW MCKEE AND REYHANEH POURSHAHAMI
Proof.
Define ˜Φ := id ⊗ Φ; it is clear that ˜Φ is an A ( G )-module map, andwell-known that k ˜Φ k cb ≤ k Φ k cb . Since Φ is G -equivariant we have, for a ∈ M, r ∈ G, ξ ∈ L ( G, H N ),˜Φ (cid:0) π α ( a ) (cid:1) ξ ( r ) = Φ (cid:0) α r − ( a ) (cid:1) ξ ( r ) = β r − (cid:0) Φ( a ) (cid:1) ξ ( r ) = π β (cid:0) Φ( a ) (cid:1) ξ ( r ) . It follows that ˜Φ( π α ( M )) ⊂ π β ( N ), so the claim about the restriction of ˜Φfollows since it acts as the identity on vN( G ) ⊗ C ⊂ M ⋊ α G . (cid:3) Approximation properties for operator modules.
The followingdefinitions are given by Crann [15, Section 7].Let X be a completely contractive Banach algebra. An operator space A will be called a left operator module over X if A is a left X -module andthe module action extends to a complete contraction X ˆ ⊗ A → A (here ˆ ⊗ denotes the operator space projective tensor product). If X (1) := X ⊕ C denotes the unitisation of X then the module action extends to a completecontraction X (1) ˆ ⊗ A → A by ( x, c ) · a := x · a + ca ( x ∈ X, a ∈ A, c ∈ C ).We will often omit the adjective “operator” in the rest of the paper, forexample writing left module over X in place of left operator module over X .Write X − mod for the category of left X -modules with completely boundedmodule maps as morphisms. Definition 2.3.
Let X be a completely contractive Banach algebra and A, B be left modules over X (respectively, left modules over X which are dualspaces). A map θ : A → B is called nuclear in X − mod (respectively weakly nuclear in X − mod ) if there are morphisms: ϕ k : A → M n k ( X ∗ (1) ) , ψ k : M n k ( X ∗ (1) ) → B, such that ψ k ◦ ϕ k converges to θ in the point-norm (respectively, the point-weak*) topology. The following definitions are natural generalisations of the usual ones foroperator spaces.
Definition 2.4.
Let X be a completely contractive Banach algebra and A be a left module over X . We say A is nuclear in X − mod if the identitymap id A : A → A is nuclear in X − mod . If A is also a dual space then wesay A is semidiscrete in X − mod if the identity map id A is weakly nuclearin X − mod . Finally, we say A is injective in X − mod if, for any two X -modules B and C , any morphism φ : B → A and any completely isometricmorphism κ : B → C , there is a morphism ˜ φ : C → A such that φ = ˜ φ ◦ κ . We will make repeated use of the obvious fact that injectivity in X − mod is preserved under taking conditional expectations which are X -modulemaps.The following results are known; we record them here for later use. Lemma 2.5. (i) For any locally compact group G the algebra B ( L ( G )) is injective in A ( G ) − mod . MENABLE AND INNER AMENABLE ACTIONS 7 (ii) Suppose that M is an injective von Neumann algebra. Then B ( L ( G )) ⊗ M , endowed with the left A ( G ) -module structure induced by that on B ( L ( G )) , is injective in A ( G ) − mod .Proof. (i) This is essentially proved in [16, Theorem 5.5]. Indeed, [31] shows,for any locally compact group G , that vN( G ) has an A ( G )-invariant state.Then the same proof as [16, Theorem 5.5] implies the claim.(ii) If M = B ( H ) this follows routinely from (i) (see e.g. [33, PropositionXV.3.2]). The general case follows. (cid:3) Amenability and injectivity.
Recall that a von Neumann algebra M is called injective if for all unital C ∗ -algebras A with a unital inclusion M ⊂ A there is a projection of norm 1 from A to M . Equivalently, if M ⊂ B ( H ) there is a projection of norm 1 from B ( H ) to M . We say thata crossed product M ⋊ α G is relatively injective in A ( G ) − mod if there isa morphism Φ : CB ( A ( G ) , M ⋊ α G ) → M ⋊ α G which is an A ( G )-modulemap and satisfies Φ ◦ ∆ = id. When M = C and α is trivial this definitionreduces to relative injectivity of vN( G ) as used by Crann–Tanko [17].Recall that a locally compact group is called amenable if there is a stateon L ∞ ( G ) which is invariant under the left translation action τ of G on L ∞ ( G ). A locally compact group is called inner amenable if there is a stateon L ∞ ( G ) which is invariant under the conjugation action of β of G on L ∞ ( G ). Crann–Tanko [17, Proposition 3.2] showed that inner amenabilityof G is equivalent to the existence of a state on vN( G ) which is invariantunder the conjugation action of G on vN( G ); it is this condition whichwe generalise in Definition 4.1. Note that there is another notion of inneramenable locally compact group, introduced by Effros [18], which is differentto the one just introduced because it excludes the inner invariant state δ e .We will mostly use amenability in the form of injectivity properties of op-erator algebras. Here we summarise what is known about amenable groupsin the language of injectivity. A discrete group is amenable if and only if itsgroup von Neumann algebra is injective (in C − mod ).For locally compact groups inner amenability of G is equivalent to relativeinjectivity of vN( G ) in A ( G ) − mod .Crann–Tanko [17, Theorem 3.4] observe that Lau–Paterson’s result [25,Corollary 3.2] can be phrased in this way, generalising the above results. Theorem 2.6.
Let G be a locally compact group. The following are equiv-alent:(i) G is amenable;(ii) vN( G ) is injective in A ( G ) − mod ;(iii) vN( G ) is relatively injective in A ( G ) − mod and injective in C − mod . Our aim is to generalise this to crossed products. Crann–Tanko [17, The-orem 3.4] show how relative module injectivity is precisely what is neededfor averaging arguments to work. This paper began with the idea to lookfor similar averaging techniques for crossed products.
ANDREW MCKEE AND REYHANEH POURSHAHAMI Amenable Actions on von Neumann Algebras
Recall Anantharaman-Delaroche’s definition of an amenable action [1].
Definition 3.1.
Let α be an action of a locally compact group G on avon Neumann algebra M . We say that α is amenable if there is a projectionof norm P : L ∞ ( G ) ⊗ M → M such that P ◦ ( τ r ⊗ α r ) = α r ◦ P, r ∈ G .That is, there is a G -equivariant conditional expectation from L ∞ ( G ) ⊗ M to M . Here τ is the natural action of G on L ∞ ( G ) by left translation. When M = C and the action is trivial the projection P is a left-invariantmean, so G is amenable.We now aim to improve [1, Proposition 3.11] by accounting for the A ( G )-module structure. Recall that the coproduct on vN( G ) is unitarily imple-mented, so extends to B ( L ( G )) where it induces a module action of A ( G ). Proposition 3.2.
Let α be an action of G on M . The following are equiv-alent:(i) α is an amenable action;(ii) there is a norm 1 projection B ( L ( G )) ⊗ M → M ⋊ α G which is an A ( G ) -module map.Proof. (i) = ⇒ (ii) Let P : L ∞ ( G ) ⊗ M → M be the norm 1 equivariantprojection from the definition of an amenable action. By Lemma 2.2 P extends to an A ( G )-module map ˜ P : B ( L ( G )) ⊗ L ∞ ( G ) ⊗ M → B ( L ( G )) ⊗ M , which restricts to a norm one A ( G )-module projection P α : (cid:0) L ∞ ( G ) ⊗ M (cid:1) ⋊ τ ⊗ α G → M ⋊ α G. By [1, Lemme 3.10] (cid:0) L ∞ ( G ) ⊗ M (cid:1) ⋊ τ ⊗ α G is isomorphic to B ( L ( G )) ⊗ M ,so P α is identified with a norm 1 projection B ( L ( G )) ⊗ M → M ⋊ α G .Moreover, for each r ∈ G the isomorphism in [1, Lemme 3.10] identifies λ r ⊗ id L ∞ ( G ) ⊗ M ∈ (cid:0) L ∞ ( G ) ⊗ M (cid:1) ⋊ τ ⊗ α G with λ r ⊗ id M ∈ B ( L ( G )) ⊗ M ,so it follows that the natural A ( G )-module structure on the domain of P α istransformed under this isomorphism to the natural A ( G )-module structureon B ( L ( G )) ⊗ M . It follows that the map which corresponds to P α underthis identification is an A ( G )-module map.(ii) = ⇒ (i) Let P α be the projection in (ii) and take x ∈ L ∞ ( G ) ⊗ M and u ∈ A ( G ); we have u ∗ P α ( x ) = P α ( u ∗ x ) = u ( e ) P α ( x ) . As this holds for all u ∈ A ( G ) we conclude P α ( x ) ∈ π α ( M ) ∼ = M . To seethat the restriction of P α to L ∞ ( G ) ⊗ M is equivariant identify P α with thecorresponding map( L ∞ ( G ) ⊗ M ) ⋊ τ ⊗ α G → ( C ⊗ M ) ⋊ id ⊗ α G MENABLE AND INNER AMENABLE ACTIONS 9 using [1, Lemme 3.10]. Then, for x ∈ L ∞ ( G ) ⊗ M , P α (cid:0) π τ ⊗ α (cid:0) ( τ r ⊗ α r ) x (cid:1)(cid:1) = P α (cid:0) λ r π τ ⊗ α ( x ) λ ∗ r (cid:1) = λ r P α (cid:0) π τ ⊗ α ( x ) (cid:1) λ ∗ r by the bimodule property of P α . Since P α ( π τ ⊗ α ( x )) ∈ π id ⊗ α ( M ) this showsthat P α is equivariant. (cid:3) The A ( G )-module structure allows us to recover some results for locallycompact groups which are only proved for discrete groups in [1]. Remark 3.3.
Anantharaman-Delaroche [1, Proposition 3.6] has shown thatamenable groups always have amenable actions. There are also many ex-amples of non-amenable groups which have amenable actions; this followsfrom e.g. [8, Theorem 5.8], since there exist non-amenable exact groups.The following result is given by Anantharaman-Delaroche [1, Proposition3.6]. We will use the implication (ii) = ⇒ (i) later in Theorem 4.6, so wegive a proof of this implication using our techniques. For a von Neumannalgebra M we write Z ( M ) for the centre of M . Proposition 3.4.
Let α be an action of G on M . The following are equiv-alent:(i) G is amenable;(ii) α is amenable and Z ( M ) has a G -invariant state.Proof. (i) = ⇒ (ii) See Remark 3.3 and [1, Proposition 3.6].(ii) = ⇒ (i) Let P α : B ( L ( G )) ⊗ M → M ⋊ α G be a norm one projection.For each x ∈ B ( L ( G )) we claim P α ( x ⊗ M ) ∈ Z ( M ) ⋊ α G . We need onlyverify the claim for x ∈ L ∞ ( G ); in this case the bimodule property of P α means that for y ∈ π α ( M ) yP α ( x ⊗ M ) = P α (cid:0) y ( x ⊗ M (cid:1) = P α (cid:0) ( x ⊗ M ) y (cid:1) = P α ( x ⊗ M ) y, so P α ( x ) ∈ M ⋊ α G ∩ π α ( M ) ′ ⊂ Z ( M ) ⋊ α G . Now let φ : Z ( M ) → C be a G -invariant state, and ˜ φ : Z ( M ) ⋊ α G → vN( G ) the corresponding map given byLemma 2.2. The composition ˜ φ ◦ P α is a norm one A ( G )-module projectionfrom B ( L ( G )) ⊗ C to vN( G ), so G is amenable by Theorem 2.6. (cid:3) Now we show how accounting for the A ( G )-module structure helps thestudy of injectivity of crossed products; this result generalises [1, Proposi-tion 3.12, Corollaire 4.2]. Part of this result was also obtained recently byCrann [15, Theorem 8.2] and Bearden–Crann [7, Theorem 5.2]. Theorem 3.5.
Let α be an action of G on M . The following are equivalent:(i) α is amenable and M is injective;(ii) M ⋊ α G is injective in A ( G ) − mod .Proof. (i) = ⇒ (ii) Since M is injective we have B ( L ( G )) ⊗ M is injectivein A ( G ) − mod by Lemma 2.5. By Proposition 3.2 there is a norm 1 A ( G )-module projection P α : B ( L ( G )) ⊗ M → M ⋊ α G . Hence (ii) holds. (ii) = ⇒ (i) It is routine to obtain a norm 1 A ( G )-module projection E : B ( L ( G )) ⊗ B ( H ) → M ⋊ α G . Clearly E restricts to a norm 1 projection B ( L ( G )) ⊗ M → M ⋊ α G giving amenability of α . Take x ∈ L ∞ ( G ) ⊗ B ( H )and u ∈ A ( G ), so that u ∗ x = u ( e ) x . Since E is an A ( G )-module map u ∗ E ( x ) = E ( u ∗ x ) = u ( e ) E ( x ); as this holds for all u ∈ A ( G ) we concludethat E ( x ) ∈ π α ( M ). Thus E restricts to a norm one projection L ∞ ( G ) ⊗B ( H ) → π α ( M ) ∼ = M , so M is injective. (cid:3) Inner Amenable Actions on von Neumann Algebras
Recall that a locally compact group G is called inner amenable if there isan inner-invariant state on L ∞ ( G ), and that Crann–Tanko [17, Proposition3.2] showed that inner amenability of G is equivalent to the existence of aninner-invariant state on vN( G ), i.e. a state φ on vN( G ) satisfying φ ( β r ( x )) = φ ( x ) ( r ∈ G, x ∈ vN( G )), where β r ( x ) := λ r xλ ∗ r , x ∈ vN( G ) , r ∈ G. It is this latter condition which we generalise to define inner amenable ac-tions.
Definition 4.1.
Let ( M, G, α ) be a w ∗ -dynamical system. We say α is inneramenable if there is a projection of norm 1 Q : vN( G ) ⊗ M → M such that Q ◦ ( β r ⊗ α r ) = α r ◦ Q, r ∈ G .That is, there is a G -equivariant conditional expectation from vN( G ) ⊗ M to M . If M = C and α is trivial then the equivariant projection Q in Defini-tion 4.1 is an inner-invariant state on vN( G ), so the above definition reducesto inner amenability of G by [17, Proposition 3.2]. Remark 4.2.
To see that inner amenable groups have inner amenable ac-tions suppose that (
M, G, α ) is a w ∗ -dynamical system, and that G is inneramenable. Let φ : vN( G ) → C be a state which is invariant under the action β and define Q : vN( G ) ⊗ M → M ; Q ( x ) := ( φ ⊗ id)( x ) , x ∈ vN( G ) ⊗ M. It is easily seen that Q satisfies Definition 4.1, hence α is inner amenable.We would like to know if there is a way to formulate our definition ofinner amenable actions in terms of an equivariant projection on L ∞ ( G ) ⊗ M ,generalising the original definition of inner amenable groups as the existenceof an inner-invariant mean on L ∞ ( G ).In order to use inner amenable actions for averaging arguments we nowshow that Definition 4.1 is equivalent to a crossed product version of rela-tive injectivity, which is exactly the property required for the arguments inTheorem 4.6 and Section 5. MENABLE AND INNER AMENABLE ACTIONS 11
Proposition 4.3.
Let ( M, G, α ) be a w ∗ -dynamical system. The followingare equivalent:(i) α is inner amenable;(ii) M ⋊ α G is relatively injective in A ( G ) − mod (i.e. there is a norm 1 A ( G ) -module projection from vN( G ) ⊗ ( M ⋊ α G ) onto π ˆ α ( M ⋊ α G ) ).Proof. (i) = ⇒ (ii) Let U ∈ vN( G ) ⊗ L ∞ ( G ) be the unitary on L ( G ) ⊗ L ( G )given by U ξ ( s, t ) := ξ ( ts, t ) and let V := σU , where σ is the flip operator.Routine calculations show that for x ∈ vN( G ) V ( x ⊗ id vN( G ) ) V ∗ = π β ( x ) and V ∗ ( x ⊗ id vN( G ) ) V = ∆ vN( G ) ( x ) . Thus we identify the A ( G )-modules vN( G ) ⋊ β G with vN( G ) ⊗ vN( G ) byconjugating with σV ∗ = σU ∗ σ (the A ( G )-module structure on vN( G ) ⊗ vN( G ) comes from applying the coproduct to the left component). Underthis identification vN( G ) ⊗ C ⊂ vN( G ) ⋊ β G is identified with ∆ vN( G ) (vN( G )).Similarly, if M acts on the Hilbert space H , conjugating by the unitary σV ∗ ⊗ id H we identify (vN( G ) ⊗ M ) ⋊ β ⊗ α G with ( σ ⊗ id)(vN( G ) ⊗ ( M ⋊ α G ))( σ ⊗ id). Under this identification ( C ⊗ M ) ⋊ β ⊗ α G is identified with π ˆ α ( M ⋊ α G ).Now let Q : vN( G ) ⊗ M → M be the norm one equivariant projectionfrom Definition 4.1, so the A ( G )-module map ˜ Q given by Lemma 2.2 givesa norm one A ( G )-module projection( V ⊗ id H ) ∗ ˜ Q ( V ⊗ id H ) : vN( G ) ⊗ ( M ⋊ α G ) → π ˆ α ( M ⋊ α G ) , as required.(ii) = ⇒ (i) Let Q α denote the projection in (ii), and as in the first part ofthe proof above conjugate with the unitary ( V ⊗ id H ) to identify Q α witha norm one A ( G )-module projection from (vN( G ) ⊗ M ) ⋊ β ⊗ α G onto ( C ⊗ M ) ⋊ id ⊗ α G ∼ = M ⋊ α G . The rest of the proof proceeds as in Proposition 3.2. (cid:3) Since a left-invariant mean on L ∞ ( G ) is automatically two-sided invariant,all amenable groups are automatically inner amenable. We can generalisethis fact to actions on injective von Neumann algebras. Proposition 4.4.
Suppose that ( M, G, α ) is a w ∗ -dynamical system, with M injective and α amenable. Then α is inner amenable.Proof. Since α is amenable and M is injective it follows from Theorem 3.5that M ⋊ α G is injective in A ( G ) − mod . It follows easily that M ⋊ α G isrelatively injective in A ( G ) − mod , so α is inner amenable. (cid:3) Examples 4.5. (i) The action τ of G on L ∞ ( G ) is always amenable [2,Section 1.4], hence always inner amenable. So B ( L ( G )) is relativelyinjective in A ( G ) − mod .(ii) Let G denote a second-countable, connected, non-amenable locallycompact group, for example G = SL(2 , R ). Such groups cannot be inner amenable, but they are exact [23, Theorem 6.8]. Brodzki–Cave–Li [8, Theorem 5.8] have shown that such G admit an amenable action,say α , on a compact space X , so it follows from Proposition 4.4 that α is inner amenable.By analogy with the class of exact groups, it may be interesting to studythe class of locally compact groups which admit an inner amenable actionon L ∞ ( X ) for some compact space X . By Remark 4.2 this class containsall inner amenable groups, and the same argument as in Example 4.5 part(ii) shows this class contains all second-countable exact groups.To close this section we generalise a result of Lau–Paterson [25, Theorem3.1] to actions on noncommutative von Neumann algebras. Theorem 4.6.
Let ( M, G, α ) be a w ∗ -dynamical system. The following areequivalent:(i) G is amenable and M is injective;(ii) M ⋊ α G is injective, α is inner amenable and Z ( M ) has a G -invariantstate.Proof. (i) = ⇒ (ii) Follows from Proposition 3.4, Theorem 3.5 and Re-mark 4.2.(ii) = ⇒ (i) Since α is inner amenable, by Proposition 4.3, we can upgradeinjectivity of M ⋊ α G to injectivity of M ⋊ α G in A ( G ) − mod as in [13,Proposition 2.3]. It follows that M is injective as in the proof of Theorem 3.5.To show G is amenable we follow a similar idea to Proposition 3.4. Let E : B ( L ( G )) ⊗ B ( H M ) → M ⋊ α G be a norm one A ( G )-module projection,and observe that for x ∈ B ( L ( G )) we have E ( x ⊗ M ) ∈ Z ( M ) ⋊ α G , asin the proof of Proposition 3.4. Let φ be a G -invariant state on Z ( M ),and ˜ φ : Z ( M ) ⋊ α G → vN( G ) the associated A ( G )-module map given byLemma 2.2. The composition ˜ φ ◦ E is a norm one A ( G )-module projectionfrom B ( L ( G )) ⊗ C to vN( G ), so G is amenable by Theorem 2.6. (cid:3) A Sample Averaging Argument on Crossed Products
Our definition of inner amenable actions is designed to enable averagingarguments. One example of such an argument has already occurred in theproof that (ii) implies (i) in Theorem 4.6, where inner amenability of theaction allows us to obtain injectivity of the crossed product in A ( G ) − mod from the assumption that the crossed product is injective in C − mod . Inthe setting of C ∗ -algebra crossed products discreteness of the acting groupis used in the same way in [27, Proposition 3.4]. In this section we brieflygive a further example of this technique. Lemma 5.1.
Let ( M, G, α ) be a w ∗ -dynamical system. Suppose that α isinner amenable, with associated norm one A ( G ) -module projection Q α :vN( G ) ⊗ M ⋊ α G → π ˆ α ( M ⋊ α G ) , and let Φ : M ⋊ α G → M ⋊ α G be acompletely bounded map. Then S Φ : M ⋊ α G → M ⋊ α G ; S Φ := π − α ◦ Q α ◦ (id ⊗ Φ) ◦ π ˆ α MENABLE AND INNER AMENABLE ACTIONS 13 is a completely bounded A ( G ) -module map, with k S Φ k cb ≤ k Φ k cb .Proof. Let Q α : vN( G ) ⊗ M ⋊ α G → π ˆ α ( M ⋊ α G ) be the A ( G )-module norm1 projection given by inner amenability of α in Proposition 4.3. It is easilychecked that π ˆ α ( u ∗ x ) = ( u ⊗ id) ∗ π ˆ α ( x ) for all x ∈ M ⋊ α G . Then, for any u ∈ A ( G ) and x ∈ M ⋊ α G we have S Φ ( u ∗ x ) = π − α ◦ Q α ◦ (id ⊗ Φ) (cid:0) ( u ⊗ id) ∗ π ˆ α ( x ) (cid:1) = π − α ◦ Q α ◦ (cid:16) ( u ⊗ id) ∗ (cid:0) id ⊗ Φ) ◦ π ˆ α ( x ) (cid:1)(cid:17) = u ∗ (cid:0) π − α ◦ Q α ◦ (id ⊗ Φ) ◦ π ˆ α ( x ) (cid:1) = u ∗ S Φ ( x ) , Hence S Φ is an A ( G )-module map. The norm inequality is obvious. (cid:3) A von Neumann algebra N is said to have the weak* completely boundedapproximation property (w*CBAP) if there exists a net of ultraweakly-continuous, finite-rank, completely bounded maps (Φ i : N → N ) i suchthat Φ i → id N in the point-ultraweak topology and a constant C for which k Φ i k cb ≤ C for each i . The Haagerup constant Λ cb ( N ) is the infimum ofthose C for which such a net (Φ i ) i exists, and Λ cb ( N ) = ∞ if N does nothave the w*CBAP. Proposition 5.2.
Let ( M, G, α ) be a w ∗ -dynamical system. Consider theconditions:(i) M has the weak* completely bounded approximation property;(ii) M ⋊ α G has the weak* completely bounded approximation property.If α is amenable then (i) implies (ii). If α is inner amenable then (ii) implies(i).Proof. (i) = ⇒ (ii) This was shown by Anantharaman-Delaroche [4, 4.10].(ii) = ⇒ (i) Let Q α : vN( G ) ⊗ M ⋊ α G → π ˆ α ( M ⋊ α G ) be the norm 1 A ( G )-module projection arising from inner amenability of α in Proposition 4.3.Let (Φ i : M ⋊ α G → M ⋊ α G ) i be a net of maps which implement thew*CBAP of M ⋊ α G , and define S Φ i : M ⋊ α G → M ⋊ α G ; S Φ i := π − α ◦ Q α ◦ (Φ i ⊗ id) ◦ π ˆ α as in Lemma 5.1. The maps ( S Φ i ) i satisfy k S Φ i k cb ≤ k Φ i k cb , also imple-ment the w*CBAP of M ⋊ α G , and they are A ( G )-module maps. Since u ∗ ( S Φ i ( x )) = S Φ i ( u ∗ x ) for all x ∈ M ⋊ α G and all u ∈ A ( G ) we have S Φ i ( π α ( M )) ⊂ π α ( M ). It follows that π α ( M ) ∼ = M has the w*CBAP andΛ cb ( M ) ≤ Λ cb ( M ⋊ α G ). (cid:3) The above proof illustrates one of our motivations for defining inneramenable actions: inner amenability of the action allows us to “average”maps which implement an approximation property of A ⋊ α,r G into A ( G )-module maps which implement the approximation property. The resulting A ( G )-module maps may be viewed as Herz–Schur multipliers, as in [26]; thisis the perspective adopted in [27], where averaging arguments are explicitlyused to produce Herz–Schur multipliers. Amenable actions on C ∗ -algebras In this section (
A, G, α ) will be a C ∗ -dynamical system. Anantharaman-Delaroche [3, D´efinition 4.1] gave a definition of amenability for actions ofdiscrete groups on C ∗ -algebras: if ( A, G, α ) is a C ∗ -dynamical system with G discrete then α is called amenable if the corresponding double dual action α ∗∗ of G on A ∗∗ is amenable in the sense of Definition 3.1. In [3, Section 4]it is shown that this definition has several nice properties.The work of Anantharaman-Delaroche leaves open the question of definingamenable actions of locally compact groups on C ∗ -algebras. It is proposedin [5, Section 9.2] that one might define an action of G on A to be amenable if A ⋊ α G = A ⋊ α,r G , that is, if the canonical quotient map A ⋊ α G → A ⋊ α,r G is an isomorphism, and several questions about the behaviourof this proposed definition are raised. Suzuki [32] constructed examplesshowing that A ⋊ α G = A ⋊ α,r G does not satisfy the functoriality propertiesin [5, Section 9.2]. We interpret the questions asked by Anantharaman-Delaroche as requirements for a sensible definition of an amenable action. Problem 6.1.
Let ( A, G, α ) be a C ∗ -dynamical system, with G locally com-pact. Give a definition of amenability of α which satisfies the followingproperties:(1) if A is nuclear and α is amenable then the crossed product A ⋊ α G and/or the reduced crossed product A ⋊ α,r G is also nuclear;(2) if α is amenable and H is a closed subgroup of G then the restrictionof α is an amenable action of H on A ;(3) if ( B, G, β ) is a C ∗ -dynamical system such that there is an equivari-ant ∗ -homomorphism Φ : A → M( B ) with Φ( A ) B dense in B and Φ( Z (M( A ))) ⊂ Z (M( B )) then amenability of α implies amenabilityof β ;(4) if A is a simple C ∗ -algebra then amenability of α implies amenabilityof G ;(5) if α is amenable then the canonical quotient map A ⋊ α G → A ⋊ α,r G is an isomorphism. We have added condition (5) to the list in [5, Section 9.2] since it willnot be part of the definition of an amenable action below. Versions of thisproblem have been considered by a number of authors; we note the recentwork of Bearden–Crann [6] and Buss–Echterhoff–Willett [10], which targets(5) above and along the way addresses a number of the other properties. Ourmain contribution is to the nuclearity problem (1): we use A ( G )-modules torelate amenable actions and nuclearity of crossed products. We also surveyknown results, showing that the notion of amenable action we use satisfiesseveral of the points in Problem 6.1.We follow Anantharaman-Delaroche [3, D´efinition 4.1] in defining an ac-tion of G on a C ∗ -algebra A to be amenable if a suitable double dual actionis amenable in the sense of Definition 3.1. The double dual action α ∗∗ on MENABLE AND INNER AMENABLE ACTIONS 15 A ∗∗ can fail to be weak*-continuous [22], but fortunately there is a suitablereplacement, which was introduced by Ikunishi [22]. Definition 6.2.
Let ( A, G, α ) be a C ∗ -dynamical system. Denote by ( i A , i G ) the covariant representation underlying the universal representation i A ⋊ i G : A ⋊ α G → B ( H u ) . We define A ∗∗ α := i A ( A ) ′′ ⊂ B ( H u ) . This definition is not the original one given by Ikunishi; we refer to Buss–Echterhoff–Willett [10, Section 2], where it is shown that Definition 6.2 isequivalent to Ikunishi’s definition, and several useful properties of this objectare shown. The von Neumann algebra A ∗∗ α carries an action of G given byAd i G ; we will abuse notation and write α ∗∗ for this action, which givesrise to a w ∗ -dynamical system ( A ∗∗ α , G, α ∗∗ ). To define an amenable actionwe follow [10, Definition 3.1], where such actions are called von Neumannamenable . Definition 6.3.
Let ( A, G, α ) be a C ∗ -dynamical system. Say that α is amenable if the corresponding universal action α ∗∗ of G on A ∗∗ α is amenablein the sense of Definition 3.1. For actions of discrete groups A ∗∗ α = A ∗∗ [10, Section 2.2], so this defini-tion extends the one given by Anantharaman-Delaroche [3, D´efinition 4.1]for discrete groups. The remainder of this section is concerned with investi-gating if Definition 6.3 has the properties in Problem 6.1.First we consider nuclearity of crossed products, and aim to generalise[3, Th´eor`eme 4.5]. The assumption that the action is inner amenable isdiscussed in Remark 6.5(iii) below. Theorem 6.4.
Let ( A, G, α ) be a C ∗ -dynamical system, and suppose that α ∗∗ is an inner amenable action of G on A ∗∗ α . The following are equivalent:(i) A ⋊ α G is nuclear;(ii) A ⋊ α,r G is nuclear;(iii) A ∗∗ α ⋊ α ∗∗ G is injective in A ( G ) − mod ;(iv) α is amenable and A ∗∗ α is injective.Proof. (i) = ⇒ (ii) Nuclearity passes to quotients.(ii) = ⇒ (iii) By hypothesis ( A ⋊ α,r G ) ∗∗ is injective (in C − mod ). Since i A is a faithful representation of A we may form the covariant pair (( i A ) α , λ )as in equation (1). It is shown in [10, Remark 2.6] that(2) A ∗∗ α ⋊ α ∗∗ G = (cid:0) ( i A ) α ⋊ λ (cid:1) ∗∗ (cid:0) ( A ⋊ α,r G ) ∗∗ (cid:1) , so it follows that A ∗∗ α ⋊ α ∗∗ G is injective (in C − mod ). Since α is assumedto be inner amenable, A ∗∗ α ⋊ α ∗∗ G is relatively injective in A ( G ) − mod by Proposition 4.3, so by a standard argument (see [13, Proposition 2.3]) A ∗∗ α ⋊ α ∗∗ G is injective in A ( G ) − mod .(iii) = ⇒ (iv) This is shown in Theorem 3.5.(iv) = ⇒ (i) Recent results allow us to adapt Anantharaman-Delaroche’sproof [3, Th´eor`eme 4.5] to the locally compact case, as we now explain. We will show that for a (non-degenerate) covariant representation ( φ, ρ ) of(
A, G, α ) on the Hilbert space H the von Neumann algebra ( φ ⋊ ρ )( A ⋊ α G ) ′′ isinjective. In fact we will show the equivalent statement that the commutant( φ ⋊ ρ )( A ⋊ α G ) ′ is injective. Observe (as in [3, Th´eor`eme 4.5]) that thelatter is the space φ ( A ) ′ G := { x ∈ φ ( A ) ′ : ρ r xρ ∗ r = x for all r ∈ G } . Applying [10, Corollary 2.3] to the map φ : A → φ ( A ) we obtain a sur-jective, equivariant extension φ ∗∗ : A ∗∗ α → φ ( A ) ′′ . Since A ∗∗ α is injective sois φ ( A ) ′′ , therefore also φ ( A ) ′ . Now, since α is amenable in our sense itis also amenable in the sense of [10, Definition 3.4], by [6, Theorem 3.6](see also [10, Section 8]); therefore, by [10, Proposition 5.9] α is commutantamenable in the sense of [10, Definition 5.7]. Therefore, by [6, Theorem 3.6],the action Ad ρ of G on φ ( A ) ′ given byAd ρ r ( x ) := ρ r xρ ∗ r , r ∈ G, x ∈ φ ( A ) ′ , is also amenable. Let R : L ∞ ( G ) ⊗ φ ( A ) ′ → φ ( A ) ′ be the correspondingnorm 1 equivariant projection and define R G : φ ( A ) ′ → φ ( A ) ′ G ; R G ( x ) := R (ˆ x ) , x ∈ φ ( A ) ′ , where ˆ x ∈ L ∞ ( G ) ⊗ φ ( A ) ′ is defined by ˆ x ( r ) := Ad ρ r ( x ). Since φ ( A ) ′ is injective and R G is a norm 1 projection we have shown that φ ( A ) ′ G isinjective, as required. (cid:3) Remarks 6.5. (i) The implication (iii) = ⇒ (iv) fails if we do not accountfor the A ( G )-module structure: there exist locally compact groups, forexample SL(2 , C ), for which the group von Neumann algebra is injec-tive but the group is not amenable [12] (see also [9, Remark 2.6.10]).Anantharaman-Delaroche works only with discrete groups, where thisdifficulty does not arise. See also Theorem 2.6.(ii) Note that if A is nuclear then A ∗∗ α is injective, so condition (iv) aboveholds if α is amenable and A is nuclear. If G is discrete then A ∗∗ α = A ∗∗ ,so (iv) is equivalent to (iv)’ α is amenable and A is nuclear . In generalwe do not see any reason for (iv) and (iv)’ to be equivalent.(iii) Our original goal was to prove a full generalisation of [3, Th´eor`eme 4.5],without the assumption of inner amenable actions but using nuclearityof A ⋊ α G as an A ( G )-module in (i) (and nuclearity of A ⋊ α,r G asan A ( G )-module in (ii)). Unfortunately it seems that such a resultwould require module versions of the deep results linking injectivity,semidiscreteness and nuclearity, which do not appear to be known ingeneral.The following shows that the class of inner amenable locally compactgroups is an answer to [10, Question 8.3]. Corollary 6.6.
Let ( A, G, α ) be a C ∗ -dynamical system. Suppose that A ⋊ α,r G is nuclear and G is inner amenable. Then α is amenable. MENABLE AND INNER AMENABLE ACTIONS 17
Proof.
By hypothesis ( A ⋊ α,r G ) ∗∗ is injective, so it follows from [10, Remark2.6] that A ∗∗ α ⋊ α ∗∗ G is injective. Since G is inner amenable it follows fromRemark 4.2 that α ∗∗ is inner amenable, that is, A ∗∗ α ⋊ α ∗∗ G is relativelyinjective in A ( G ) − mod . Now [13, Proposition 2.3] shows that A ∗∗ α ⋊ α ∗∗ G is injective in A ( G ) − mod . By Theorem 3.5 α is amenable. (cid:3) For question (2) on restriction to closed subgroups we refer to Ozawa–Suzuki [29, Corollary 3.4], where the following result was recently shown.
Proposition 6.7.
Let ( A, G, α ) be a C ∗ -dynamical system with α amenable.If H is a closed subgroup of G then the restriction α | H is also amenable. A solution to question (3) follows from work of Buss–Echterhoff–Willettand Bearden–Crann.
Proposition 6.8.
Let ( A, G, α ) and ( B, G, β ) be C ∗ -dynamical systems and Φ : A → M( B ) a G -equivariant ∗ -homomorphism such that Φ( A ) B is densein B and Φ( Z (M( A ))) ⊂ Z (M( B )) . If α is amenable then so is β .Proof. By [6, Theorem 3.6] α is amenable in our sense if and only if it isamenable in the sense of [10, Definition 3.4] (see also [10, Section 8]). Thusthe claim is equivalent to [10, Lemma 3.17]. (cid:3) Examples given by Suzuki [32] show that our definition of amenable ac-tion does not satisfy the requirements of problem (4) — see [11, Section3] and [10, Corollary 3.13]. However, we are able to solve problem (4) inthe special case where A is the compact operators on some Hilbert space(see [10, Observation 5.24]), as this is the only case for which we know that A ∗∗ α is a factor. It would be interesting to have other examples where A ∗∗ α isa factor. Proposition 6.9.
Let ( K ( H ) , G, α ) be a C ∗ -dynamical system with α amenableand K ( H ) the C ∗ -algebra of compact operators on the Hilbert space H . Then G is amenable.Proof. Ikunishi [22, Example 1] shows that K ( H ) ∗∗ α = B ( H ), so the claimfollows from Proposition 3.4. (cid:3) A solution to question (5) can also be deduced from the work of Bearden–Crann and Buss–Echterhoff–Willett.
Proposition 6.10.
Let ( A, G, α ) be a C ∗ -dynamical system with α amenable.Then the canonical quotient map A ⋊ α G → A ⋊ α,r G is an isomorphism.Proof. It follows from the result of Bearden–Crann [6, Theorem 3.6] that if α is amenable in our sense then it is amenable in the sense of [10, Definition3.4]. The conclusion then follows from [10, Theorem 5.9]. (cid:3) Remark 6.11.
Buss–Echterhoff–Willett [10, Proposition 5.28, Example5.29] have shown that the converse to Proposition 6.10 does not hold forgeneral locally compact groups. We refer to [10, Section 5] for an investiga-tion of other conditions which are related to this weak containment property . Acknowledgements.
The second author thanks Alireza Medghalchi for hiscontinuous encouragement and the Department of Mathematics of KharazmiUniversity for support. Parts of this work were completed when the secondauthor was visiting the first author at Chalmers University of Technologyand the University of Gothenburg; she would like to thank Departmentof Mathematical Sciences at Chalmers University of Technology and theUniversity of Gothenburg for their warm hospitality. Further progress wasmade when the authors attended the 7th Workshop on Operator Algebrasand their Applications in Tehran, in January 2020; we are very gratefulto the organisers for their hospitality. We thank Lyudmila Turowska andMassoud Amini for their valuable comments on early versions of our resultsand Fatemeh Khosravi for helpful discussions during this work. Finally,thanks to Siegfried Echterhoff for pointing out a mistake in a previous versionof Section 6.
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