Amenability and approximation properties for partial actions and Fell bundles
aa r X i v : . [ m a t h . OA ] N ov AMENABILITY AND APPROXIMATION PROPERTIES FORPARTIAL ACTIONS AND FELL BUNDLES
FERNANDO ABADIE, ALCIDES BUSS, AND DAMIÁN FERRARO
Abstract.
Building on previous papers by Anantharaman-Delaroche we in-troduce and study the notion of AD-amenability for partial actions and Fellbundles over discrete groups. If the Fell bundle is AD-amenable, the full and re-duced crossed products coincide. We prove that the cross-sectional C*-algebraof the Fell bundle is nuclear if and only if the underlying unit fibre is nuclearand the Fell bundle is AD-amenable. If a partial action is globalisable, then itis AD-amenable if and only if its globalisation is AD-amenable. Moreover, weprove that AD-amenabity is invariant under (weak) equivalence of Fell bundlesand show that AD-amenabity is equivalent to a weak form of the approxima-tion property introduced by Exel. For Fell bundles whose unit fibre is (Moritaequivalent to) a commutative C*-algebra we prove that AD-amenability isequivalent to the approximation property.
Contents
1. Introduction 22. Partial actions on von Neumann algebras 53. Amenability of partial actions 94. Morita equivalence of partial actions 125. AD-amenability of Fell bundles 135.1. W*-enveloping Fell bundles 145.2. Central partial actions of W*-Fell bundles 155.3. Cross-sectional W*-algebras of W*-Fell bundles 165.4. The W*-algebra of kernels 195.5. The dual coaction: another picture for the W*-algebra of kernels 226. Exel’s approximation property and AD-amenability 256.1. Invariance under equivalences 337. Cross-sectional C*-algebras and the WAP 368. Fell bundles with commutative unit fibre 41Appendix A. W*-bimodules and their representations 49Appendix B. Induction of representations and tensor products 53Appendix C. Biduals of Hilbert bimodules 54References 56
Date : November 11, 2019.2010
Mathematics Subject Classification. Introduction
In her seminal paper [10] Anantharaman-Delaroche introduced a notion of amenabil-ity (that we here call AD-amenability) for actions of discrete groups on C*-algebras.Her definition is based on previous papers [7, 8] where she studies amenability forgroup actions on W*-algebras (i.e von Neumann algebras). More precisely, an ac-tion γ of a discrete group G on a W*-algebra N is said to be amenable in the senseof Anantharaman-Delaroche (or just W*AD-amenable for short) if there exists a G -equivariant conditional expectation P : ℓ ∞ ( G, N ) ։ N with respect to the diagonal G -action ˜ γ on ℓ ∞ ( G, N ) = ℓ ∞ ( G ) ¯ ⊗ N where G acts on ℓ ∞ ( G ) by (left) translations;the map P should be interpreted as a G -invariant mean for the action. An action α of G on a C ∗ -algebra A is then said to be AD-amenable if the induced action α ′′ on the enveloping (bidual) W*-algebra A ′′ is W*AD-amenable.One of the main results in [10] (namely Theorem 3.3) shows that an invariantmean P : ℓ ∞ ( G, N ) → N can be always approximated with respect to the pointwiseweak* (i.e. ultraweak) topology by using certain nets of functions from G to Z ( N ).One precise form of such approximation that will be specially important to us inthis paper is given by a net of functions of finite support { a i : G → Z ( N ) } i ∈ I whichis bounded when viewed as a net of the Hilbert N -module ℓ ( G, N ) and satisfies(1.1) h a i | ˜ γ g ( a i ) i = X h ∈ G a i ( h ) ∗ γ g ( a i ( g − h )) → g ∈ G . This condition indeed charac-terises amenability and shows, among other things, that γ is W*AD-amenable if andonly if so is its restriction to the centre Z ( N ). Moreover, W*AD-amenability be-haves well with respect to injectivity of W*-algebras and nuclearity of C*-algebras:if N is injective then γ is W*AD-amenable if and only if the W*-crossed product N ¯ ⋊ γ G is injective. And similarly, if A is a nuclear C*-algebra, then the (reduced)C*-crossed product A ⋊ α, r G is nuclear if and only if α is AD-amenable.Notice that the AD-amenability of an action on a C*-algebra A requires (andis equivalent to) the existence of a net as above with values in Z ( A ′′ ). While thisis a huge commutative algebra in general, finding explicitly such an approximateinvariant mean might be a very difficult task – if not impossible. Hoping for moreconcrete realisations of such approximate means one might wonder whether it isnot always possible to find a net with values in Z ( A ) or at least in Z M ( A ) (thecentral multiplier algebra). This is indeed possible for commutative A (by [10, The-orem 4.9]) and hence more generally for A admitting (nondegenerate) G -equivariant ∗ -homomorphism C ( X ) → Z M ( A ) for some amenable G -space X . Unfortunatelythis is not possible in general: striking recent results by Suzuki in [33] show thatevery exact group admits an AD-amenable action on a unital simple nuclear C*-algebra A (and one can even choose such algebra for which the crossed product isin the same class). For such an A we have Z ( A ) = Z M ( A ) = C · Z ( A ) forces G to beamenable. On the other hand, dropping the commutativity completely and askingonly for a net { a i } i ∈ I ⊂ ℓ ( G, N ) satisfying (1.1) is also not a good idea becausethen one adds undesirable actions. For instance the adjoint action γ = Ad λ of theleft regular representation on N = L ( ℓ G ) = K ( ℓ G ) ′′ has this weaker propertybecause ℓ ∞ ( G ) ֒ → L ( ℓ G ) equivariantly. But this action is AD-amenable only if G is amenable. MENABILITY FOR PARTIAL ACTIONS AND FELL BUNDLES 3
Fortunately there is an alternative out of this: we only need to change (1.1)slightly, requiring instead the existence of a bounded net of finitely supported func-tions { a i } i ∈ I ⊂ ℓ ( G, N ) satisfying(1.2) h a i | b ˜ γ g ( a i ) i = X h ∈ G a i ( h ) ∗ bγ g ( a i ( g − h )) → b for the weak*-topology for all g ∈ G and b ∈ N . It turns out that this is equivalentto the W*AD-amenability of γ (and hence to the existence of a central net satisfy-ing (1.1)). Moreover, we prove that if A is a weak*-dense G -invariant C*-subalgebraof N , then the above condition (hence the W*AD-amenability of N ) is equivalentto the existence of a bounded net of finitely supported functions { a i } i ∈ I ⊂ ℓ ( G, A )satisfying (1.2). In particular an action α on a C*-algebra A is AD-amenable ifand only if there exists a bounded net { a i } i ∈ I ⊂ ℓ ( G, A ) of functions with finitesupports satisfying(1.3) h a i | b ˜ α g ( a i ) i = X h ∈ G a i ( h ) ∗ bα g ( a i ( g − h )) → b with respect to the weak topology on A for all g ∈ G and b ∈ A . This now bringsus to a close connection with the approximation property (AP) as defined by Exelin [22] for Fell bundles over discrete groups. If B α = A × G is the semidirect-product Fell bundle over G associated with α , then the AP for B α is equivalent tothe existence of a bounded net of finitely supported functions { a i } i ∈ I ⊂ ℓ ( G, A )satisfying exactly the same condition (1.3) except that the weak convergence isreplaced by the convergence with respect to the norm on A . In particular the APof an action (in the sense that its associated Fell bundle has the AP) always impliesits AD-amenability. It seems that this simple fact has not been recognised beforeexcept for the case of actions on nuclear C ∗ -algebras, where ones uses the fact thatthe AP implies nuclearity of the crossed product and that this is equivalent toAD-amenability, see for instance [26, Corollary 4.5].Since the AP makes sense for general Fell bundles (in particular for partialactions) and since it is so close to the AD-amenability of actions, it is a naturaltask trying to extend the notion of AD-amenability also to Fell bundles. This isone of our main goals in this paper. It is indeed not difficult to give a possibledefinition of AD-amenability for a general Fell bundle B . One can use for instancethe C*-algebra of kernels k ( B ) of B . This carries a canonical global action whoseassociated Fell bundle is (weakly) equivalent to B , see [5]. One can then say that B is AD-amenable if the action on k ( B ) is AD-amenable.Of course, in practice one does not want to go to an equivalent Fell bundlein order to check its AD-amenability. In order to have a direct description ofAD-amenability for Fell bundles we proceed as in the case of ordinary actions‘transporting’ everything to a von Neumann algebraic context. For this we introducethe notion of W*-Fell bundles and prove that every Fell bundle B has an envelopingW*-Fell bundle B ′′ ; indeed, the fibres B ′′ t of B ′′ are just the bidual Banach spacesof the original fibres B t of B . We then define a W*-version of the approximationproperty of Exel: we say that a W*-Fell bundle M = ( M t ) t ∈ G has the W*AP ifthere is a bounded net { a i } i ∈ I ⊂ ℓ ( G, M e ) of finitely supported functions satisfyingan approximation condition very similar (1.3); it is indeed the same condition asin the original definition of Exel in [22] except that we replace the convergence in FERNANDO ABADIE, ALCIDES BUSS, AND DAMIÁN FERRARO norm by the weak*-convergence, see our Definition 6.3 for details. We then saythat a (C*-algebraic) Fell bundle B has the WAP if B ′′ has the W*AP.Our main results show that all these notions behave nicely and have the expectedproperties. We prove that the W*AP of M is equivalent to its W*AD-amenabilityin the sense that the canonical action on its W*-algebra of kernels k w ∗ ( M ) (a cer-tain W*-completion of its C*-algebra of kernels) is W*AD-amenable. Moreover,we prove that the WAP of a Fell bundle B is equivalent to a weak form of Exel’sapproximation property – the only difference, again, is with respect to the conver-gence: for the WAP we have weak convergence while for the AP we have normconvergence. This weak form of the AP is still enough to prove some of the maindesirable properties: for instance, we prove the coincidence of full and reducedcross-sectional C*-algebras C ∗ ( B ) = C ∗ r ( B ) whenever B has the WAP.The advantage of the WAP is that it corresponds exactly to nuclearity of cross-sectional C*-algebras for Fell bundles with nuclear unit fibre: we prove that C ∗ r ( B )is nuclear if and only if B e is nuclear and B has the WAP. This equivalence is unclearfor the AP of Exel. Indeed, we know that the AP always implies the WAP but theconverse is not clear in general. In particular it is unclear whether the nuclearityof C ∗ r ( B ) implies the AP of B . This is an open question already raised by Exel.Our methods and results can be viewed as a partial answer to this question. Theonly remaining question is then to see whether the weak convergence appearingin the WAP can be always replaced by norm convergence, hence showing that theWAP and the AP are equivalent. We prove that this is indeed true for a huge classof Fell bundles, namely all Fell bundles whose unit fibres are C*-algebras Moritaequivalent to a commutative C*-algebra. In particular this applies to the importantcase of partial actions on commutative C*-algebras and shows that a partial crossedproduct C ( X ) ⋊ r G is nuclear if and only if the underlying partial action has theAP.The structure of the paper is organised as follows. In Section 2 we introduceand study the notion of partial actions of groups on W*-algebras. It seems this hasnot been studied before, but it will be important to us here as it opens a canonicalgeneral link between the C*- and W*-theory of partial actions. In particular weshow that every partial action α on a C*-algebra A extends to a canonical envelopingW*-partial action α ′′ on A ′′ . One of the main results of this section states thatevery W*-partial action admits an enveloping W*-global action. This in particularallows us to canonically extend Anantharaman-Delaroche’s notion of amenabilityto partial actions on C*- and W*-algebras. This is done in Section 3. We givesome basic examples and prove that amenability in this sense behaves well withrespect to taking restrictions and enveloping actions. In Section 4 we prove thatAD-amenability is invariant under equivalences of partial actions, both for C*- andW*-algebras.In Section 5 we start to extend the theory of AD-amenability to Fell bundles. Weintroduce the notion of W*-Fell bundles and prove that every Fell bundle admits acanonical enveloping W*-Fell bundle. We also introduce the W*-algebra of kernelsin this section. This algebra always carries a global W*-action and allows us toextend AD-amenability to W*-Fell bundles and hence also to C*-algebraic Fellbundles by taking W*-envelopings.In Section 6 we study Exel’s approximation property and translate it to thecontext of W*-Fell bundles. We prove that this new notion, the W*AP, gives an MENABILITY FOR PARTIAL ACTIONS AND FELL BUNDLES 5 alternative description of the W*AD-amenability. We also define the WAP forC*-algebraic Fell bundles and give alternative characterisations, proving that it isequivalent to AD-amenability and to a weak form of Exel’s AP. Moreover, in thefinal part of Section 6 we show that the WAP behaves well with respect to (weak)equivalences of Fell bundles. In Section 7 we give some elementary properties forAD-amenability and the approximation property. In particular we prove that theWAP is enough to conclude C ∗ ( B ) = C ∗ r ( B ) and that nuclearity of this algebra isequivalent to the WAP in case B e is already nuclear. We also prove an analogousproperty for W*-Fell bundles using injectivity in place of nuclearity.Finally, we add an appendix at the end of the paper where we review some basictheory of Hilbert W*-modules and W*-equivalences. This makes the paper moreself-contained – with the disadvantage of making it longer – and probably easierto follow to people not used to these aspects of von Neumann algebra theory. Inprinciple all the stuff added in our appendix is already known, at least to specialistsfrom the W*-community. But it is probably not very common to people workingwith C*-algebras and it is certainly not easy to grasp all things we need from theliterature.We shall only consider discrete groups in this paper although probably many ofthe things we do here also extend to locally compact groups. The approximationproperty of Fell bundles has been extended to locally compact groups by Exeland Ng in [26]. The notions of equivalences of Fell bundles are also available tolocally compact groups [5, 6]. On the other hand, the theory of amenable actions oflocally compact groups on C*-algebras has not been touched yet. Anantharaman-Delaroche only defines it for discrete groups in [10] although she actually considerslocally compact groups when acting on von Neumann algebras [7, 8].2. Partial actions on von Neumann algebras
A lot is already known about partial actions of groups on C ∗ -algebras but itseems that partial actions on W ∗ -algebras (i.e. von Neumann algebras) have neverbeen studied. Probably the main reason is that every W ∗ -algebra is unital, so thatevery partial action in this setting automatically has an enveloping action, that is,they are always restrictions of a global action on a W ∗ -algebra (see Proposition 2.7).This means that W ∗ -partial actions are not as interesting as their C ∗ -companions.However, starting with a partial action α on a C ∗ -algebra A , its bidual A ′′ vonNeumann algebra carries a natural partial action α ′′ that will serve as one of ourmain tools in this paper. This is the reason why we develop the basic theory ofpartial actions on von Neumann algebras here.We start by recalling some basic facts about partial actions and their globalisa-tions. Definition 2.1.
A partial action of a group G on a set X is a family of functions σ = { σ t : X t − → X t } t ∈ G such that:(i) For every t ∈ G , X t is a subset of X .(ii) X e = X and α e is the identity of X .(iii) Given s, t ∈ G and x ∈ X t − such that σ t ( x ) ∈ X s − , it follows that x ∈ X ( st ) − and σ st ( x ) = σ s ( σ t ( x )).An action is just a partial action σ such that X t = X for every t ∈ G . In thiscase we also say that σ is a global action. FERNANDO ABADIE, ALCIDES BUSS, AND DAMIÁN FERRARO
Given two partial actions σ and τ of G on sets X and Y respectively, a morphism f : σ → τ is a function f : X → Y such that f ( X t ) ⊆ Y t and f ( σ t ( x )) = τ t ( f ( x ))for all t ∈ G and x ∈ X t − . The composition of morphisms is just the compositionof functions.The restriction of σ to a subset Y ⊆ X is σ | Y := { σ | Y,t : Y t − → Y t } , where Y t := Y ∩ σ t ( X t − ∩ Y ) and σ | Y,t ( y ) = σ t ( y ). It follows that σ | Y is a partial actionof G on the set Y and, if Z ⊆ Y , then σ | Y | Z = σ | Z . When a partial action τ can beexpressed as a restriction τ = σ | Y of a global action σ , we say that τ is globalisable,and that σ is a globalisation of τ .In this article we work with discrete groups exclusively, so here “group” actuallymeans “discrete group”.A partial action σ of a group G on a C ∗ -algebra A is a partial action of G on theset A for which each A t is a closed two-sided ideal of A , and each σ t : A t − → A t is an isomorphism of C ∗ -algebras. If β is a global action of G on a C ∗ -algebra B and A is a closed two-sided ideal of B , then the restriction α := β | A is a partialaction on the C ∗ -algebra. We then say that β is a globalisation of α , and that α isglobalisable. Definition 2.2. A W ∗ -partial action of a group G on a W ∗ -algebra M is a settheoretic partial action of G on M , γ := { γ t : M t − → M t } t ∈ G , where each M t isa W ∗ -ideal of M (possibly { } ) and each γ t is a W ∗ -isomorphism. A morphism of W ∗ -partial actions is just a morphism of set theoretic partial actions which is alsoa morphism of W ∗ -algebras (a w ∗ -continuous morphism of *-algebras).Here we always view W ∗ -algebras as Banach space duals, M ∼ = ( M ∗ ) ′ , endowedwith the w ∗ -topology. A W ∗ -ideal is then just a ∗ -ideal of M that is closed for the w ∗ -topology. And a W ∗ -isomorphism between two W ∗ -algebras is a ∗ -isomorphism thatis w ∗ -continuous. Actually, every ∗ -isomorphism between W ∗ -algebras is normal(preserves suprema of increasing bounded nets) and it is therefore automatically a W ∗ -isomorphism, see [12, Proposition III.2.2.2].As in the case of actions on sets or on C*-algebras, we have suitable notions ofrestriction and globalisation of partial actions in the category of W ∗ -algebras: Example . Given an ordinary (global) W ∗ -action γ of G on a W ∗ -algebra N and a W ∗ -ideal M E N , the restriction γ | M is a W ∗ -partialaction. When a given W ∗ -partial action α on a W ∗ -algebra M can be written as α := γ | M , where γ is a global W ∗ -action on a W ∗ -algebra that contains M as a W ∗ -ideal, then we say that α is globalisable, and that γ is a globalisation of α . Moregenerally, the restriction of a W ∗ -partial action to a W ∗ -ideal is again a W ∗ -partialaction.We will see below in Proposition 2.7 that, unlike the case of partial actions on C ∗ -algebras, any W ∗ -partial action has a globalisation, the so called W ∗ -envelopingaction, which is essentially unique when a certain natural minimality condition isrequired to hold. Example . Given a C ∗ -partial action α = { α t : A t − → A t } t ∈ G of G on a C ∗ -algebra A , the double dual (enveloping) W ∗ -algebra A ′′ of A carries a canonical W ∗ -partial action α ′′ := { α ′′ t : A ′′ t − → A ′′ t } t ∈ G which is theunique W ∗ -partial action such that α ′′ | A = α . Here we view the bidual algebra A ′′ t as a W ∗ -ideal of A ′′ and α ′′ t as the unique w ∗ -continuous extension of α t . MENABILITY FOR PARTIAL ACTIONS AND FELL BUNDLES 7
One of our goals is to show that every W ∗ -partial action is (isomorphic to) arestriction of a global W ∗ -action as in Example 2.3, that is, every W ∗ -partial actionwill automatically have an enveloping action. One may think that this is trivialsince every von Neumann algebra has a unit, so that all the ideals of a partialaction are unital (possibly zero). It is well known from the C ∗ -algebra theory ofpartial actions that in this situation the partial action has an enveloping actionin the C ∗ -algebra category (see for instance [27]). However the following exampleshows that the C ∗ -enveloping action might be not a W ∗ -algebra. Example . Consider the “trivial” partial action of G on the W ∗ -algebra M := C in which all the ideals are zero except for M e := M . This can also be viewed asthe restriction of the global action of G by (left) translations on the C ∗ -algebra C ( G ) to the ideal C ∼ = C δ e ⊆ C ( G ). Moreover, since the linear orbit of thisideal is dense in the entire algebra C ( G ), this action is (up to isomorphism) theenveloping action of the original partial action on C . But if G is infinite, C ( G ) isnot a W ∗ -algebra. On the other hand, we may also view C ∼ = C δ e as a W ∗ -ideal ofthe W ∗ -algebra ℓ ∞ ( G ). And since the linear orbit of this ideal is w ∗ -dense, this isa W ∗ -enveloping action in the following sense. Definition 2.6. A W ∗ -enveloping action of a W ∗ -partial action γ of a group G ona W ∗ -algebra M is a W ∗ -global action σ of G on a W ∗ -algebra N together with a W ∗ -ideal ˜ M of N and an isomorphism of W ∗ -partial actions ι : γ → σ | ˜ M , such thatthe linear σ -orbit of ˜ M is w ∗ -dense in N . We summarise this situation by sayingthat ( N, σ ) is a W ∗ -enveloping action of ( M, γ ).Note that a W ∗ -enveloping action of ( M, γ ) is essentially a minimal W ∗ -global-isation of γ (see Example 2.3). The reader should not confuse the W ∗ -envelopingactions we defined above with the enveloping actions or the Morita enveloping ac-tions defined in [2]. Proposition 2.7.
Every W ∗ -partial action γ of G on a W ∗ -algebra M has a W ∗ -enveloping action that is unique up to isomorphism.Proof. We define ι : M → ℓ ∞ ( G, M ) by ι ( x )( t ) := γ t − ( x · t ), where 1 t denotesthe unit of the W ∗ -algebra M t (1 t = 0 if M t = { } ). This unit is a centralprojection of M because M t is a W ∗ -ideal of M . The map ι is an injective w ∗ -continuous ∗ -homomorphism whose image consists of functions f ∈ ℓ ∞ ( G, M ) with f ( t ) = γ t − ( f ( e )1 t ). The image ˜ M := ι ( M ) is a w ∗ -closed W ∗ -subalgebra of ℓ ∞ ( G, M ) which is therefore isomorphic to M via ι . We now endow ℓ ∞ ( G, M )with the G -action τ by left translations: τ t ( f )( s ) := f ( t − s ). This is a W ∗ -globalaction and ι : γ → τ is a morphism. Let N be the w ∗ -closure of the linear τ -orbitof ˜ M , that is, the w ∗ -closure of span { τ t ( f ) : f ∈ ˜ M , t ∈ G } . Moreover, N is τ -invariant, so that τ restricts to a W ∗ -global action σ of G on N and this is thedesired W ∗ -enveloping action of ( M, γ ) , as we now show.It is important to note that it follows from Definition 2.1 that γ t (1 t − s ) = 1 t ts for all s, t ∈ G, because 1 t − s and 1 t ts are the units of M t − ∩ M s = M t − M s and M t − ∩ M s = M t − M s , respectively. This implies ι : γ → σ is a morphism because,for all s, t ∈ G and x ∈ M t : τ t ( ι ( x ))( s ) = γ s − t ( x t − s ) = γ s − ( γ t ( x t − t − s )) = γ s − ( γ t ( x )1 s ) = ι ( γ t ( x ))( s ) . FERNANDO ABADIE, ALCIDES BUSS, AND DAMIÁN FERRARO
In order to prove that ˜ M is a W ∗ -ideal of N and that ι : γ → σ | ˜ M is an isomor-phism it suffices to show that τ t ( ˜ M ) ∩ ˜ M = ι ( M t ) , for all t ∈ G. For all x, y ∈ M :[ τ t ( ι ( a )) ι ( b )]( s ) = γ s − t ( a t − s ) γ s − ( b s ) = γ s − t ( a t − s )1 s − t s − γ s − ( b s )= γ s − t ( a t − s ) γ s − t (1 t − s t ) γ s − ( b s )= γ s − t ( a t − s t − ) γ s − ( b s ) = γ s − ( γ t ( a t − ) b s − )= ι ( γ t ( a t − ) b )( s ) . We then conclude that τ t ( ˜ M ) ∩ ˜ M = τ t ( ˜ M ) ˜ M ⊂ ι ( M t ) ⊂ τ t ( ˜ M ) ∩ ˜ M , where thelast inclusion follows from the fact that ι : γ → σ is a morphism. At this point weknow ( N, σ ) is a W ∗ -enveloping action for ( M, γ ) . For uniqueness, assume that M is a W ∗ -ideal of a W ∗ -algebra ˜ N carrying a W ∗ -global action ˜ σ whose restriction to M is γ and such that the linear ˜ σ -orbitof M is w ∗ -dense in ˜ N (for simplicity, we omit the inclusion map M ֒ → ˜ N here,that is, we already assume M ⊆ ˜ N ). Then we extend ι to ˜ ι : ˜ N → ℓ ∞ ( G, M ) by˜ ι ( x )( t ) := ˜ σ t − ( x )1 e . First we show that ˜ ι is in fact an extension of ι . For x ∈ M :˜ ι ( x )( t ) = ˜ σ t − ( x )1 e = ˜ σ t − ( x )1 t − = ˜ σ t − ( x t ) = σ t − ( x t ) = ι ( x )( t )because ˜ σ t − ( x )1 e ∈ ˜ σ t − ( M ) M = M t . A similar computation shows that ˜ ι isequivariant. Observe that ˜ ι is injective (hence isometric) because, if ˜ ι ( x ) = 0, then x ˜ σ t (1 e ) = 0 for all t ∈ G . This is equivalent to xy = 0 for all y in the linear ˜ σ -orbitof M , which is w ∗ -dense in ˜ N by assumption. Since ˜ ι is an isometry and it is w ∗ -continuous in { x ∈ ˜ N : k x k ≤ } , its range is w ∗ -closed (hence a W ∗ -subalgebra)and it is a W ∗ -isomorphism over its image. Finally, ˜ ι ( ˜ N ) is the w ∗ -closure of thelinear span of ˜ ι ( M ) = ι ( M ). Thus ˜ ι ( ˜ N ) = N and ˜ σ is isomorphic, as a W ∗ -partialaction, to σ . (cid:3) Proposition 2.8.
Let γ = { γ t : M t − → M t } t ∈ G be a W ∗ -partial action of G ona W ∗ -algebra M and let ( N, σ ) be its enveloping W ∗ -action. Then the restriction γ | Z ( M ) = { γ t : Z ( M t − ) → Z ( M t ) } of γ to Z ( M ) is a W ∗ -partial action whoseenveloping W ∗ -action is the restriction of σ to Z ( N ) .Proof. First notice that Z ( M t ) is indeed a W ∗ -ideal of Z ( M ). In fact, if M t = 1 t M ,where 1 t ∈ Z ( M ) is the central projection of M representing the unit of M t , then Z ( M t ) = 1 t Z ( M ). It is then clear that the restriction of γ to Z ( M ) defines a W ∗ -partial action. For the same reason, viewing M as a W ∗ -ideal of N , M is thenthe ideal generated by the central projection p = 1 e , and then Z ( M ) = pZ ( N ) isthe W ∗ -ideal of Z ( N ) generated by the same projection. The restriction σ | Z ( N ) is clearly σ | Z ( N ) | Z ( M ) = σ | Z ( M ) = σ | M | Z ( M ) = γ | Z ( M ) . To see that σ | Z ( N ) is theenveloping action of γ | Z ( M ) it remains to show that the linear σ -orbit of Z ( M ) isw ∗ -dense in Z ( N ). For each finite subset F ⊆ G , we define M F := P t ∈ F σ t ( M ).This is a W ∗ -ideal of N (being a finite sum of such) and the union of all these idealsis the linear σ -orbit of M , so it is w ∗ -dense in N since ( N, σ ) is the enveloping actionof (
M, γ ). On the other hand the linear σ -orbit of Z ( M ) is the w ∗ -closure P of theideal ∪ F Z ( M ) F , where Z ( M ) F = P t ∈ F σ t ( Z ( M )). Note that P is a W ∗ -ideal of Z ( N ). To see that P = Z ( N ) it is enough to show that the unit of N is containedin P . For this notice that the unit 1 F of M F is a (finite) linear combination of 1 t ,and this is also the unit of Z ( M ) F . The net (1 F ) F is increasing and bounded andits w ∗ -limit is the unit of N because ∪ F M F is w ∗ -dense in N . However this limitis also the unit of P . (cid:3) MENABILITY FOR PARTIAL ACTIONS AND FELL BUNDLES 9
Remark . If (
N, σ ) is a W ∗ -enveloping action of ( M, γ ), then N is abelian if andonly if M is abelian. Indeed, clearly M is abelian if N is. For the converse observethat, by the proof of the Proposition 2.7, N is isomorphic to a W ∗ -subalgebra ofthe abelian algebra ℓ ∞ ( G, M ).Another property that is preserved by taking enveloping actions is injectivity inthe sense that if N is the W ∗ -enveloping action of M , then M is injective if and onlyif N is injective. Indeed, since injectivity passes to ideals, the reverse direction isclear. For the converse one uses that injectivity passes to (finite) sums and directedunions of ideals and the description of N as the w ∗ -closure of ∪ F M F as in the proofof the previous proposition.3. Amenability of partial actions
First let us recall the notion of amenability for (global) actions of groups on C ∗ -algebras and W ∗ -algebras introduced by Anantharaman-Delaroche, see [7,8,10]. Definition 3.1.
A (global) action of a group G on a W ∗ -algebra M is Anantharaman-Delaroche amenable (or just
W*AD-amenable for short ) if there exists a linearpositive contractive and G -equivariant map P : ℓ ∞ ( G, M ) ։ M whose composi-tion with the canonical embedding (by constant functions) M ֒ → ℓ ∞ ( G, M ) is theidentity map M → M . Here ℓ ∞ ( G, M ) is endowed with the diagonal G -action:˜ γ t ( f )( r ) = γ t ( f ( t − r )), where γ denotes the G -action on M .An action α of G on a C ∗ -algebra A is AD-amenable if the corresponding doubledual W ∗ -action α ′′ on A ′′ is W*AD-amenable.Let us recall some basic examples of amenable actions. Example . The translation G -action on itself, viewed as a G -action on the C ∗ -al-gebra C ( G ) , is always AD-amenable (this action is even proper). By definition, thismeans that the translation action on C ( G ) ′′ ∼ = ℓ ∞ ( G ) is always W*AD-amenableas a W ∗ -action. However the translation action on ℓ ∞ ( G ) is AD-amenable as a C ∗ -action if and only if G is exact, see [14, Theorem 5.1.7].More generally, if M is a G - W ∗ -algebra, the G - W ∗ -algebra ℓ ∞ ( G, M ) endowedwith the diagonal G -action is always W*AD-amenable because we have a canonical G -equivariant unital embedding ℓ ∞ ( G ) ֒ → Zℓ ∞ ( G, M ) = ℓ ∞ ( G, Z ( M )) (see [8,Corollary 3.8]). As before, here ℓ ∞ ( G ) carries the translation G -action.Before we proceed, let us highlight some of the most important characterisationsof AD-amenability obtained by Anantharaman-Delaroche in her papers [7, 8, 10]. Theorem 3.3 (Anantharaman-Delaroche) . The following are equivalent for a globalaction γ of a group G on a W ∗ -algebra M :(i) γ is W*AD-amenable;(ii) the restriction of γ to the center Z ( M ) is W*AD-amenable, that is, there isa G -equivariant norm-one projection ℓ ∞ ( G, Z ( M )) ։ Z ( M ) ;(iii) there is a net { a i : G → Z ( M ) } i ∈ I of finitely supported functions with h a i | a i i := P g ∈ G a i ( g ) ∗ a i ( g ) ≤ for all i and h a i | ˜ γ g ( a i ) i → ultraweakly forall g ∈ G .Moreover, if M is injective as a W ∗ -algebra, then the above are also equivalent to(iv) the W ∗ -crossed product M ¯ ⋊ G is an injective W ∗ -algebra. If α is an AD-amenable action of G on a C ∗ -algebra A , then the full and reduced C ∗ -crossed products coincide, that is, A ⋊ α G = A ⋊ α, r G . And if A is nuclear, then α is AD-amenable if and only if A ⋊ α, r G is a nuclear C ∗ -algebra. Let us also remark that for an action on a commutative C ∗ -algebra A = C ( X ),its AD-amenability is equivalent to amenability of the associated transformationgroupoid X ⋊ G in the sense of Anantharaman-Delaroche and Renault, see [11].Moreover, the AD-amenability in this case is equivalent to the existence of a net { a i : G → Z ( A ) = A } i ∈ I with the same properties as in (iii) above, except that theultraweak convergence in (iii) can be strengthened to the convergence with respectto the strict topology on A ⊆ M ( A ) (the multiplier algebra), see [10, Théoréme 4.9].This cannot be expected – and indeed it is not true – for noncommutative algebrasbecause simple unital C ∗ -algebras can carry AD-amenable actions of non-amenablegroups, see Remark 3.6.We are now ready to introduce the notion of amenability for partial actions on C ∗ -algebras and W ∗ -algebras: Definition 3.4.
We say that a partial action of a group G on a W ∗ -algebra M is W*AD-amenable if its enveloping W ∗ -action (provided by Proposition 2.7) isW*AD-amenable.We say that a partial action of G on a C ∗ -algebra A is AD-amenable if theinduced W ∗ -partial action on A ′′ is W*AD-amenable.Of course, a global action is AD-amenable if and only if it is AD-amenable as apartial action. Before we give some proper examples of amenable partial actions,we observe the following general fact: Proposition 3.5. A W ∗ -partial action ( M, γ ) is W*AD-amenable if and only ifits restriction to the center Z ( M ) is W*AD-amenable.Proof. This follows directly from the definition, Proposition 2.8 and Theorem 3.3. (cid:3)
Remark . The above result does not hold for partial actions on C ∗ -algebras, noteven for global actions. Indeed, the results of Suzuki in [33] show that every exactgroup admits an AD-amenable action on a unital simple (and nuclear) C ∗ -algebra.Such an algebra has trivial center and a trivial global action can only be AD-amenable if the group is amenable. Example . The “trivial” partial action of G on A = C appearing in Example 2.5is AD-amenable, both in C ∗ - and W ∗ -sense. This is because A = A ′′ has as itsenveloping W ∗ -action the global translation G -action on ℓ ∞ ( G ) as explained inExample 2.5, and this W ∗ -action is W*AD-amenable. In the same way, we canconsider the partial action on a W ∗ -algebra M as in Example 2.5 with all domainideals M g = 0 except for M e = M . This partial action is always W*AD-amenablebecause its enveloping W ∗ -action is the translation action on ℓ ∞ ( G, M ), which is W ∗ -amenable by Example 3.2. For the same reason, any C ∗ -algebra A endowedwith the “trivial” partial action (in which all the domain ideals are A g = 0 exceptfor A e = A ) is always AD-amenable because then A ′′ carries the “trivial” partial G -action which is W*AD-amenable. Example . More generally, the following holds: take an amenable subgroup H ⊆ G acting (globally) on a C ∗ -algebra A (or on a W ∗ -algebra M ) and “extend” this MENABILITY FOR PARTIAL ACTIONS AND FELL BUNDLES 11 to a partial G -action on A “by zero” in the sense that A h = A for h ∈ H , A g = 0for g ∈ G \ H and α g : A g − → A g acts via the original H -action for g ∈ H and byzero otherwise. This partial action (which is global only if H = G ) is always AD-amenable. Indeed, the canonical action of G on ℓ ∞ ( G/H ), ν t ( f )( sH ) = f ( t − sH ),plays an important role here. This W ∗ -action is W*AD-amenable if (and only if) H is amenable. Indeed, ℓ ∞ ( G/H ) = C ( G/H ) ′′ and the crossed product C ( G/H ) ⋊ r G is Morita equivalent to C ∗ r ( H ) by Green’s imprimitivity theorem.To see that the partial action of G defined above is AD-amenable, it is enoughto consider the von Neumann algebraic situation. For this, let us write γ for theaction of H on a W ∗ -algebra M and name ¯ γ its extension to G .To prove amenability of this partial G -action, we give an explicit description ofits W ∗ -enveloping action. Consider the W ∗ -subalgebra of ℓ ∞ ( G, M ) N := { f ∈ ℓ ∞ ( G, M ) : f ( s ) = γ s − t ( f ( t )) if sH = tH } . This subalgebra is invariant under the action τ of G on ℓ ∞ ( G, M ) given by τ t ( f )( s ) = f ( t − s ). We name δ the restriction of τ to N . In order to view M as a W ∗ -ideal of N in such a way that δ is the W ∗ -globalisation of ¯ γ , we consider the map ι : M → N given by ι ( a )( s ) = ( γ s − ( a ) if s ∈ H s / ∈ H. Note that in case M = C , we have N = ℓ ∞ ( G/H ) and τ = ν . In any case, wemay view ℓ ∞ ( G/H ) as a unital δ -invariant subalgebra of Z ( N ) by considering theinclusion κ : ℓ ∞ ( G/H ) → N , κ ( f )( t ) = f ( tH ). Moreover, the restriction of δ to ℓ ∞ ( G/H ) is ν , from which it follows that δ is W*AD-amenable, hence so is ¯ γ . Example . If (
M, γ ) is an AD-amenable partial W ∗ -action of G and H ⊆ G is any subgroup, then the restriction of the partial G -action on M to H , namely, γ | H = { γ h : M h − → M h } h ∈ H is also AD-amenable. A similar assertion holdsfor C ∗ -partial actions. Indeed, it is clearly enough to check this for W ∗ -partialactions. This is known to hold for global W ∗ -actions (it follows trivially from(iii) in Theorem 3.3). Now for a general W ∗ -partial action ( M, γ ) of G , take itsglobalisation W ∗ -action ( N, σ ). For simplicity we identify M as a W ∗ -ideal of N .Let H · M = P t ∈ H σ t ( M ) be the H -linear orbit of M in N . Notice that this is anideal of N and its w ∗ -closure N H := H · M w ∗ is an H -invariant W ∗ -ideal of N whichcan be viewed as an H -globalisation of γ | H . If ( M, γ ) is W*AD-amenable, then bydefinition (
N, σ ) is W*AD-amenable, and then so is (
N, σ | H ) and hence also every H -invariant W ∗ -ideal, like ( N H , σ | H ). Therefore ( M, γ | H ) is W*AD-amenable.Next we look at restrictions of partial actions to ideals and prove that amenabilitybehaves nicely also in this direction. Proposition 3.10.
The restriction of a W*AD-amenable W ∗ -partial action of agroup to a W ∗ -ideal is again W*AD-amenable. Moreover, the analogous statementholds for AD-amenability on C*-algebras, that is, AD-amenability is also preservedby restrictions.Proof. First we deal with W ∗ -partial actions. Let γ be a W*AD-amenable W ∗ -par-tial action of the group G on M and let J be a W ∗ -ideal of M . We know that M can be viewed as a W ∗ -ideal of a W ∗ -algebra N carrying a W*AD-amenable W ∗ -global action σ of G with σ | M = γ . Then J is a W ∗ -ideal of N and the w ∗ -closure of P t ∈ G σ t ( J ), denoted by [ J ], is a σ -invariant W ∗ -ideal of N . Moreover, σ | [ J ] is the W ∗ -enveloping action of γ | J because σ | [ J ] | J = σ | J = σ | M | J = γ | J andit is also W*AD-amenable because it is a restriction of a global W*AD-amenable W ∗ -action to a G -invariant W ∗ -ideal.Now let β be an AD-amenable C ∗ -partial action of G on B and let A be a C ∗ -idealof B . Then we may view A ′′ as the w ∗ -closure of A in B ′′ . Note that ( β | A ) ′′ is theunique W ∗ -partial action of G on A ′′ extending β | A . But β ′′ | A ′′ is a W ∗ -partialaction such that β ′′ | A ′′ | A = β ′′ | A = β ′′ | B | A = β | A . Thus β ′′ | A ′′ = ( β | A ) ′′ . By theprevious paragraph, β | A is AD-amenable if β is. (cid:3) Proposition 3.11.
Let β be a C ∗ -global action of a group G on B and let A be a C ∗ -ideal of B such that the norm closure of P t ∈ G β t ( A ) is B . In other words, β isthe C ∗ -enveloping action of α := β | A . Then β ′′ is the W ∗ -enveloping action of α ′′ and α is AD-amenable if and only if β is AD-amenable.Proof. We view A ′′ as the w ∗ -closure of A in B ′′ , thus A ′′ is a W ∗ -ideal of B ′′ . Inthe proof of Proposition 3.10 we showed that β ′′ | A ′′ = α ′′ . Thus, to show that β ′′ is a W ∗ -enveloping action of α ′′ , it suffices to prove that B ′′ is the w ∗ -closure of J := P t ∈ G β ′′ t ( A ′′ ); let us write J for this closure. Note that P t ∈ G β t ( A ) ⊆ J ⊆ J and, taking norm closure, this implies B ⊆ J . Now taking w ∗ -closure we get J = B ′′ .The rest of the proof follows directly from the definition of AD-amenability forpartial actions. (cid:3) Morita equivalence of partial actions
Many C ∗ -partial actions do not admit a C ∗ -enveloping action, but every C ∗ -par-tial action has a Morita enveloping action, as defined in [2], which is unique upto Morita equivalence of actions. It is therefore important to see how amenabilitybehaves in terms of Morita equivalences.Equivalences of partial actions are defined in [2]. We shortly recall the definition:two partial actions α = { α t : A t − → A t } and β = { β t : B t − → B t } of G on C ∗ -algebras A and B are equivalent if there exist an equivalence Hilbert A - B -bimodule X carrying a (set theoretic) partial action γ = { γ t : X t − → X t } of G bylinear maps on X such that X t ⊆ X are Hilbert A - B -submodules implementing anequivalence between the domain ideals A t and B t , that is, the images of X t by theleft and right inner products are contained and generate A t and B t as C ∗ -algebras,and the usual compatibility between the actions holds: h γ t ( x ) | γ t ( y ) i B = β t ( h x | y i B )and A h γ t ( x ) | γ t ( y ) i = α t ( A h x | y i ) for all x, y ∈ X t − . Using W ∗ -equivalencebimodules (see the Appendix) one defines equivalences between W ∗ -partial actionsin a similar way.Both notions of equivalences can be conveniently described in terms of linkingalgebras: given an equivalence bimodule X as above, one considers its linking C ∗ -algebra L . This carries a partial action of G where the domain ideal L t is(isomorphic to) the linking algebra of X t . If X is an equivalence between W ∗ -partialactions, then L is a W ∗ -algebra carrying a W ∗ -action encoding the W ∗ -equivalence. Proposition 4.1.
Let µ and ν be W*-Morita equivalent W*-partial actions of agroup G on the algebras M and N , respectively. Then the restrictions σ := µ | Z ( M ) and τ := ν | Z ( N ) are isomorphic (as W*-partial actions). MENABILITY FOR PARTIAL ACTIONS AND FELL BUNDLES 13
Proof.
Let X be a W*-Morita equivalence bimodule between M and N equippedwith a W*-partial action γ of G inducing µ and ν . More precisely: • γ = ( { X t } t ∈ G , { γ t } t ∈ G ) is a set theoretic partial action on W*-ideals bylinear isometries. • For every t ∈ G , M t ( N t , respectively) is the w ∗ -closure of the space spannedby M h X t , X t i ( h X t , X t i N , respectively). • For every t ∈ G and x, y ∈ X t − , M h γ t ( x ) , γ t ( y ) i = µ t ( M h x, y i ) and h γ t ( x ) , γ t ( y ) i N = ν t ( h x, y i N )The conditions above imply that, for all t ∈ G , x ∈ X t − and a ∈ M t − , γ t ( ax ) = µ t ( a ) γ t ( x ). The same holds for γ and ν .Proposition A.9 yields a unique W ∗ -isomorphism π : Z ( N ) → Z ( M ) such that xa = π ( a ) x for all a ∈ Z ( N ) and x ∈ X . To finish the proof we check that theisomorphism π : Z ( N ) → Z ( M ) above intertwines the partial actions σ and τ . Firstof all, if p t and q t are the units of M t and N t (respectively), then for all x ∈ X : p t x = ( p t x ) q t = p t ( xq t ) = xq t . Hence π ( q t ) = p t and π ( Z ( N ) t ) = Z ( M ) t .Now fix t ∈ G and a ∈ Z ( N ) t − . For every x ∈ X we have π ( ν t ( a )) x = xν t ( a ) = xq t ν t ( a ) = γ t ( γ t − ( xq t ) a )= γ t ( π ( a ) γ t − ( p t x )) = µ t ( π ( a )) p t x = µ t ( π ( a )) x. This implies π ( µ t ( a )) = ν t ( π ( a )) and the proof is complete. (cid:3) As a consequence we derive the following important result:
Proposition 4.2.
AD-amenability is preserved by Morita equivalence of partialactions, both in C ∗ - and W ∗ -contexts.Proof. A C ∗ -equivalence A - B -bimodule X induces a W ∗ -equivalence A ′′ - B ′′ -bimo-dule X ′′ , so it is enough to deal with the W ∗ -case. But this follows directly as acombination of Propositions 4.1 and 3.5. (cid:3) Remark . The above result applies in particular to global actions and shows thatAD-amenability is invariant under Morita equivalence of group actions. We believethat this is known for specialists but we could not find a reference.
Corollary 4.4. A C ∗ -partial action is AD-amenable if and only if one (hence all)of its Morita enveloping actions is AD-amenable.Proof. Let α be a C ∗ -partial action of a group and let β be one of its Moritaenveloping actions. This means that α is Morita equivalent to a restriction γ of β and β is the C ∗ -enveloping action of γ . By Propositions 4.2 and 3.11, α isAD-amenable if and only if γ is AD-amenable if and only if β is AD-amenable. (cid:3) AD-amenability of Fell bundles
One of our goals in this paper is to extend Anantharaman-Delaroche’s notionof amenability to Fell bundles over discrete groups. For this we need some prepa-ration because, as the case of partial actions already indicates, the definition ofAD-amenability requires going to the W ∗ -setting. W*-enveloping Fell bundles.
Given a Fell bundle B over a group G , wewant to turn the bundle of biduals B ′′ := { B ′′ t } t ∈ G into a Fell bundle in such away that B becomes a Fell subbundle of B ′′ and there is a certain continuity ofthe operations with respect to the w ∗ -topology. This requires to equip every fibre B ′′ t with a Hilbert bimodule structure over B ′′ e (or a ternary W ∗ -ring structure)extending that of B t .The machinery described here is not new: biduals of Hilbert modules are knownto be Hilbert W ∗ -modules (see for instance [13]), biduals of ternary W ∗ -rings aredescribed in [35], and biduals of Fell bundles over inverse semigroups are alreadydescribed in [18, Section 3], although there only saturated Fell bundles are consid-ered. For the convenience of the reader and to make this article as self-containedas possible, we provide the complete constructions here. Lemma 5.1.
Let B be a Fell bundle over a group G . For every nondegeneraterepresentation π : B e → L ( H ) there exists a nondegenerate representation ψ : B →L ( K ) such that π is a sub-representation of ψ | B e .Proof. Define H t := B t ⊗ π H , where we view B t as a Hilbert B e -module. We wantto construct, for every s, t ∈ G and b ∈ B s , a linear bounded map ψ b,t : B t ⊗ π H → B st ⊗ π H such that ψ b,t ( a ⊗ h ) = ba ⊗ h . For this it suffices to show that, given P nj =1 a j ⊗ h j ∈ B t ⊗ H , we have k P nj =1 ba j ⊗ h j k ≤ k b k k P nj =1 a j ⊗ h j k . Viewing B t as a left B e -Hilbert module we get k n X j =1 ba j ⊗ h j k = n X j,l =1 h h j , a ∗ j b ∗ ba l h l i = k n X j =1 ( b ∗ b ) / a j ⊗ h j k ≤ k b k k n X j =1 a j ⊗ h j k . The computations above show that ψ t,b is well defined and k ψ t,b k ≤ k b k .Define K as the ℓ -direct sum L t ∈ G H t . We claim that there exists a represen-tation ψ : B → L ( K ) such that, for all s, t ∈ G , b ∈ B s and f ∈ K , ψ ( b )( f )( t ) = ψ b,s − t f ( s − t ). First we show that ψ is well defined. Take f ∈ K , b ∈ B s and a fi-nite set λ ⊆ G . Then P t ∈ λ k ψ b,s − t f ( s − t ) k ≤ k b k P t ∈ λ k f ( s − t ) k ≤ k b k k f k .This shows that t ψ b,s − t f ( s − t ) is square summable in norm.The facts that ψ ( ab ) = ψ ( a ) ψ ( b ) and ψ ( b ) ∗ = ψ ( b ∗ ) follow from the facts that ψ b,t ∗ = ψ b ∗ ,st and ψ a,st ψ b,t = ψ ab,t for all s, t ∈ G , a ∈ B and b ∈ B t . These lastidentities are straightforward to prove and are left to the reader.The natural identification of H with B e ⊗ π H provides an inclusion ι : H → K .Under this inclusion we see that ψ ( B e ) H = H and, for all a, b ∈ B e and h ∈ H : ψ ( b )( ι ( π ( a ) h )) = ba ⊗ h = ι ( π ( b ) π ( a ) h ) . This clearly implies that π is a sub-representation of ψ . Finally, ψ is nondegen-erate because for every approximate unit ( e j ) j ∈ J of B e and a ⊗ h ∈ H t we havelim j ψ ( e j ) a ⊗ h = e j a ⊗ h = a ⊗ h . (cid:3) Theorem 5.2.
Let B = { B t } t ∈ G be a Fell bundle. Then the bundle of biduals B ′′ := { B ′′ t } t ∈ G has a unique Fell bundle structure extending that of B and suchthat, for every r, s ∈ G and a ∈ B r , the functions B ′′ s → B rs , b ab , and B ′′ s → B ′′ s , b b ∗ , are w ∗ -continuous. MENABILITY FOR PARTIAL ACTIONS AND FELL BUNDLES 15
Proof.
Let ψ : B → L ( H ) be a nondegenerate representation such that ψ | B e containsthe universal representation of B e as a sub-representation. Then we view the bidual B ′′ e as the weak closure (or the bicommutant) of ψ ( B e ) ⊆ L ( H ). For every t ∈ G we view the bidual I ′′ t of the ideal I t := B t B ∗ t ⊆ B e as the weak closure of ψ ( I t ).We claim that there exists a fibre-wise linear and w ∗ -continuous extension ψ ′′ : B ′′ →L ( H ). For this we view the fibre B t as a I t − I t − -equivalence bimodule with theoperations inherited from B . Let 1 t be the unit of I ′′ t and H t := 1 t H . Notethat B t ⊗ ψ | It − H t − = ψ ( B t ) H t − . Since every b ∈ B t factors as cc ∗ c for some c ∈ B t , we also have ψ ( B t ) H t − = H t = ψ ( B t ) H . For every b ∈ B t we have h ψ ( b )( H t − ⊥ ) , H i = h H t − ⊥ , ψ ( b ∗ ) H i ⊆ h H t − ⊥ , H t − i = 0. This shows that ψ ( b )vanishes on H t − ⊥ and its image is contained in H t . Thinking of ψ ( b ) as an op-erator from H t − to H t = B t ⊗ ψ | It − H t − , ψ ( b ) becomes the function h b ⊗ h .By Corollary C.3, ψ | B t has a unique w ∗ -continuous extension ( ψ | B t ) ′′ : B ′′ t → L ( H ),which is faithful. The map ψ ′′ is then given by ψ ′′ ( b ) := ( ψ | B t ) ′′ ( b ) for every b ∈ B t .Notice that ψ ′′ is faithful on each fiber. Indeed, viewing ψ ′′ | B ′′ t as a map withimage in L ( H t − , H t ) , it follows that ( ψ ′′ | B ′′ t ) r = ( ψ | I t ) ′′ is faithful because ( ψ | B e ) ′′ is faithful. Hence, by Proposition B.1, ψ ′′ | B ′′ t is faithful.Using w ∗ -density arguments we get ψ ′′ ( B ′′ s ) ψ ′′ ( B ′′ t ) ⊆ ψ ′′ ( B ′′ st ) and ψ ′′ ( B ′′ s ) ∗ = ψ ′′ ( B ′′ s − ) . Thus we may define the multiplication in such a way that, for x ∈ B ′′ s and y ∈ B ′′ s , xy is the unique element of B ′′ st such that ψ ′′ ( xy ) = ψ ′′ ( x ) ψ ′′ ( y ). The involutionis determined by the condition ψ ′′ ( x ∗ ) = ψ ′′ ( x ) ∗ and the norm is k x k := k ψ ′′ ( x ) k .These operations clearly extend those of B and B ′′ is a Fell bundle. Finally, ( B ′′ t ) ∗ B ′′ t is a W ∗ -ideal because ψ ′′ (( B ′′ t ) ∗ B ′′ t ) = I ′′ t − . (cid:3) Remark . Let α be a partial action of G on a C ∗ -algebra A . Then ( B α ) ′′ iscanonically W ∗ -isomorphic to B α ′′ .Inspired by the previous result we introduce the following definition, which isthe natural W ∗ -analogue of Fell bundles – also called C ∗ -algebraic bundles in [20]. Definition 5.4. A W ∗ -Fell bundle (or W ∗ -algebraic bundle ) over the group G is a Fell bundle M = { M t } t ∈ G such that each M t is isometrically isomorphic tothe dual of a Banach space and, for every s, t ∈ G and a ∈ M s , the functions M t → M t − , b b ∗ , and M t → M st , b ab , are w ∗ -continuous.By [35] the predual of each fibre M t is unique because M t is a M e - M e -Hilbertbimodule (not necessarily full) with respect to the canonical operations comingfrom the product and involution of M .5.2. Central partial actions of W*-Fell bundles.
Take a W ∗ -Fell bundle M over a group G . For every t ∈ G we define I t as the W ∗ -algebra generated by M t M ∗ t in M e . Note that I t is in fact a W ∗ -ideal of M e . The fiber M t has a natural W ∗ -equivalence I t - I t − -bimodule structure with the multiplication of M definingthe left and right actions and the inner products I t h x, y i := xy ∗ and h x, y i I t − := x ∗ y . Then Proposition A.9 provides an isomorphism σ t : Z ( I t − ) → Z ( I t ). Weclaim that σ := { σ t : Z ( I t − ) → Z ( I t ) } t ∈ G is a W ∗ -partial action of G on Z ( M e ).To prove this it suffices to show that σ is a set theoretic partial action. To simplifythe notation we write Z t instead of Z ( I t ). It is clear that I e = M e . Moreover, σ e is the isomorphism corresponding to the W ∗ -algebra M e viewed as the identity W ∗ -equivalence M e - M e -bimodule, hence σ e is the identity of Z e .Lets show that σ t ( Z t − ∩ Z s ) ⊆ Z t ∩ Z ts . Writing p t for the unit of I t , it sufficesto prove that σ t ( p t − p s ) = p t p ts . For every x ∈ M t we have σ t ( p t − p s ) x = xp t − p s ,hence σ t ( p t − p s ) is the unit ofspan w ∗ M t p t − p s ( M t p t − p s ) ∗ = span w ∗ M t p t − p s M ∗ t = span w ∗ M t M ∗ t M t M s M ∗ s M ∗ t ⊆ span w ∗ M t M ∗ t M ts M ∗ ts ⊆ I t ∩ I ts ⊆ span w ∗ M t M ∗ t M ts M ∗ ts p t = span w ∗ M t M ∗ t M ts M ∗ ts M t M ∗ t ⊆ span w ∗ M t p t − M s M ∗ s M ∗ t ⊆ span w ∗ M t p t − p s M ∗ t . Thus σ t ( p t − p s ) is the unit of I t ∩ I ts and we have σ t ( p t − p s ) = p t p ts .Now take x ∈ Z t − ∩ Z t − s − . We already know that σ t ( x ) ∈ Z t ∩ Z s − and σ s ( σ t ( x )) ∈ Z st ∩ Z s . Also σ st ( x ) ∈ Z st ∩ Z s . We can write p s as a w ∗ -limit of theform p s = lim i P n i j =1 u i,j v ∗ i,j with u i,j , v i,j ∈ M s . Then, for all z ∈ M st : σ s ( σ t ( x )) z = σ s ( σ t ( x )) p s z = σ s ( σ t ( x )) lim i n i X j =1 u i,j v ∗ i,j z = lim i n i X j =1 σ s ( σ t ( x )) u i,j v ∗ i,j z = lim i n i X j =1 u i,j σ t ( x ) v ∗ i,j z = lim i n i X j =1 u i,j v ∗ i,j zx = p s zx = σ st ( x ) p s z = σ st ( x ) z. This implies σ st ( x ) = σ s ( σ t ( x )). Definition 5.5.
Let M be a W ∗ -Fell bundle over a group G . The central partialaction of M is the W ∗ -partial action σ of G on Z ( M e ) constructed above. Example . Let γ = ( { M t } t ∈ G , { γ t } t ∈ G ) be a W ∗ -partial action of a group G ona W ∗ -algebra M . If M is the semidirect product bundle of γ , which is a W ∗ -Fellbundle, then the central partial action of M is the restriction of γ to Z ( M ).To prove the claim above note that the W ∗ -ideals I t of M = M δ e generated by( M t δ t )( M t δ t ) ∗ = γ t ( γ t − ( M t M ∗ t )) δ e = M t δ e are just M t seen as a subalgebra of M = M δ e . Then the domains of σ and γ | Z ( M ) agree. If x ∈ Z ( M t − ) and y ∈ M t ,then γ t ( x ) δ e yδ t = γ t ( x ) yδ t = yγ t ( x ) δ t = γ t ( γ t − ( y ) x ) δ t = yδ t xδ e = σ t ( x ) δ e yδ t and this implies γ t ( x ) = σ t ( x ) (because we identify x ∈ M e with xδ e ∈ M e δ e ).5.3. Cross-sectional W*-algebras of W*-Fell bundles.
To a W ∗ -Fell bundle M one can naturally assign a cross-sectional W ∗ -algebra W ∗ r ( M ) as follows: theusual Hilbert M e -module ℓ ( M ) is not suitable here because it might not be a W ∗ -module, that is, it is possibly not self-dual. We look at its self-dual completionthat can be concretely described as follows. Let ℓ ∗ ( M ) be the space of sections ξ : G → M for which the net of finite sums P t ∈ F ξ ( t ) ∗ ξ ( t ) (for F ⊆ G finite)is bounded; since this net is increasing and consists of positive elements, it w ∗ -converges to some element h ξ | ξ i M e := P t ∈ G ξ ( t ) ∗ ξ ( t ) ∈ M e . The space ℓ ∗ ( M ) isthen a right W*-Hilbert M e -module when endowed with right M e -action ( ξ · b )( t ) := ξ ( t ) · b and inner product h ξ | η i M e := P t ∈ G ξ ( t ) ∗ η ( t ), the limit of this sum beingwith respect to the w ∗ -topology, for all ξ, η ∈ ℓ ∗ ( M ) . MENABILITY FOR PARTIAL ACTIONS AND FELL BUNDLES 17
Next we define the left regular representation of M . This is done as in the C ∗ -case, except that we now act on ℓ ∗ ( M ). More precisely, for each t ∈ G we definethe map Λ t : M t → L ( ℓ ∗ ( M )) by (Λ t ( a ) ξ )( s ) := a · ξ ( t − s ) (the multiplicationperformed in M ) for all s ∈ G , a ∈ M t and ξ ∈ ℓ ∗ ( M ). As in the C ∗ -setting,a routine argument shows that Λ t ( a ) is a well-defined adjointable operator withΛ t ( a ) ∗ = Λ t − ( a ∗ ) and that Λ = (Λ t ) t ∈ G is a representation of M . Note that L ( ℓ ∗ ( M )) is a W ∗ -algebra, see for example [31, Proposition 3.10]. Definition 5.7.
The cross-sectional W ∗ -algebra of M is the W ∗ -subalgebra W ∗ r ( M )of L ( ℓ ∗ ( M )) generated by the image of its regular representation Λ.The linear span of the image of Λ is already a ∗ -subalgebra, so that W ∗ r ( M ) isjust the w ∗ -closure of that subalgebra. We also observe that the cross-sectional C ∗ -algebra C ∗ r ( M ) embeds as a w ∗ -dense C ∗ -subalgebra of W ∗ r ( M ). Moreover,since ℓ ∗ ( M ) is the self-dual completion of ℓ ( M ), every adjointable operator on ℓ ( M ) extends to an adjointable operator on ℓ ∗ ( M ) and this gives a C ∗ -embedding L ( ℓ ( M )) ֒ → L ( ℓ ∗ ( M )) that restricts to the embedding C ∗ r ( M ) ֒ → W ∗ r ( M ).The (reduced) W ∗ -algebra of a W ∗ -Fell bundle M is exactly the W ∗ -counterpartof the reduced C ∗ -algebra as defined by Exel and Ng in [26]. Proposition 5.8.
Let M be a W ∗ -Fell bundle over a group G and π : M e → L ( H ) be a weak*-continuous representation. Then the map Λ π : M → L ( ℓ ∗ ( M ) ⊗ π H ) , b Λ( b ) ⊗ id is a representation which is w ∗ -continuous on each fiber. The integrated form ˜Λ π factors through a representation of C ∗ r ( M ) that can be extended to a weak* contin-uous representation ˜Λ w ∗ π of W ∗ r ( M ) in a unique way. Moreover, ˜Λ w ∗ π is unital and ˜Λ w ∗ π ( W ∗ r ( M )) = ˜Λ w ∗ π ( M ) ′′ (the bicommutant). If π is injective then so is ˜Λ w ∗ π . Proof.
Consider the map ρ : L ( ℓ ∗ ( M )) → L ( ℓ ∗ ( M ) ⊗ π H ) , ρ ( R ) = R ⊗ id , ofLemma A.2. Then Λ π := ρ ◦ Λ :
M → L ( ℓ ∗ ( M ) ⊗ π H ) is clearly a representationthat, when restricted to the closed unit ball of a fiber, is w ∗ -continuous by LemmaA.2. Hence Λ π is a representation which is w ∗ -continuous on each fiber. In case π is faithful then so is ρ and hence Λ π is faithful if π is.Note that ℓ ∗ ( M ) ⊗ π H = ℓ ( M ) ⊗ π H. Thus we may very well think of Λ π : M →L ( ℓ ( M ) ⊗ π H ) as the composition of the C ∗ -regular representation M → L ( ℓ ( M ))with C ∗ r ( M ) ⊂ L ( ℓ ( M )) → L ( ℓ ( M ) ⊗ π H ) , T T ⊗ id . This clearly implies thatΛ π factors through C ∗ r ( M ) . The restriction ρ | W ∗ r ( M ) : W ∗ r ( M ) → L ( ℓ ∗ ( M ) ⊗ π H ) is a w ∗ -continuous rep-resentation that clearly extends the integrated form of Λ π . Hence this integratedform can be extended to W ∗ r ( M ) (as ρ | W ∗ r ( M ) ) and the extension is faithful if π is.The rest of the proof follows by the Bicommutant Theorem. (cid:3) Theorem 5.9 (c.f. [26, Corollary 2.15]) . Assume that M is a W ∗ -Fell bundleover G and write λ for the left regular representation of G by unitary operatorson ℓ ( G ) . Let T : M → L ( H ) be a nondegenerate representation which is weak*continuous on each fiber and let µ λ,T : M → L ( ℓ ( G, H )) be the representationsuch that µ λ,T ( b ) = λ t ⊗ T b , for every b ∈ B t and t ∈ G. Then the integrated formof µ λ,T , denoted ˜ µ λ,T , factors through a representation of C ∗ r ( M ) that has a unique w ∗ -continuous extension to a representation ˜ µ w ∗ λ,T of W ∗ r ( M ) . Moreover, if T | B e isfaithful then so is ˜ µ w ∗ λ,T . Proof.
It was shown in [26] that ˜ µ λ,T factors through a representation of C ∗ r ( M ) . To extend ˜ µ λ,T to W ∗ r ( M ) take R ∈ W ∗ r ( M ) . Then, by [19], there exists a boundednet ( f j ) j ∈ J ⊂ C ∗ r ( M ) such that R = w ∗ lim j ˜ λ w ∗ M ( f j ) . Every closed ball of L ( ℓ ( G, H )) is compact in the weak* topology, thus thereexists S ∈ L ( ℓ ( G, H )) and a subnet ( f j l ) l ∈ L such that S = w ∗ lim l ˜ µ λ,T ( f j l ) . Inorder to prove that S = w ∗ lim j ˜ µ λ,T ( f j ) it suffices to show the existence of a set X ⊂ ℓ ( G, H ) spanning a dense subset of ℓ ( G, H ) and such that, for every x, y ∈ X, ( h x, ˜ µ λ,T ( f j ) y i ) j ∈ J is a convergent net.Using the notation of [26, Propositon 2.13] we define X := [ r ∈ G { ( ρ r ⊗ ◦ V u : u ∈ ℓ ( M ) ⊗ T H, h ∈ H } . Recall that ρ : G → ℓ ( G ) is the right regular representation and that ρ r ⊗ µ λ,T ( C ∗ r ( M )) . Recall also that V : ℓ ( M ) ⊗ T H → ℓ ( G, H )is an isometry such that V ( z ⊗ h )( t ) = T z ( t ) h. Take x = ρ r ⊗ ◦ V u ∈ X and y = ρ r ⊗ ◦ V v ∈ X. Then, by [26, Proposition 2.6],lim j h x, ˜ µ λ,T ( f j ) y i = lim j h V u, ( ρ r − ⊗ µ λ,T ( f j )( ρ s ⊗ V v i = lim j h V u, ( ρ r − s ⊗ V ( λ M ( f j ) ⊗ v i = h V u, ( ρ r − s ⊗ V ( R ⊗ v i . This not only shows that (˜ µ λ,T ( f j )) j ∈ J converges in the weak (and weak*) topol-ogy, but also that its limit is completely determined by R = w ∗ lim j ˜ λ w ∗ M ( f j ) . Ofcourse, we define ˜ µ w ∗ λ,T ( R ) := S. Define V r := ( ρ r ⊗ ◦ V. Then ˜ µ w ∗ λ,T : W ∗ r ( M ) → L ( ℓ ( G, H )) is uniquely deter-mined by the condition(5.10) h V r u, ˜ µ λ,T ( R ) V s v i = h V s − r u, V ( R ⊗ v i , ∀ u, v ∈ ℓ ( M ) , r, s ∈ G. This condition immediately implies that ˜ µ w ∗ λ,T is linear and w ∗ -continuous in anyclosed ball. Moreover, it is also straightforward to prove that ˜ µ w ∗ λ,T preserves theinvolution. To show that ˜ µ w ∗ λ,T is multiplicative take R, S ∈ W ∗ r ( M ) and boundednets ( f j ) j ∈ J , ( g l ) l ∈ L ⊂ C ∗ r ( M ) weak* converging to R and S, respectively. Then,using that multiplication is separately weakly continuous, we deduce˜ µ w ∗ λ,T ( RS ) = lim j ˜ µ w ∗ λ,T ( f j S ) = lim j lim l ˜ µ w ∗ λ,T ( f j g l ) = lim j lim l ˜ µ w ∗ λ,T ( f j )˜ µ w ∗ λ,T ( g l )= lim j ˜ µ w ∗ λ,T ( f j )˜ µ w ∗ λ,T ( S ) = ˜ µ w ∗ λ,T ( R )˜ µ w ∗ λ,T ( S ) . Assume T | B e is faithful and ˜ µ w ∗ λ,T ( R ) = 0 . Then (5.10) implies (with r = s = e )that h u, ( R ⊗ v i = 0 for all u, v ∈ ℓ ∗ ( M ) ⊗ T | Be H. Since T | B e is faithful, we have R = 0 . (cid:3) Definition 5.11.
Let M be a W ∗ -Fell bundle over the discrete group G. We saythat the subset
N ⊂ M is a W ∗ -Fell subbundle if it is a Fell subbundle and the w ∗ topology of N e is the restriction of the w ∗ topology of M e . Since each fiber of N is a W ∗ -equivalence bimodule between W ∗ -ideals of N e , the definition above actually implies that the w ∗ topology of each fiber N t is therestriction of the w ∗ topology of B t . MENABILITY FOR PARTIAL ACTIONS AND FELL BUNDLES 19
Proposition 5.12.
Let
N ⊂ M be a W ∗ -Fell subbundle. If we view C ∗ r ( N ) asa C ∗ -subalgebra of C ∗ r ( M ) ⊂ W ∗ r ( M ) as in [2, Proposition 3.2] , then W ∗ r ( N ) isisomorphic to the w ∗ -closure of C ∗ r ( N ) in W ∗ r ( M ) . Proof.
Our proof is a slight modification of that of [2, Proposition 3.2]. By Propo-sition 5.8 there exists a representation T : M → L ( H ) with T | M e faithful andw ∗ -continuous on each fiber. Define H := T N H, where 1 N is the unit of N e , andthe restriction map R : N → L ( H ) by R a := T a | H . Then R is a representationw ∗ -continuous on each fiber and with R | N e faithful.In terms of the decomposition ℓ ( G, H ) = ℓ ( G, H ) ⊕ ℓ ( G, H ) ⊥ , we have(5.13) µ λ,T ( a ) = (cid:18) µ λ,R ( a ) 00 0 (cid:19) , ∀ a ∈ N . We get the desired result by considering the integrated forms of µ λ,T and µ λ,R and the respective w ∗ − continuous extensions to W ∗ r ( M ) and W ∗ r ( N ) , respectively. (cid:3) Remark . If we add to the hypotheses of the last theorem the condition that N is hereditary in M (that is N MN ⊂ N ) then W ∗ r ( N ) is hereditary in W ∗ r ( M ) . Indeed, it follows from separate w ∗ -continuity of the product and the fact that C ∗ r ( N ) is hereditary in C ∗ r ( M ) . The W*-algebra of kernels.
Let M be a W*-Fell bundle over a group G .A kernel of M is a function k : G × G → M such that k ( r, s ) ∈ M rs − . As usual wedenote by k ( M ) the C ∗ -algebra of kernels of M and k c ( M ) the kernels of compactsupport, see [2] for more details. Recall that there exists a canonical action of G on k ( M ), given by β t k ( r, s ) = k ( rt, st ). We are going to define a W ∗ -version of k ( M )and also of β .Consider the canonical representation π : k ( M ) → L ( ℓ ( M )) given by π ( k ) f ( s ) = P s ∈ G k ( s, t ) f ( t ) for every k ∈ k c ( M ) and f ∈ ℓ ( M ) with finite support. Thisrepresentation has been already considered in [2]. Using the canonical embed-ding L ( ℓ ( M )) ֒ → L ( ℓ ∗ ( M )), we may view π as a representation π : k ( M ) →L ( ℓ ∗ ( M )).With the canonical action of G on k ( M ) we construct the *-homomorphism π β : k ( M ) → ℓ ∞ ( G, L ( ℓ ∗ ( M ))) defined by π β ( f ) | t = π ( β − t ( f )). Note that π β isequivariant with respect to the translation W ∗ -action γ on ℓ ∞ ( G, L ( ℓ ∗ ( M ))).Recall that we may view the algebra of (generalised) compact operators K ( M ) := K ( ℓ ( M )) as an ideal of k ( M ) and β is the enveloping action of β | K ( M ) . In the C ∗ -case we know that π : k ( M ) → L ( ℓ ( M )) is the identity when restricted to K ( M ). In particular π : k ( M ) → L ( ℓ ∗ ( M )) is injective on K ( M ).Now we can prove that π β is injective. Indeed, π β ( f ) = 0 implies that π ( β t ( f ) x ) =0 for every t ∈ G and x ∈ K ( M ) and since π is faithful on K ( M ), this implies f β t ( x ) = 0 for every t ∈ G and x ∈ K ( M ), and this is equivalent to f = 0 becausethe linear G -orbit of K ( M ) is dense in k ( M ).Let k w ∗ ( M ) and K w ∗ ( M ) be the w ∗ -closures of π β ( k ( M )) and π β ( K ( M )), re-spectively. Then clearly K w ∗ ( M ) is a W ∗ -ideal of k w ∗ ( M ) and β w ∗ := γ | k w ∗ ( M ) isthe W ∗ -enveloping action of β w ∗ | K w ∗ ( M ) = γ | K w ∗ ( M ) .Our construction implies that β w ∗ is a quotient of β ′′ . This quotient is suchthat we can faithfully view β as a restriction of β w ∗ . Notice that K ( ℓ ∗ ( M )) is w ∗ -dense in N := L ( ℓ ∗ ( M )) (this follows, for instance, from [13, Lemma 8.5.23]). We claim that K w ∗ ( M ) is canonically isomorphic to N . Indeed, the evaluation at e ∈ G , ev e : ℓ ∞ ( G, N ) → N, is a surjective w ∗ -continuous *-homomorphism. Moreover, ev e is injective when restricted to K w ∗ ( M ) because ev e ◦ π β | K w ∗ ( M ) is just π | K w ∗ ( M ) .Thus ev e | K w ∗ ( M ) is an isomorphism between K w ∗ ( M ) and N = L ( ℓ ∗ ( M )). Definition 5.15.
The W ∗ -algebra k w ∗ ( M ) constructed above will be called the W ∗ -algebra of kernels of M . It will be always endowed with the canonical W ∗ -ac-tion β w ∗ of G defined above. Definition 5.16.
We say that two W ∗ -Fell bundles are weakly W ∗ -equivalent ifthe canonical actions on their W ∗ -algebras of kernels are W ∗ -Morita equivalent. Remark . W ∗ -equivalence of W ∗ -Fell bundles is an equivalence relation because,as in the C ∗ -case, we have inner tensor products of W ∗ -equivalence bimodules. Theorem 5.18.
Let M be a W ∗ -Fell bundle over a group G . Then the W ∗ -en-veloping action of the central partial action σ of M is the restriction of β w ∗ to thecentre of k w ∗ ( M ) .Proof. By Proposition 2.8, β w ∗ | Z ( k w ∗ ( M )) is the W ∗ -enveloping action of τ := β w ∗ | Z ( K w ∗ ( M )) . Hence all we need is to show that τ is isomorphic to σ .The module ℓ ∗ ( M ) is a W ∗ -equivalence bimodule between K w ∗ ( M ) and M e ,hence it induces a W ∗ -isomorphism µ : Z ( M e ) → Z ( K w ∗ ( M )) which we claim is anisomorphism of W ∗ -partial actions between σ and τ .To simplify our notation we write ZM and Z M instead of Z ( M e ) and Z ( K w ∗ ( M )),respectively. Consequently, the domains of σ and τ will be denoted ZM t and Z M t for t ∈ G .We must show that µ ( ZM t ) = Z M t or, equivalently, that ℓ ∗ ( M ) induces theideal I t = span w ∗ M t M ∗ t to J t := K w ∗ ( M ) ∩ β w ∗ t ( K w ∗ ( M )). From the proof of[5, Theorem 3.5] we know that ℓ ( M ) ⊆ ℓ ∗ ( M ) induces span kk M t M ∗ t to K ( M ) ∩ β t ( K ( M )). By taking w ∗ -closures in M e and ℓ ∞ ( G, L ( ℓ ∗ ( M ))), respectively, weget the desired induction.The composition µ ◦ σ t equals the composition µ | t ◦ σ t , where µ | t representsthe restriction and co-restriction of µ to ZM t (in the domain) and Z M t (in theco-domain). But µ | t is the isomorphism corresponding to the bimodule X t := J t ℓ ∗ ( M ) I t = ℓ ∗ ( M ) I t = J t ℓ ∗ ( M ) . Hence we may view µ ◦ σ t as the isomorphism corresponding to the bimodule X t ⊗ w ∗ I t M t . In the same way we may view τ t ◦ µ as the isomorphism correspondingto the bimodule J t δ t ⊗ w ∗ J t − δ e X t − , where J t δ t is the fiber over t of the semidirectproduct bundle of β w ∗ | K w ∗ ( M ) , B w ∗ , and J t − δ e is the ideal J t − seen as an idealof the unit fiber of that bundle. Once again we will make use of the C ∗ -version ofall these constructions.The semidirect product bundle of β | K ( ℓ ( M )) will be denoted B , and we willthink of it as a Fell subbundle of B ′′ . The fibre over t of B is K ( M ) t δ t and K ( M ) t δ t ( K ( M ) t δ t ) ∗ = K ( M ) t δ e ⊆ K ( M ) δ e Define I kk t and J kk t as the C ∗ -algebras generated by M t M ∗ t and( K ( M ) t δ t )( K ( M ) t δ t ) ∗ in M e and K ( M ) δ e , respectively. If we set X kk t := J kk t ℓ ( M ) I kk t = ℓ ( M ) I kk t = J kk t ℓ ( M ) , MENABILITY FOR PARTIAL ACTIONS AND FELL BUNDLES 21 then X kk t ⊗ I kk t M t and K ( M ) t δ t ⊗ J kk t − X t − are isomorphic as C ∗ -trings. Toprove this claim consider the canonical L -bundle of M , LM = { L t } t ∈ G , whichestablishes a strong equivalence between B and M [5]. Then X kk t is exactly J kk t L e = L e I kk t = J kk t L e I kk t , and we have canonical injective maps ν : X kk t ⊗ I kk t M t → L t , x ⊗ y xy,ν : K ( M ) t δ t ⊗ J kk t − X t − → L t , T ⊗ x T x, where the actions used are the actions of B and M on LM . The images of ν and ν are L e M t ⊆ L t and K ( M ) t δ t L e ⊆ L t , respectively, because M t = I kk t M t and K ( M ) t = K ( M ) t J kk t − (due to Cohen-Hewitt Theorem we do not need closed linearspans here). Recalling the definition of strong equivalence and understanding theproducts below as norm closed linear spans of products, we obtain: L e M t = L e M t M ∗ t M t ⊆ L e h L t − , L t − i M M t ⊆ LB h L e , L t − i L t − M t ⊆ LB h L e , L t − i L e ⊆ K ( M ) t δ t L e ⊆ . . . ⊆ L e M t . It can be directly shown that ν rj and ν lj ( j = 1 ,
2) are the natural inclusions of I kk t and J kk t on M e and K ( M ) δ e . This is due to the fact that we are allowed to use theinner products of LM in the computations of the tensor product. Hence ν − ◦ ν : X kk t ⊗ I kk t M t → K ( M ) t δ t ⊗ J kk t − X t − is an isomorphism of ternary C ∗ -trings with ( ν − ◦ ν ) r and ( ν − ◦ ν ) l being theidentities on I kk t − and J kk t , respectively.The question is now if we can extend ν − ◦ ν to an isomorphism(5.19) ν − ◦ ν : X t ⊗ w ∗ I t M t → J t δ t ⊗ w ∗ J t − δ e X t − In fact we can follow the same line of reasoning we used when constructing theinner W ∗ -tensor product (see the construction preceding Definition B.3). The ideais to represent the W ∗ -equivalence modules of (5.19) using the same representationof I t − and then to translate wot continuity into continuity of inner products. Atthat point everything will follow immediately because ( ν − ◦ ν ) r is the identityoperator.After constructing the isomorphism ν − ◦ ν as a w ∗ -extension of ν − ◦ ν itfollows directly that ν − ◦ ν r and ν − ◦ ν l are the identities (on J t and I t , respec-tively). Now the isomorphism in (5.19) and Corollary A.10 imply µ ◦ σ t = τ t ◦ µ . (cid:3) Corollary 5.20.
For a W ∗ -Fell bundle M the following are equivalent:(i) The canonical action β w ∗ on k w ∗ ( M ) is W*AD-amenable.(ii) The restriction of β w ∗ to Z ( K w ∗ ( M )) is W*AD-amenable.(iii) The central partial action of M is W*AD-amenable.Proof. Follows at once from the definition of W*AD-amenability of partial actions,Theorem 5.18 and Theorem 3.3. (cid:3)
Definition 5.21. A W ∗ -Fell bundle is said to be W*AD-amenable if the equivalentconditions of the corollary above are satisfied. A Fell bundle B is AD-amenable ifthe enveloping W ∗ -Fell bundle B ′′ is W*AD-amenable. Remark . W*AD-amenability is preserved by weak equivalence of W ∗ -Fell bun-dles. Remark . Proposition 3.5 and Example 5.6 imply that a W*-partial action isW*AD-amenable if and only if its semidirect product bundle (which is a W*-Fellbundle) is W*AD-amenable. Hence the same conclusion holds for C*-partial actionsand AD-amenability.
Theorem 5.24.
Let B be a Fell bundle over a group and let B ′′ be the enveloping W ∗ -Fell bundle of B . Then the canonical action β w ∗ on k w ∗ ( B ′′ ) and the bidual ofthe canonical action β on k ( B ) , β ′′ , are isomorphic as W ∗ -actions. In particular, k w ∗ ( B ′′ ) is isomorphic to k ( B ) ′′ .Proof. We first show that ℓ ∗ ( B ′′ ) ∼ = ℓ ( B ) ′′ as W*-Hilbert M e -modules. This isthe crucial point if we follow the original construction of k w ∗ ( B ′′ ) at the beginningof Section 5.4.Let ρ : B e → L ( H ) be the universal representation; we extend it to the bidualand view it as a faithful W*-representation ρ ′′ : B ′′ e → L ( H ). We may then view ℓ ∗ ( B ′′ ) as a wot-closed subspace of L ( H, ℓ ∗ ( B ′′ ) ⊗ ρ ′′ H ). But ℓ ∗ ( B ′′ ) ⊗ ρ ′′ H = ℓ ( B ) ⊗ ρ H =: K and we have a faithful representation U : ℓ ∗ ( B ′′ ) → L ( H, K ) suchthat U ( x ) h = x ⊗ h . Moreover, ℓ ∗ ( B ′′ ) is the wot-closure of U ( ℓ ( B )), i.e. ℓ ( B ) ′′ .Looking at the linking algebra L of ℓ ( B e ), we may view K w ∗ ( G, B ′′ ) as the W ∗ -completion of K ( B ) in L ′′ . But this closure is also equal to K ( B ) ′′ . Now wehave( β ′′ | K ( B ) ′′ ) | K ( B ) = β ′′ | k ( B ) | K ( B ) = β | K ( B ) = β w ∗ | k ( B ′′ ) | K ( B ) = ( β w ∗ | K ( B ) ′′ ) | K ( B ) . Hence we have two W ∗ -partial actions on K ( B ) ′′ , namely β ′′ | K ( B ) ′′ and β w ∗ | K w ∗ ( B ′′ ) ,which are the unique W ∗ -actions extending the C ∗ -partial action β | K ( B ) . There-fore β ′′ | K ( B ) ′′ = β w ∗ | K w ∗ ( B ′′ ) . But β ′′ and β w ∗ are both W ∗ -enveloping actions of β ′′ | K ( B ) ′′ , then uniqueness of W ∗ -enveloping actions implies that β ′′ is isomorphicto β w ∗ . (cid:3) Corollary 5.25.
If two Fell bundles A and B over a group are weakly equivalentthen their enveloping W ∗ -Fell bundles A ′′ and B ′′ are weakly W ∗ -equivalent. Inparticular, AD-amenability of Fell bundles is preserved by weak equivalence of Fellbundles.Proof. The canonical partial actions on k ( A ) and k ( B ), α and β respectively, areMorita equivalent through a partial action γ on a k ( A )- k ( B )-equivalence bimodule X . If L is the linking partial action of X and ν the linking partial action of γ , thenthe w ∗ -closure of X in L ′′ and the restriction γ ′′ := ν ′′ | X ′′ provide a W ∗ -equivalencebetween α ′′ and β ′′ . The rest follows from Theorem 5.24 and Definitions 5.16and 5.21. (cid:3) Corollary 5.26.
A Fell bundle B is AD-amenable if and only if the canonicalaction on k ( B ) , β, is AD-amenable.Proof. Just recall that B is weakly equivalent to B β [5, 6] and use the Corollaryabove. (cid:3) The dual coaction: another picture for the W*-algebra of kernels.
Inthis section we want to show that the sectional W ∗ -algebra W ∗ r ( M ) of a W ∗ -Fellbundle M over G carries a canonical G -coaction and identify the crossed productby this coaction with the W ∗ -algebra of kernels k w ∗ ( M ). MENABILITY FOR PARTIAL ACTIONS AND FELL BUNDLES 23
Recall that a coaction of G on a W ∗ -algebra N is a faithful unital W ∗ -homo-morphism δ : N → N ¯ ⊗ W ∗ r ( G ) satisfying ( δ ⊗ id) δ = (id ⊗ δ G ) δ , were ¯ ⊗ denotes the(spatial) tensor product of W ∗ -algebras. Given such a coaction, the W ∗ -crossedproduct is defined as the W ∗ -subalgebra of N ¯ ⊗L ( ℓ ( G )) generated by δ ( N ) and1 ⊗ ℓ ∞ ( G ), where, as usual, ℓ ∞ ( G ) is represented as a W ∗ -subalgebra of L ( ℓ G )via multiplication operators. We omit this representation here for simplicity, thatis, we view ℓ ∞ ( G ) as a subalgebra of L ( ℓ G ). It turns out that N ¯ ⋊ δ G = span w ∗ { δ ( n )(1 ⊗ f ) : n ∈ N, f ∈ ℓ ∞ ( G ) } . Representing N on a Hilbert space or, more generally, on a self-dual Hilbert mod-ule H , the W ∗ -crossed product N ¯ ⋊ δ G gets represented as a W ∗ -subalgebra of L ( H ) ¯ ⊗L ( ℓ G ) = L ( H ⊗ ℓ G ). This crossed product carries a canonical G -action,the so called dual action b δ . It is given on a generator δ ( n )(1 ⊗ f ) by b δ t ( δ ( n )(1 ⊗ f )) = δ ( n )(1 ⊗ τ t ( f )), where τ t ( f )( s ) := f ( st ) denotes the right translation G -actionon ℓ ∞ ( G ). This can also be described as b δ t ( x ) = (1 ⊗ ρ t ) x (1 ⊗ ρ − t ), where ρ : G → L ( ℓ G ) denotes the right regular representation of G .Now, returning to the case of a W ∗ -Fell bundle M , we want to define a coaction δ M : W ∗ r ( M ) → W ∗ r ( M ) ¯ ⊗ W ∗ r ( G ) that acts on generators Λ t ( a ) ∈ M t with a ∈ M t by the formula(5.27) δ M (Λ t ( a )) = Λ t ( a ) ⊗ λ t . This is therefore an extension of the usual dual coaction on C ∗ r ( M ) ⊆ W ∗ r ( M ).Here W ∗ r ( G ) denotes the group W ∗ -algebra of G , that is, the W ∗ -subalgebra of L ( ℓ G ) generated by the left regular representation λ : G → L ( ℓ G ) . To prove that δ M exists, we proceed as in the C ∗ -algebra situation (see [2,Section 8] or [26]): Let M × G be the pullback of M along the first coordinateprojection G × G → G . This is a W ∗ -Fell bundle over G × G whose W ∗ -algebra iscanonically isomorphic to W ∗ r ( M × G ) = W ∗ r ( M ) ¯ ⊗ W ∗ r ( G ), in particular we havea canonical W ∗ -embedding W ∗ r ( M ) ¯ ⊗ W ∗ r ( G ) ⊆ L ( ℓ ∗ ( M × G )) . Now we define a unitary operator V on the Hilbert W ∗ -module ℓ ∗ ( M × G ) by theformula V ζ ( s, t ) := ζ ( s, s − t ) for all ζ ∈ ℓ ∗ ( M × G ) , s, t ∈ G. Straightforward computations show that this is indeed a unitary operator withadjoint V ∗ ζ ( s, t ) = ζ ( s, st ). Now we define a w ∗ -continuous injective unital homo-morphism δ M : L ( ℓ ∗ ( M )) → L ( ℓ ∗ ( M × G )) by δ M ( a ) := V ( a ⊗ V ∗ , a ∈ W ∗ r ( M ) . It is easy to see that (5.27) is satisfied. Moreover, since the elements Λ t ( a ) with a ∈ M t generate W ∗ r ( M ) as a W ∗ -algebra, the above formula restricts to an injectivew ∗ -continuous unital homomorphism δ M : W ∗ r ( M ) → W ∗ r ( M ) ¯ ⊗ W ∗ r ( G ) . This is indeed a coaction, that is, the coassociativity identity ( δ M ⊗ id) ◦ δ M = (id ⊗ δ G ) ◦ δ M holds, where δ G : W ∗ r ( G ) → W ∗ r ( G ) ¯ ⊗ W ∗ r ( G ) denotes the comultiplicationof W ∗ r ( G ) (which, incidentally, is the coaction δ M for the trivial one-dimensionalFell bundle M = C × G ). Remark . There is a canonical normal conditional expectation E : W ∗ r ( M ) ։ M e given on generators by E (Λ( a )) = δ t,e ( a ) for all a ∈ M t . This can be provedas in the C ∗ -case, or it can be deduced from the existence of the dual coaction δ M above as follows: Consider the canonical tracial state τ : W ∗ r ( G ) → C given by τ ( x ) = h δ e | xδ e i . Then E = (id ⊗ τ ) ◦ δ M is the desired conditional expectation. Proposition 5.29.
For a W ∗ -Fell bundle M , we have a canonical isomorphism W ∗ r ( M ) ¯ ⋊ δ M G ∼ = k w ∗ ( M ) , that identifies a generator δ M ( a )(1 ⊗ f ) ∈ W ∗ r ( M ) ¯ ⋊ δ M G with the kernel k a,f ( s, t ) := a ( st − ) f ( t ) for a ∈ C c ( M ) and f ∈ ℓ ∞ ( G ) . This isomorphism is G -equivariantwith respect to the dual G -action on W ∗ r ( M ) ¯ ⋊ δ M G and the canonical G -action on k w ∗ ( M ) .Proof. Let N := W ∗ r ( M ) and δ := δ M . We show how to turn ℓ ∗ ( M ) into a W ∗ -Hilbert N ¯ ⋊ δ G - M e -bimodule.Consider the map ι : C c ( M ) → N ⊗ alg C c ( G ) ⊆ N ¯ ⊗ ℓ ( G ) defined by ι ( ξ ) = δ ( ξ )(1 ⊗ δ e ) = P s ∈ G Λ( ξ ( s )) ⊗ δ s . Here and throughout this proof ( δ s ) s ∈ G will alsodenote the standard ortonormal basis of ℓ ( G ) – apologies for the overuse of thesymbol δ here! Let X be the w ∗ -closure of the image of ι in N ¯ ⊗ ℓ ( G ). Notice thatwith respect to the N -inner product on N ¯ ⊗ ℓ ( G ) we have h ι ( ξ ) | ι ( η ) i N = X s,t ∈ G h Λ( ξ ( s ) ⊗ δ s ) | Λ( η ( t )) ⊗ δ t i = Λ( h ξ | η i M e ) , for all ξ, η ∈ C c ( M ), where h ξ | η i M e denotes the M e -valued inner product on C c ( M ) ⊆ ℓ ∗ ( M ). Since Λ is a W ∗ -embedding M e ֒ → N , it follows that theimage of the N -valued inner product on X takes values in Λ( M e ) ∼ = M e so that X can be viewed as a right W ∗ -Hilbert M e -module and ι extends to an isomorphism ℓ ∗ ( M ) ∼ = X of W ∗ -Hilbert M e -modules. The advantage of this picture is that X is also canonically a left W ∗ -Hilbert N ¯ ⋊ δ G -module, where the left inner productis defined by I h ξ | η i := δ ( ξ )(1 ⊗ χ e ) δ ( η ∗ ) ∈ I for ξ, η ∈ C c ( M ). The image of thisinner product generates a W ∗ -ideal I of N ¯ ⋊ δ G , namely the W ∗ -ideal generated bythe projection p := χ e . It follows that I ∼ = L ( ℓ ∗ ( M )); this isomorphism identifies δ ( ξ )(1 ⊗ p ) δ ( η ∗ ) with θ ξ,η = | ξ ih η | ∈ K ( ℓ ∗ ( M )) ⊆ L ( ℓ ∗ ( M )), and it is determinedby this formula and the fact that it is w ∗ -continuous.Next, considering the dual G -action b δ on Q := N ¯ ⋊ δ G , we notice that the linear G -orbit of I is w ∗ -dense. This is because b δ t − ( χ e ) = χ t , so that b δ t − ( I ) is the W ∗ -ideal of N ¯ ⋊ δ G generated by the projection p t = χ t , and these projectionsgenerate ℓ ∞ ( G ) as a W ∗ -algebra. Therefore b δ can be viewed as the W ∗ -envelopingaction of its restriction b δ | I . On the other hand, the G -action on the W ∗ -algebra ofkernels k w ∗ ( M ) is also enveloping for a partial action on L ( ℓ ∗ ( M )). By uniquenessof enveloping W ∗ -actions (Proposition 2.7), to see that k w ∗ ( M ) ∼ = Q , it is enoughto see that the restriction of b δ to I coincides with the partial action on L ( ℓ ∗ ( M ))obtained as restriction of the G -action β w ∗ on k w ∗ ( M ). But by definition, β w ∗ isthe unique w ∗ -continuous extension of the G -action β on the C ∗ -algebra of kernels k ( M ) given by β r ( k )( s, t ) = k ( sr, tr ) for a kernel k ∈ k c ( M ). An elementarycompact operator θ ξ,η ∈ K ( ℓ ( M )) ⊆ K ( ℓ ∗ ( M )) identifies with the kernel function k ξ,η ( s, t ) := ξ ( s ) η ( t ) ∗ . And by [2, Proposition 8.1] we have a C ∗ -isomorphism k ( M ) ∼ = B := C ∗ r ( M ) ⋊ δ G that is G -equivariant for the dual G -action b δ on B and β on k ( M ). Here δ also denotes the dual coaction of G on C ∗ r ( M ); this is MENABILITY FOR PARTIAL ACTIONS AND FELL BUNDLES 25 a restriction of the dual coaction on N = W ∗ r ( M ), denoted by the same symbol δ . The isomorphism k ( M ) ∼ = B is given as in the statement (see the proof ofProposition 8.1 in [2]). The C ∗ -algebra of compact operators K ( ℓ ( M )) identifies,as above, with the C ∗ -ideal J of B generated by p = χ e . This is w ∗ -dense in I . Sincethe partial G -action on J we get from viewing it as an ideal of k ( M ) coincides withthe partial action coming from the dual action on B , the same has to be true forthe w ∗ -closures, that is, via the isomorphism L ( ℓ ∗ ( M )) ∼ = I the partial action on I we get by restriction of b δ is the partial action we get from k w ∗ ( M ) by restrictingit to the W ∗ -ideal L ( ℓ ∗ ( M )). (cid:3) Corollary 5.30.
For every W*-Fell bundle M we have a canonical isomorphism k w ∗ ( M ) ⋊ β w ∗ G ∼ = W ∗ r ( M ) ¯ ⊗L ( ℓ G ) . Proof.
This follows from Proposition 5.29 and general duality theory for crossedproducts by W*-coactions, see [30]. (cid:3)
We recall from [5] that given a Fell subbundle A of B we can identify k ( A ) withthe norm closure of k c ( A ) in k ( B ) . This inclusion has a W ∗ -counterpart. Corollary 5.31. If N is a W ∗ -Fell subbundle of M and we view k ( N ) as a C ∗ -sub-algebra of k ( M ) ⊂ k w ∗ ( M ) , then k w ∗ ( N ) is isomorphic to the w ∗ − closure of k ( N ) in k w ∗ ( M ) . Proof.
The inclusion k w ∗ ( N ) ⊂ k w ∗ ( M ) is just the inclusion W ∗ r ( N ) ¯ ⋊ δ N G ⊂ W ∗ r ( M ) ¯ ⋊ δ M G provided by Proposition 5.29. (cid:3) Exel’s approximation property and AD-amenability
The main goal of this section is to compare the notion of amenability in the senseof Anantharaman-Delaroche with the approximation property introduced by Exelin [22]. We start by recalling Exel’s approximation property:
Definition 6.1.
A Fell bundle B = { B t } t ∈ G has the approximation property (AP)if there exists a net { a i } i ∈ I of functions a i : G → B e with finite support such that(i) sup i ∈ I k P r ∈ G a i ( r ) ∗ a i ( r ) k < ∞ .(ii) For every t ∈ G and b ∈ B t , lim i k b − P r ∈ G a i ( tr ) ∗ ba i ( r ) k = 0.A partial action α on a C*-algebra has the AP if the semidirect product bundle B α has the AP. Remark . Notice that (1) above means that { a i } i ∈ I is a bounded net when viewedas a net in the Hilbert B e -module ℓ ( G, B e ). Indeed, the original definition of theAP in [22] uses such nets and Proposition 4.5 in [22] says that both definitions areequivalent (the difference being whether the supports of the functions are requiredto be finite or not).Condition (2) can also be weakened: it is enough to check the norm convergencein (2) for b in total subsets of B t , that is, for b in a subset B t spanning a norm-densesubset of B t for each t ∈ G .As a way of combining Exel’s approximation property and amenability in thesense of Anantharaman-Delaroche ([7, 8, 10]), we introduce the following: Definition 6.3. A W ∗ -Fell bundle M = { M t } t ∈ G has the W ∗ -approximationproperty (W*AP) if there exists a net of functions { a i : G → M e } i ∈ I with finitesupport such that (i) sup i ∈ I k P r ∈ G a i ( r ) ∗ a i ( r ) k < ∞ , and(ii) for every t ∈ G and b ∈ M t ,lim i X r ∈ G a i ( tr ) ∗ ba i ( r ) = b in the w ∗ -topology of M t .We say that a W ∗ -partial action γ has the W*AP if the associated W ∗ -Fell bundle B γ has the W*AP.A ( C ∗ -algebraic) Fell bundle B has the WAP if its W ∗ -enveloping Fell bundle B ′′ has the W*AP and a C ∗ -partial action α has the WAP if α ′′ has the W*AP.The reader should read WAP as “weak approximation property”, the reason forthis will be clear after Theorem 6.11. We recommend to read the statement of thattheorem at this point to get a feeling of what we want to do next.Remark 5.3 implies that a C ∗ -partial action α has the WAP if and only if B α has the WAP. We shall prove in what follows that AD-amenability and the WAPare equivalent notions, first for global actions and later also for general Fell bun-dles. This is not trivial, even for global actions, because the AD-amenability of aglobal action requires the existence of a certain net that takes central values (seeTheorem 3.3) while for the WAP this is not explicitly necessary (Definition 6.3).Let γ be a (global) action of G on the W ∗ -algebra N . As usual, we write ˜ γ forthe action of G on ℓ ∞ ( G, N ) given by ˜ γ t ( f )( r ) = γ t ( f ( t − r )) and view N as thesubalgebra of constant functions in ℓ ∞ ( G, N ). Abusing the notation we also usethe same notation for the G -action on functions f ∈ ℓ ( G, N ). The following resultgives an explicit characterisation of the W*AP for global actions.
Proposition 6.4.
Let γ be a global action of G on a W*-algebra N . Then γ has theW*AP if and only if there exists a net { a i } i ∈ I of finitely supported functions a i : G → N such that { a i } i ∈ I is bounded in ℓ ( G, N ) and {h a i , b ˜ γ t ( a i ) i } i ∈ I w ∗ -converges to b for all b ∈ N and t ∈ G .Proof. We view the fibre of B γ at t as N δ t and denote its elements by xδ t . Withthis notation xδ t yδ s = xγ t ( y ) δ ts and ( xδ t ) ∗ = γ t − ( x ∗ ) δ t − . Viewing x ∈ N as xδ e ,we can then think of a function a : G → N as a function from G to the unit fibre N δ e ≡ N . If a : G → N has finite support, then for every t ∈ G and b ∈ N we have: P r ∈ G ( a ( r ) δ e ) ∗ ( a ( r ) δ e ) = P r ∈ G a ( r ) ∗ a ( r ) and X r ∈ G ( a ( tr ) δ e ) ∗ bδ t a ( r ) δ e = X r ∈ G a ( tr ) ∗ bγ t ( a ( r ))) δ t = X r ∈ G a ( r ) ∗ bγ t ( a ( t − r ))) δ t = h a, b ˜ γ t ( a ) i δ t . The proof follows directly from the computations above and from the fact thatunder the identification N → N δ t , x xδ t , the w ∗ -topology of N δ t is just thew ∗ -topology of N . (cid:3) In order to show that the W*AD-amenability is equivalent to W*AP for W*-Fellbundles we shall need the following result:
Lemma 6.5.
Let γ be a W*-global action of G on N. Then γ is AD-amenable ifand only if there exists a γ -invariant w ∗ -dense *-subalgebra A ⊂ N and a boundednet { a i } i ∈ I ⊂ ℓ ( G, N ) of functions with finite support such that for all b ∈ A and t ∈ G, {h a i , b ˜ γ t ( a i ) i } i ∈ I w ∗ -converges to b . MENABILITY FOR PARTIAL ACTIONS AND FELL BUNDLES 27
Proof.
The direct implication follows from Theorem 3.3 (with A = N ). For theconverse we view N as a concrete (unital) von Neumann algebra of operators onsome Hilbert space H , N ⊂ L ( H ) , and take a *-subalgebra A ⊆ N and a net { a i } i ∈ I as in the statement. For t ∈ G and i ∈ I we define ϕ ti : N → N by ϕ ti ( b ) := h a i | b ˜ γ t ( a i ) i . Then ( ϕ ti ) i ∈ I is a net of uniformly bounded linear maps(with uniform bound c := sup i k a i k < ∞ ). By assumption ϕ ti ( b ) → b in the weak*-topology for every b ∈ A and t ∈ G . A standard argument shows that the samehappens for all b in the norm closure of A which is then a (w*-dense) C*-subalgebraof N . Hence we may assume, without loss of generality, that A is already a C*-algebra. In particular we may assume that A is closed by continuous functionalcalculus and that Λ := { x ∈ A : 0 ≤ x ≤ } is an approximate unit for A and thusw ∗ -converges to 1 N (this follows from the assumption that A is w*-dense in N ).For each ( i, λ ) ∈ I × Λ we define P i,λ : ℓ ∞ ( G, N ) → N, P i,λ ( f ) = h λ / a i , f λ / a i i , where the product f λ / a i represents the diagonal action of f λ / ∈ ℓ ∞ ( G, N )on ℓ ( G, N ). Each P i,λ is a completely positive linear map with norm k P i,λ k = k λ / a i k ≤ c := sup i ∈ I k a i k < ∞ .Let K be the Hilbert space ℓ (Λ , H ) and define, for each i ∈ I, the function P i : ℓ ∞ ( G, N ) → L ( K ) , P i ( f ) g | λ = P i,λ ( f )( g | λ ) . If we view K = ℓ (Λ , H ) as the direct sum of Λ-copies of H , then P i ( f ) is the“diagonal” operator formed by the family ( P i,λ ) λ . Thus P i is completely positiveand k P i k ≤ c for all i ∈ I .The set of completely positive maps Q : ℓ ∞ ( G, N ) → L ( K ) with k Q k ≤ c iscompact with respect to the topology of pointwise w ∗ -convergence, thus there existsa completely positive map P : ℓ ∞ ( G, N ) → L ( K ) and a subnet { P i j } j ∈ J such that P ( f ) = lim j P i j ( f ) in the w ∗ -topology for every f ∈ ℓ ∞ ( G, N ) . By passing to asubnet we may therefore assume that { P i } i ∈ I converges to P for the pointwisew ∗ -topology.As a consequence of the last paragraph we get that, for each f ∈ ℓ ∞ ( G, N ) and λ ∈ Λ , { P i,λ ( f ) } i ∈ I w ∗ -converges to some P λ ( f ) . In fact, the map P λ : ℓ ∞ ( G, N ) → N, f P λ ( f ) , is completely positive and k P λ k ≤ c for all λ ∈ Λ . For each λ ∈ Λ we define Q λ : ℓ ∞ ( G, Z ( N )) → N ⊆ L ( H ) as the restriction of P λ . We claim that { Q λ } λ ∈ Λ converges w ∗ -pointwise. Indeed, it suffices to provethat for each positive f ∈ ℓ ∞ ( G, Z ( N )) the net { Q λ ( f ) } λ ∈ Λ is increasing. Take λ, µ ∈ Λ with λ ≤ µ. Then, for every h ∈ H and i ∈ I : h h, P i,λ ( f ) h i = X r ∈ G h a i ( r ) h, λ / f ( r ) λ / a i ( r ) h i = X r ∈ G h a i ( r ) h, f / ( r ) λf / ( r ) a i ( r ) h i≤ X r ∈ G h a i ( r ) h, f / ( r ) µf / ( r ) a i ( r ) h i = h h, P i,µ ( f ) h i . Taking limit in i we get h h, Q λ ( f ) h i ≤ h h, Q µ ( f ) h i and it follows Q λ ( f ) ≤ Q µ ( f ) . Let Q : ℓ ∞ ( G, Z ( N )) → N be the pointwise w ∗ -limit of { Q λ } λ ∈ Λ . Let us prove that the image of Q is contained in Z ( N ) . It suffices to show thatfor f ∈ ℓ ∞ ( G, Z ( N )) + and a self-adjoint b ∈ A, Q ( f ) b is self-adjoint. Let { P λ j } j ∈ J be a pointwise w ∗ -convergent subnet of { P λ } λ ∈ Λ . Clearly, both f b and P ( f b ) are self-adjoint. We claim that Q ( f ) b = P ( f b ) . Fix h, k ∈ H. Using the inner productof ℓ ∗ ( G, N ) ⊗ N H in the following computations, we deduce |h h, ( Q ( f ) b − P ( f b )) k i| = lim j lim i |h λ / j a i ⊗ h, f (cid:16) λ / j a i ⊗ bk − bλ / j a i ⊗ k (cid:17) i|≤ lim j lim i √ c k h kk f kk λ / j a i ⊗ bk − bλ / j a i ⊗ k k . The double limit above is zero because lim j lim i k λ / j a i ⊗ bk − bλ / j a i ⊗ k k islim j lim i h bk, h a i , λ j a i i bk i + h k, h a i , λ / j b λ / j a i i k i−− lim j lim i h bk, h a i , λ / j bλ / j a i i k i − h k, h a i , λ / j bλ / j a i i bk i = lim j h bk, λ j bk i + h k, λ / j b λ / j k i − h bk, λ / j bλ / j k i − h k, λ / j bλ / j bk i = h bk, bk i + h k,b k i − h bk, bk i − h k, bbk i = 0 . This shows that Q ( f ) b = P ( f b ). From now on we think of Q as a completelypositive map from ℓ ∞ ( G, Z ( N )) to Z ( N ) . We claim that Q is a projection. Take a ∈ Z ( N ) . Then, in the wot topology: Q ( a ) = lim λ lim i P i,λ ( a ) = lim λ lim i h a i , λaa i i = lim λ lim i a h a i , λa i i = lim λ aλ = a. In particular Q (1) = 1 , so Q is a norm one projection. The proof will be completedonce we show Q is equivariant.Suppose we can prove, for all f ∈ ℓ ∞ ( G, Z ( N )) , that(6.6) Q λ (˜ γ t ( f )) = γ t ( Q γ t − ( λ ) ( f )) . Since { γ t ( λ ) } λ ∈ Λ is a subnet of { λ } λ ∈ Λ , if we take the w ∗ -limit in (6.6) we obtain Q (˜ γ t ( f )) = lim λ Q λ (˜ γ t ( f )) = γ t (lim λ Q γ t − ( λ ) ( f )) = γ t ( Q ( f )) . Hence the proof will be complete after we show (6.6).Fix f ∈ ℓ ∞ ( G, Z ( N )) + , λ ∈ Λ and t ∈ G. In the w ∗ -topology: Q λ (˜ γ t ( f )) = lim i h λ / a i , ˜ γ t ( f ) λ / a i i = lim i γ t ( h ˜ γ t − ( a i ) , γ t − ( λ ) f ˜ γ t − ( a i ) i )We know that the net {h ˜ γ t − ( a i ) , γ t − ( λ ) f ˜ γ t − ( a i ) i} i ∈ I w ∗ -converges. We only needto prove it wot-converges to Q γ t − ( λ ) ( f ) . To avoid the annoying inverse t − , wechange t by t − . Note that h ˜ γ t ( a i ) , γ t ( λ ) f ˜ γ t ( a i ) i = h f / ˜ γ t ( λ / a i ) , f / ˜ γ t ( λ / a i ) i is self-adjoint,and so it is Q γ t ( λ ) ( f ) . Thus, it suffices to show that, for all h ∈ H, lim i h h, h ˜ γ t ( a i ) , γ t ( λ ) f ˜ γ t ( a i ) i h i = h h, Q γ t ( λ ) ( f ) h i In any Hilbert space we have |k x k − k y k | ≤ ( k x k + k y k ) k x − y k . In particularwe use this inequality in ℓ ∗ ( G, N ) ⊗ N H :lim i |h h, h ˜ γ t ( a i ) ,γ t ( λ ) f ˜ γ t ( a i ) i h i − h h, Q γ t ( λ ) ( f ) h i| == lim i |k f / ˜ γ t ( λ / a i ) ⊗ h k − k f / γ t ( λ / ) a i ⊗ h k |≤ lim i k f k / k a i k k h kk f / (˜ γ t ( λ / a i ) − γ t ( λ / ) a i ) ⊗ h k≤ lim i k f kk a i k k h kk γ t ( λ / )(˜ γ t ( a i ) − a i ) ⊗ h k MENABILITY FOR PARTIAL ACTIONS AND FELL BUNDLES 29
Moreover, {k a i k } i ∈ I is bounded and the limit of {k γ t ( λ / )(˜ γ t ( a i ) − a i ) ⊗ h k } i ∈ I is lim i h h, γ t ( h a i , λa i i ) h i + h h, h a i , γ t ( λ ) a i i h i−− lim i h h, h ˜ γ t ( a i ) , γ t ( λ ) a i i h i + h h, h a i , γ t ( λ )˜ γ t ( a i ) i h i = h h, γ t ( λ ) h i + h h, γ t ( λ ) h i − h h, γ t ( λ ) h i − h h, γ t ( λ ) h i = 0 . This implies (6.6) and the proof is complete. (cid:3)
The next remark will be extremely useful to show that W*AD-amenability andthe W*AP are equivalent.
Remark . For a Fell bundle B = { B t } t ∈ G and an ordered set F = { t , . . . , t n } ⊂ G , the algebra M F ( B ) formed by the n × n matrices M = ( M i,j ) ni,j =1 such that M i,j ∈ B t i t − j , is a C*-algebra with usual matrix involution and multiplication [6, Lemma2.8]. The C*-norm is equivalent to k M k ∞ := max i,j k M i,j k and each B t i t − j may beisometrically identified with a subspace of M F ( B ). Moreover, M F ( B ) ′′ = M F ( B ′′ ) . Theorem 6.8.
A W*-Fell bundle is W*AD-amenable if and only if it has theW*AP.Proof.
Assume that the W*-Fell bundle M over the group G is W*AD-amenable.By Corollary 5.20 the central partial action γ on Z := Z ( M e ) is W*AD-amenable.Let δ be the W*-enveloping action of γ, acting on the commutative W*-algebra Y. We know that Z is a W*-ideal of Y and that δ is W*AD-amenable.Let { ξ i } i ∈ I ⊂ ℓ ( G, Y ) be a net for γ as in Theorem 3.3 and let p ∈ Y be theunit of Z. We define a i := ξ i p and claim that { a i } i ∈ I ⊂ ℓ ( G, Z ) is a net as in thedefinition of the W*AP.First of all note that { a i } i ∈ I is bounded because h a i , a i i = p h ξ i , ξ i i , for all i ∈ I. If p t is the unit of Z t = Z ∩ δ t ( Z ) , then for every t ∈ G and x ∈ M t we have (bythe definition of the central partial action):lim i X r ∈ G a i ( tr ) ∗ xa i ( r ) = lim i X r ∈ G a i ( tr ) ∗ xp t − a i ( r ) = lim i X r ∈ G a i ( tr ) ∗ γ t ( p t − a i ( r )) x = lim i X r ∈ G pξ i ( tr ) ∗ p t δ t ( p ) δ t ( ξ i ( r )) x = lim i p t h ξ i , ˜ δ t ( ξ i ) i x = p t x = x, where the limits are taken in the w ∗ -topology. This shows that M has the W*AP.Now assume that M has the W*AP. We will show that the canonical W*-action β w ∗ on k w ∗ ( M ) is W*AD-amenable using Lemma 6.5 and Theorem 3.3. We set γ := β w ∗ , N := k w ∗ ( M ) and A := k c ( M ) . Recall that N is a W*-completion of k ( M ) and that A is norm dense in k ( M ) . Hence A is w ∗ -dense in N. Moreover, A is γ invariant because A is invariant under the canonical action on k ( M ) . We claim that M F ( M ) is a W*-subalgebra of N. This is important because insuch a case the convergence in the topology of M F ( M ) relative to the w ∗ -topologyof N is just entrywise w ∗ -convergence on the matrix algebra M F ( M ) . Recall that we defined N = k w ∗ ( M ) as the w ∗ -closure of the image of the map π β : k ( M ) → ℓ ∞ ( G, L ( ℓ ∗ ( M ))) (see section 5.4). Thus it suffices to prove thatthe image of ρ : M F ( M ) → L ( ℓ ∗ ( M )) , ρ ( k ) f ( r ) = P s ∈ G k ( r, s ) f ( s ) , is a W*-subalgebra. Here we think of the matrix k as a kernel of compact support. For all f, g ∈ ℓ ∗ ( M ) the map M F ( M ) → M e , k
7→ h f, ρ ( k ) g i is w ∗ -continuous. Hence theclosed unit ball ρ ( M F ( M )) is w ∗ -compact by Lemma A.2, so we conclude that theimage of ρ is a W*-subalgebra.Take a net of functions { a j } j ∈ J as in the definition of W*AP for M . Let F bethe set of finite subsets of G and consider Ξ := F × J as a directed set with the order( U, j ) ≤ ( V, i ) ⇔ U ⊆ V and j ≤ i . For each ξ = ( U, j ) ∈ Ξ let a ξ : G → M U ( M )be such that for every r ∈ G , a ξ ( r ) is the diagonal matrix with all the entriesin the diagonal equal to a j ( r ). Note kh a ξ , a ξ ik = kh a j , a j ik . Observe also that γ t ( M U ( M )) = M Ut − ( M ).Fix t ∈ G and k ∈ A. Take a finite set U ⊆ G such that supp( k ) ⊆ U × U . If ξ = ( U, i ) ∈ Ξ is such that U ⊇ U ∪ U t , that is, U ⊆ U ∩ U t − , then h a ξ , k ˜ γ t ( a ξ ) i = X r ∈ G a ξ ( tr ) ∗ kγ t ( a ξ ( r )) , and a ξ ( tr ) ∗ kγ t ( a ξ ( r )) ∈ M supp( k ) ( M ). Moreover, considering the left and rightentrywise action of M e on M supp( k ) ( M ), a ξ ( tr ) ∗ kγ t ( a ξ ( r )) = a j ( tr ) ∗ ka j ( r ). It isthen clear that lim ξ h b ξ , k ˜ γ t ( b ξ ) i = k w ∗ -entrywise and hence w ∗ in N. (cid:3) Remark . In the proof above we incidentally showed that the net { a i } i ∈ I in theDefinition of the W*AP can be taken in the unit ball of ℓ ( G, Z ( M e )) , withoutaltering the definition. Corollary 6.10.
A W*-partial (resp. C*-partial) action has the W*AP (resp.WAP) if and only if it is W*AD-amenable (resp. AD-amenable).Proof.
Follows at once from the Theorem above and Remark 5.23. (cid:3)
Our next goal is to give alternative characterisations of the WAP for Fell bundles.
Theorem 6.11.
For every Fell bundle B over a group G the following are equiva-lent:(i) B is AD-amenable.(ii) B has the WAP.(iii) B ′′ is the W*AD-amenable.(iv) B ′′ has the W*AP.(v) There exists a bounded net { a i } i ∈ I ⊂ ℓ ( G, Z ( B ′′ e )) of functions with finitesupport such that, for every t ∈ G and b ∈ B t , lim i P r ∈ G a i ( tr ) ∗ ba i ( r ) = b in B ′′ t with respect to the w ∗ -topology.(vi) There exists a bounded net { a i } i ∈ I ⊂ ℓ ( G, B e ) of functions with finite supportsuch that, for every t ∈ G and b ∈ B t , lim i P r ∈ G a i ( tr ) ∗ ba i ( r ) = b in the weaktopology of B t . Proof.
The equivalences between (i) and (iii) and between (ii) and (iv), followdirectly from the definitions of AD and W*AD amenability, and of WAP and W*APproperties. We know from the previous theorem that B ′′ is W*AD-amenable if andonly if it has the W*AP, hence (iii) and (iv) are equivalent. By Remark 6.9, (iv)and (v) are equivalent. To prove that (vi) implies (iv) we can proceed exactlyas in the proof of the converse in Theorem 6.8, noticing that convergence in theweak topology of M F ( B ) is entrywise convergence in the weak topology and, also,w ∗ -convergence in M F ( B ) ′′ = M F ( B ′′ ) . We now prove that (iv) implies (vi). First we indicate how to approximate ele-ments of ℓ ( G, B ′′ e ) by elements of ℓ ( G, B e ) in a certain particular way. We start by MENABILITY FOR PARTIAL ACTIONS AND FELL BUNDLES 31 representing ℓ ( G, B ′′ e ) and ℓ ( G, B e ) faithfully. Let π : B ′′ → L ( H ) be a nondegen-erate *-representation, fiber-wise faithful and w ∗ -continuous (we constructed onesuch representation in the proof of Theorem 5.2). Define ρ := π | B ′′ e : B ′′ e → L ( H )and note that we have canonical identifications K := ℓ ( G, H ) = ℓ ( G, B e ) ⊗ ρ H = ℓ ( G, B ′′ e ) ⊗ ρ H. The map ˆ π : ℓ ( G, B ′′ e ) → L ( H, K ), ˆ π ( f ) h ( r ) = π ( f ( r )) h is a faithful representa-tion of the C ∗ -ternary ring ℓ ( G, B ′′ e ). Then we have a canonical nondegeneraterepresentation ˆ π l : K ( ℓ ( G, B ′′ e )) → L ( K ) such that ˆ π l ( T ) π ( f ) = π ( T f ), and thuswe get a nondegenerate representation of the linking algebra L of ℓ ( G, B ′′ e ):ˆ π L : L → L ( K ⊕ H ) ˆ π L (cid:18) T f ˜ g S (cid:19) = (cid:18) ˆ π l ( T ) ˆ π ( f )ˆ π ( g ) ∗ ˆ π r ( S ) (cid:19) , where ˆ π r : B ′′ e → L ( H ) is just ρ .Fix an element c ∈ C c ( G, B ′′ e ). Using a net in B e to approximate c ( t ) (for each t in the finite support of c ) with respect to the w*-topology, we can construct anet { c i } i ∈ I ⊂ C c ( G, B e ) such that supp( c i ) ⊂ supp( c ) and c i ( t ) → c ( t ) in the w ∗ -topology for every t ∈ G . This construction implies that { ˆ π ( c i ) } i ∈ I wot − convergesto ˆ π ( c ) because, for all h ∈ H and k ∈ K :lim i h ˆ π ( c i ) h, k i = lim i X r ∈ supp( c ) h π ( c i ( r )) h, k ( r ) i = h ˆ π ( c ) h, k i . It follows from the previous comments that ˆ π ( c ) ∈ ˆ π ( C c ( G, B e )) wot . Now, accordingto [36, Theorem 4.8] and [19, Part I Ch. 3], the unit ball of ˆ π L ( L ) is *-stronglydense in the unit ball of ˆ π L ( L ) ′′ , and this bicommutant is the wot − closure of ˆ π L ( L ) . Hence there exists a net { (cid:16) T j a j ˜ b j S j (cid:17) } j ∈ J ⊆ L in the closed ball of radius k ˆ π ( c ) k = k c k such that { ˆ π L (cid:16) T j a j ˜ b j S j (cid:17) } j ∈ J converges to (cid:0) π ( c )0 0 (cid:1) *-strongly. Then, in the strongoperator topology:lim j (cid:0) π ( a j )0 0 (cid:1) = lim j ( ) ˆ π L (cid:16) T j a j ˜ b j S j (cid:17) ( ) = ( ) (cid:0) π ( c )0 0 (cid:1) ( ) = (cid:0) π ( c )0 0 (cid:1) . Now we arrange the supports of the a j ’s to be contained in supp( c ). Let P ∈L ( K ) = L ( ℓ ( G, H )) be the multiplication by the indicator function of supp( c ).Then, in the strong operator topology: lim j P ˆ π ( a j ) = P ˆ π ( c ) = ˆ π ( c ) and P ˆ π ( a j ) =ˆ π ( a j | supp c ). Thus we are allowed to assume supp( a j ) ⊆ supp( c ) for all j ∈ J . Wemust retain the following facts about the net { a j } j ∈ J ⊆ ℓ ( G, B e ): • supp( a j ) ⊆ supp( c ) for all j ∈ J . • k a j k ≤ k c k for all j ∈ J, with the norm of ℓ ( G, B e ) . • { ˆ π ( a j ) } j ∈ J converges strongly to ˆ π ( c ).We claim that these conditions imply, for every t ∈ G , b ∈ B t and ϕ ∈ B ′ t , that(6.12) lim j ϕ X r ∈ G a j ( tr ) ∗ ba j ( r ) ! = ϕ X r ∈ G c ( tr ) ∗ bc ( r ) ! . In other words, we claim that the net { P r ∈ G a j ( tr ) ∗ ba j ( r ) } k ∈ J weakly converges to P r ∈ G c ( tr ) ∗ bc ( r ) in B t . Indeed, since π | B ′′ t is an isomorphism over its image, and ahomeomorphism considering in B ′′ e and in L ( H ) the w*-topology and the ultraweaktopology σw respectively, it is enough to prove that π ( P r ∈ G a j ( tr ) ∗ ba j ( r )) σw → π ( P r ∈ G c ( tr ) ∗ bc ( r )). Let U : G → L ( K ) = L ( ℓ ( G, H )) be the unitary representation given by U t f ( r ) = f ( t − r ), and π G : B ′′ → L ( K ) be the ℓ -direct sum of G copies of π , that is, π G ( b ) f ( r ) := π ( b ) f ( r ). Note that { P r ∈ G a j ( tr ) ∗ ba j ( r ) } j ∈ J is bounded because,for all u, v ∈ H ,(6.13) h u, π X r ∈ G a j ( tr ) ∗ ba j ( r ) ! v i = h U ∗ t ˆ π ( a j ) u, π G ( b )ˆ π ( a j ) v i . Since the ultraweak topology coincides with the weak operator topology on boundedsets, to prove (6.12) it is enough to show that { π ( P r ∈ G a j ( tr ) ∗ ba j ( r )) } j ∈ J convergesto π ( P r ∈ G c ( tr ) ∗ bc ( r )) in the wot topology. But our construction of { a j } j ∈ J and(6.13) implieslim j h u, π X r ∈ G a j ( tr ) ∗ ba j ( r ) ! v i = lim j h U ∗ t ˆ π ( a j ) u, π G ( b )ˆ π ( a j ) v i = h U ∗ t ˆ π ( c ) u, π G ( b )ˆ π ( c ) v i = h u, π X r ∈ G c ( tr ) ∗ bc ( r ) ! v i . Therefore (6.12) holds (note that (6.13) does not imply (6.12) if we only know that π ( a i ) wot → π ( c )).Now assume that B ′′ has the W*AP and take a net { c i } i ∈ I as in the defini-tion of the W*AP for B ′′ , with all the c i ’s with compact support. Set M :=sup i ∈ I k P t ∈ G c i ( t ) ∗ c i ( t ) k and let F and F ′ be the families of finite subsets of B and ⊎ t ∈ G B ′ t, , respectively, where B ′ t, is the closed unit ball of B ′ t . On Λ :=(0 , × F × F ′ we consider the canonical order ( ε, U, V ) ≤ ( δ, Y, Z ) ⇔ δ ≤ ε , U ⊆ Y and V ⊆ Z . For every λ = ( ε, U, V ) ∈ Λ there exists i ∈ I such that, for every t ∈ G , b ∈ B t ∩ U and ϕ ∈ B ′ t ∩ V , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ϕ b − X r ∈ G c i ( tr ) ∗ bc i ( r ) !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < ε/ . Our approximation procedure ensures the existence of a λ ∈ ℓ ( G, B e ) such that:supp( a λ ) ⊆ supp( c i ), k a λ k ≤ k c i k ≤ M and (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ϕ b − X r ∈ G a λ ( tr ) ∗ ba λ ( r ) !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < ε, for every t ∈ G , b ∈ B t ∩ U and ϕ ∈ B ′ t ∩ V . It is then clear that { a λ } λ ∈ Λ is a netsatisfying (vi). (cid:3) Remark . By the proof above and Remark 6.9, we could replace the condition“bounded net” in the the last Theorem by “net in the closed unit ball” withoutchanging the conclusions.
Corollary 6.15.
A Fell bundle B has the WAP if and only if the canonical actionon its C*-algebra of kernels k ( B ) is AD-amenable.Proof. Follows from Theorem 6.11 and Corollary 5.26. (cid:3)
Corollary 6.16.
The AP implies the WAP.Proof.
The AP clearly implies condition (vi) of Theorem 6.11. (cid:3)
MENABILITY FOR PARTIAL ACTIONS AND FELL BUNDLES 33
Notice that by Example 3.7 every group partially acts AD-amenably on C , so thatthe above situation does happen for every group. For global actions the situationis different: no non-amenable group can act globally AD-amenably on a finitedimensional non-zero C*-algebra. Remark . We do not know if the WAP implies (and hence is equivalent to) theAP in general. We will show in Section 8 that this is true at least in the case ofFell bundles whose unit fibre is (Morita equivalent to) a commutative C*-algebra.6.1.
Invariance under equivalences.
We have shown that AD-amenability ofFell bundles is equivalent to the WAP and both are preserved by the weak equiva-lence of Fell bundles. But, is the AP preserved by weak equivalence of Fell bundles?Every Fell bundle B is weakly equivalent to the semidirect product bundle of anaction α on a C*-algebra, see [5]. Moreover, α is unique up to Morita equivalence ofactions and the equivalence class is that of the canonical action on the C*-algebraof kernels of B , see [2].In order to show that the AP is preserved by weak equivalences we decomposesuch equivalences into “elementary” pieces. By [5], every weak equivalence (repre-sented by ∼ ) between the Fell bundles A and B (over G ) can be decomposed as achain of equivalences(6.18) A ≈ B α ∼ B γ ≈ B σ ∼ B β ≈ B , where ≈ represents strong equivalence and • α and β are partial actions of G on C*-algebras. • γ ( σ ) is the canonical action of G on the C*-algebra of kernels of A ( B ,respectively) and it is also the C*-enveloping action of α ( β , respectively).The advantage of this decomposition is that we have changed a weak equivalencefor some strong equivalences and a very specific type of weak equivalence: that ofC*-enveloping actions. Lemma 6.19.
The AP is preserved by strong equivalence of Fell bundles.Proof.
Suppose A and B are Fell bundles over G , A has the AP and X is an A - B -weak equivalence bundle. Let { a i } i ∈ I be a set of functions for A as in the definitionof the AP.Let F be the collection of finite subsets of X and consider in Λ := F × (0 , + ∞ )the order ( U, ε ) ≤ ( V, δ ) ⇔ U ⊆ V and δ ≤ ε . We will construct a net of functions { b λ } λ ∈ Λ , b λ : G → B e , with finite supports such that • sup λ ∈ Λ k P r ∈ G b λ ( r ) ∗ b λ ( r ) k ≤ sup i ∈ I k P r ∈ G a i ( r ) ∗ a i ( r ) k < ∞• For every λ = ( U, V, ε ) ∈ Λ and u, v ∈ U , if h u, v i B ∈ B t then (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X r ∈ G b λ ( tr ) ∗ h u, v i B b λ ( r ) − h u, v i B (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) < ε. This will clearly suffice to complete the proof because, for every t ∈ G ,span {h u, v i B : u ∈ X r , v ∈ X rt , r ∈ G } = B t . Fix λ = ( U, ε ) ∈ Λ. Take a positive c ∈ B e such that k c k < k c h u, v i B c −h u, v i B k < ε for all u, v ∈ U . By [6] we can assume c = P nj =1 h x j , x j i B for some x , . . . , x n ∈ X e . Define, for every i ∈ I , b i : G → B e as b i ( r ) := P nj =1 h x j , a i ( r ) x j i B .The function b λ will be one of the b i ’s, that we will indicate how to choose. We claim that k P r ∈ G b i ( r ) ∗ b i ( r ) k ≤ k P r ∈ G a i ( r ) ∗ a i ( r ) k . Let F = { t , . . . , t n } be such that x j ∈ X t j ( j = 1 , . . . , n ). Define M := M F ( A ) as in Remark 6.7. Thedirect sum E := X t ⊕ · · · ⊕ X t n is an M - B e -Hilbert bimodule (not full in general).If we think of the elements of E as column matrices, the action of M is given bymatrix multiplication and the left inner product is M h ξ, η i = ξη ∗ = ( A h ξ i , η j i ) ni,j =1 . If x is the column vector ( x , . . . , x n ) t ∈ E , then k x k = k c k < (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X r ∈ G b i ( r ) ∗ b i ( r ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X r ∈ G n X j,k =1 h a i ( r ) ∗A h x k , x j i a i ( r ) x j , x k i B (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X r ∈ G h diag( a i ( r )) ∗ M h x, x i diag( a i ( r )) x, x i B e (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) M h x, x ikh diag( X r ∈ G a i ( r ) ∗ a i ( r )) x, x i B e (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ k x k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X r ∈ G a i ( r ) ∗ a i ( r ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) k x k ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X r ∈ G a i ( r ) ∗ a i ( r ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) . Now take u, v ∈ U and let t ∈ G be such that h u, v i B ∈ B t . For all i ∈ I we have X r ∈ G b i ( tr ) ∗ h u, v i B b i ( r ) = X r ∈ G n X j,k =1 h x j , a i ( tr ) x j i B∗ h u, v i B h x k , a i ( r ) x k i B = X r ∈ G n X j,k =1 h u h x j , a i ( tr ) x j i B , v h x k , a i ( r ) x k i B i B = X r ∈ G n X j,k =1 h A h u, x j i a i ( tr ) x j , A h v, x k i a i ( r ) x k i B = n X j,k =1 h X r ∈ G a i ( t − r ) ∗A h x k , v i A h u, x j i a i ( r ) x j , x k i B . Note that A h x k , v i A h u, x j i = A h x k h u, v i ∗B , x j i ∈ A t − . Then, taking limit in i ,lim i X r ∈ G b i ( tr ) ∗ h u, v i B b i ( r ) = n X j,k =1 h lim i X r ∈ G a i ( r ) ∗A h x k , v i A h u, x j i a i ( r ) x j , x k i B = n X j,k =1 h A h x k , v i A h u, x j i x j , x k i B = n X j,k =1 h u h x j , x j i B , v h x k x k i B i B = c h u, v i B c. We then can choose i ∈ I such that k P r ∈ G b i ( tr ) ∗ h u, v i B b i ( r ) − h u, v i B k < ε forall u, v ∈ U . Thus we take b λ := b i . (cid:3) Lemma 6.20.
Let B be a Fell bundle. If B has the AP and β is the canonicalaction on the C*-algebra of kernels of B , then β has the AP.Proof. Take a net of functions { b j } j ∈ J ⊂ ℓ ( G, B e ) as in the definition of the APand construct a net { b ξ } ξ ∈ Ξ ⊂ ℓ ( G, k ( B )) exaclty as in the proof of Theorem 6.8. MENABILITY FOR PARTIAL ACTIONS AND FELL BUNDLES 35
This time we can ensure that, for every k ∈ k c ( B ) , {h b λ , k ˜ β t b λ i} λ ∈ Λ converges(entrywise) in norm to k. The rest follows from Remark 6.2. (cid:3)
Lemma 6.21.
Let α be a partial action of a group G on a C*-algebra A . If aMorita enveloping action of α has the AP then α has the AP.Proof. Suppose β is a Morita enveloping action of α . By definition α is Moritaequivalent to a restriction of β , but Lemma 6.19 implies the AP is preserved underMorita equivalence of actions. Thus we may assume β is an enveloping action of α . We assume β is an action of G on B and recall that A t = β t ( A ) ∩ A and α t ( a ) = β t ( a ). Moreover, B α is a Fell subbundle of B β . We think of A and B asthe unit fibres of these bundles.Let { b i } i ∈ I be a net of functions, as in the definition of the AP for the bundle B β . Take a positive c ∈ A with k c k < aδ t ∈ B α . By cb i we mean thefunction r cb i ( r ). Note that cb i is an A valued function. Making the followingcomputations in B β we deduce(6.22) lim i X r ∈ G ( cb i )( tr ) ∗ ( aδ t )( cb i )( r ) = lim i X r ∈ G b i ( tr ) ∗ caβ t ( c ) b i ( r ) δ t = caβ t ( c ) . Since a ∈ A t , if we replace c by an approximate unit of A and take limit then caβ t ( c ) converges to a . Imitating the ideas we used to prove Lemma 6.19 to showthat B α has the AP, we can construct a net of functions (for B α ) indexed over I × F × (0 , + ∞ ), where F is the family of finite subsets of B α . We leave this taskto the reader. (cid:3) Now we use Lemmas 6.19 and 6.21 to prove our next Theorem, which in turnimplies those lemmas.
Theorem 6.23.
The AP is preserved by the weak equivalences of Fell bundles.Proof.
Let A and B be weakly equivalent Fell bundles and consider the equivalencesin (6.18). Recall that γ and σ are the canonical actions on the algebras of kernelsof A and B , respectively. If A has the AP then, by Lemmas 6.20 and 6.19, B σ hasthe AP. Now Lemmas 6.21 and 6.19 imply that B has the AP. (cid:3) Remark . We know that Theorem 6.23 (and hence also Corollary 6.25, Lem-mas 6.21, 6.20 and 6.19) hold if we replace the AP by the WAP because, by The-orem 6.11 and Corollary 5.25, the WAP is equivalent to AD-amenability and bothare preserved by weak equivalence of Fell bundles.The converse of Lemma 6.21 is also true.
Corollary 6.25.
Let α be a partial action of G on a C*-algebra A . Then α hasthe AP if and only if one (hence all) Morita enveloping action of α has the AP. Inparticular the double dual (global) action of G on A ⋊ α G ⋊ b α G has the AP if andonly if α has the AP.Proof. This follow from Theorem 6.23 and the fact that B α is weakly equivalentto the semidirect product bundle of each Morita enveloping action of α . Moreover,the double dual action on A ⋊ α G ⋊ b α G is a Morita enveloping action of α , whichis isomorphic to the canonical action on the C ∗ -algebras of kernels under the iso-morphism k ( B α ) ∼ = C ∗ ( B α ) ⋊ δ B G ∼ = A ⋊ α G ⋊ b α G by [2, Proposition 8.1], where δ B denotes the dual coaction on C ∗ ( B α ) as in Section 5.5. (cid:3) Corollary 6.26.
If a partial action α of G on A admits an enveloping global action β of G on B , then α has the AP if and only if β has the AP.Remark . By Remark 6.24 the AP and the WAP are equivalent if and onlyif they are equivalent for actions on C*-algebras. Thus the AP and the WAPagree if and only if every AD-amenable action has the AP. This is an importantquestion that will be left open. A positive answer would solve another importantquestion raised by Exel: if the reduced cross-sectional C*-algebra C ∗ r ( B ) of a Fellbundle (over a discrete group) is nuclear, does it follow that B has the AP? Byour Proposition 7.3 below this would follow if we know that the AD-amenabilityimplies the AP. Corollary 6.28.
Let B be a Fell bundle and β the canonical action on the C ∗ -al-gebra of kernels of B . Then B has the AP if and only if β has the AP.Proof. Recall that B is weakly equivalent to B β , as we discussed at the beginningof Section 6.1. The conclusion now follows from Theorem 6.23. (cid:3) Cross-sectional C*-algebras and the WAP
After Theorem 6.11 we can think of the WAP as the Fell bundle counterpartof AD-amenability of noncommutative C ∗ -dynamical systems. In fact many well-known results about AD-amenable actions hold for Fell bundles with the WAP.We start with a result involving W*-Fell bundles. Proposition 7.1.
Let M = { M t } t ∈ G be a W*-Fell bundle. Then the followingassertions are equivalent.(i) M e injective and M has the W*AP (or, equivalently, M is W*AD-amenable);(ii) k w ∗ ( M ) is injective and its canonical W ∗ -action β w ∗ has the W*AP (or isW*AD-amenable);(iii) k w ∗ ( M ) ¯ ⋊ β w ∗ G is injective;(iv) W ∗ r ( M ) is injective.Proof. First notice that M e is injective if and only if the W ∗ -algebra of kernels k w ∗ ( M ) is injective. Indeed, M e is W ∗ -Morita equivalent to L ( ℓ ∗ ( M )) (via the W ∗ -equivalence bimodule ℓ ∗ ( M ))); it follows that M e is injective if and only if L ( ℓ ∗ ( M )) is injective. But k w ∗ ( M ) carries a W*-action that is enveloping for apartial W ∗ -action on L ( ℓ ∗ ( M )). The claim now follows from Remark 2.9. Alsoobserve that M e is injective if W ∗ r ( M ) is injective because we have a canonical(normal) conditional expectation W ∗ r ( M ) ։ M e (Remark 5.28).The discussion above implies that (1) is equivalent to (2). Since (2) involves a W ∗ -action, (2) ⇔ (3) by Theorem 3.3. Finally, (3) ⇔ (4) by Corollary 5.30. (cid:3) Proposition 7.2.
Let B be a Fell bundle and let π : C ∗ ( B ) → C ∗ r ( B ) be the canon-ical map between the full and reduced cross-sectional C ∗ -algebras [20, 22] . If B hasthe WAP then π is an isomorphism.Proof. Since B has the WAP, the canonical action β on the C ∗ -algebra of kernelsof B is AD-amenable (Corollary 6.15). Hence the full and reduced cross-sectionalalgebras C ∗ ( B β ) and C ∗ r ( B β ) (i.e. the full and reduced β -crossed product) arecanonically isomorphic. This implies that π is an isomorphism, see [5, 6]. (cid:3) MENABILITY FOR PARTIAL ACTIONS AND FELL BUNDLES 37
Proposition 7.3 (c.f. [10, Théorème 4.5]) . Let B be a Fell bundle over a group G with B e nuclear. Then the following are equivalent:(i) C ∗ ( B ) is nuclear.(ii) C ∗ r ( B ) is nuclear.(iii) B has the WAP.Proof. Let k ( B ) be the C ∗ -algebra of kernels and β the canonical action of G on k ( B ). By the proof of [5, Theorem 6.3], k ( B ) is nuclear. Moreover, by Corollary 6.15and [10, Théorème 4.5], (3) is equivalent to any of the following:(1’) k ( B ) ⋊ β G := C ∗ ( B β ) is nuclear.(2’) k ( B ) ⋊ r,β G := C ∗ r ( B β ) is nuclear.(3’) β is AD-amenable.Since nuclearity is preserved by Morita equivalence of C ∗ -algebras, by [5] and [6]we know that ( n ) is equivalent to ( n ’), for n = 1 , , (cid:3) When specialised to partial actions the above proposition takes the followingform:
Corollary 7.4.
Let α be a partial action of the group G on a nuclear C ∗ -algebra A . Then the following are equivalent:(i) The full crossed product A ⋊ α G is nuclear.(ii) The reduced crossed product A ⋊ α, r G is nuclear.(iii) α is AD-amenable.Proof. Follows directly from the last Proposition and Corollary 6.10. (cid:3)
The last two results are examples of a general way of extending known resultsfrom C ∗ -actions to Fell bundles. The trick is to use the weak equivalence of Fellbundles and the canonical action on the C ∗ -algebra of kernels. We use this verysame idea to treat exactness of cross sectional C ∗ -algebras, but first we introducethe spatial tensor product of a Fell bundle (over a discrete group) and a C ∗ -algebra.This construction is a special case of the tensor products of Fell bundles developedin [4]. We recall the basic facts here for the convenience of the reader.Take a Fell bundle B and a C ∗ -algebra C. Let L t be the linking algebra of B t and define B t ¯ ⊗ C as the closure of the algebraic tensor product B t ⊙ C in L t ¯ ⊗ C. We claim that
B ⊗ C := { B t ⊗ C } t ∈ G is a Fell bundle with a multiplication andinvolution such that ( a ⊗ x )( b ⊗ y ) = ab ⊗ xy and ( a ⊗ x ) ∗ = a ∗ ⊗ x ∗ . For future purposes, and to prove
B ⊗ C is actually a Fell bundle, it is convenientto indicate how to construct this bundle using a representation of B . Let T : B →L ( H ) be a nondegenerate *-representation (in the sense of [21]) with T | B e faithfuland take a nondegenerate and faithful representation π : C → L ( K ) (here H and K are Hilbert spaces). Consider, for each t ∈ G, the map ρ t : L t → L ( H ⊕ H ) = M ( L ( H )) such that ρ t (cid:18) a bc d (cid:19) = (cid:18) T a T b T ∗ c T d (cid:19) , a ∈ span( B t B ∗ t ) , b, c ∈ B t , d ∈ span( B ∗ t B t ) . Then we get the canonical (linear and injective) map ρ t ⊗ π : L t ¯ ⊗ C → L ( H ⊗ K ) . If we restrict this map to B t ¯ ⊗ C we note that H ⊗ K = ( H ⊕ ⊗ K ⊂ H ⊗ K is aninvariant subspace and that the compression (of the restriction) to this subspace isan isometric linear representation of B t ¯ ⊗ C. Hence we obtain the map T ¯ ⊗ π : B⊗ C → L ( H ⊗ K ) which is linear isometric on each fiber and T ¯ ⊗ π ( a ⊗ c ) = T a ⊗ π ( c ) . Note that T ¯ ⊗ π ( B s ¯ ⊗ C ) T ¯ ⊗ π ( B t ¯ ⊗ C ) ⊂ T ¯ ⊗ π ( B st ¯ ⊗ C ) and that T ¯ ⊗ π ( B s ¯ ⊗ C ) ∗ = T ¯ ⊗ π ( B s − ¯ ⊗ C ) . Thus we are forced to define the multiplication and involution of B ¯ ⊗ C in such a way that T ¯ ⊗ π ( xy ) = T ¯ ⊗ π ( x ) T ¯ ⊗ π ( y ) and T ¯ ⊗ π ( x ∗ ) = T ¯ ⊗ π ( x ) ∗ . Proposition 7.5. If B is a Fell bundle, then B e is exact if and only if k ( B ) isexact.Proof. In the proof of Theorem 6.8 we constructed an inclusion M F ( M ) ⊂ k w ∗ ( M ) . That inclusion can be used to prove that k ( B ) is the direct limit of { M F ( B ) } F , where F runs over the finite subsets of G. Hence k ( B ) is exact if and only if M F ( B ) is exactfor every finite set F ⊂ G. Notice that if we take F = { e } we conclude that B e isexact if k ( B ) is.Assume B e is exact and take a finite set F ⊂ G and a short exact sequence(SES) of C ∗ -algebras I ֒ → A ։ A/I.
For every t ∈ G the linking algebra of B t , L t , is an exact C ∗ -algebra because it is Morita equivalent to the ideal span( B ∗ t B t ) of B e . Thus we get the SES L t ¯ ⊗ I ֒ → L t ¯ ⊗ A ։ L t ¯ ⊗ A/I and, by our construction ofspatial tensor products, we obtain the following SES of Fell bundles B ¯ ⊗ I ֒ → B ¯ ⊗ A ։ B ¯ ⊗ A/I.
By entrywise computation of the maps above we get the SES(7.6) M F ( B ¯ ⊗ I ) ֒ → M F ( B ¯ ⊗ A ) ։ M F ( B ¯ ⊗ A/I ) . Using a nondegenerate representation T : B → L ( H ) with T | B e faithful andrepresenting I faithfully, I ⊂ L ( K ) , we can think of M F ( B ¯ ⊗ I ) as a subalgebra of L ( H n ⊗ K ) = L (( H ⊗ K ) n ) . But we can also represent B ¯ ⊗ I in L ( H ⊗ K ) and thuswe get a faithful representation M F ( B ¯ ⊗ I ) ⊂ L (( H ⊗ K ) n ) . It turns out that, aftertaking faithful representations, M F ( B ¯ ⊗ I ) = M F ( B ) ¯ ⊗ I. The same argument holdsfor the tensor products with A and A/I.
Moreover, the identifications M F ( B ¯ ⊗ Z ) = M F ( B ) ¯ ⊗ Z (for Z = I, A, A/I ) are compatible with the maps of (7.6) and thus weget the SES M F ( B ) ¯ ⊗ I ֒ → M F ( B ) ¯ ⊗ A ։ M F ( B ) ¯ ⊗ A/I.
This proves M F ( B ) is exact. (cid:3) Corollary 7.7. If B is the enveloping C ∗ -algebra of a partial action α of a locallycompact and Hausdorff group on a C ∗ -algebra A, then B is exact if and only if A is exact.Proof. We can drop the topology of the group and work with the discrete onebecause this does not affect the enveloping action [27]. The the proof follows directlyfrom Proposition 7.5 because B is Morita equivalent to k ( B α ) and A is the unitfiber of B α . (cid:3) It is known [28, Proposition 7.1] that the crossed product of an amenable groupacting on an exact C ∗ -algebra is exact. Recently this was proved in [15, Theorem6.1] for AD-amenable actions. For the convenience of the reader we give a directproof here. If we go back to the ideas of Kirchberg [28] we see that the followingLemma is the key point. Lemma 7.8.
Let α be an AD-amenable action of G on the C ∗ -algebra A and B aC*-algebra endowed with the trivial G -action id . Then the tensor product G -action α ⊗ id on A ⊗ B is AD-amenable. MENABILITY FOR PARTIAL ACTIONS AND FELL BUNDLES 39
Proof.
Decomposing the universal representation π : A ⊗ B → L ( H ) as π = π A × π B as in [14, Theorem 3.2.6] we get a w ∗ − continuous *-homomorphism A ′′ → ( A ⊗ B ) ′′ ,a a ⊗ . Let { a i } i ∈ I be a net of functions as in (5) of Theorem 6.11. We claim that { a i ⊗ } i ∈ I is a net of functions giving the WAP for α ⊗ id . For this we use Lemma 6.5with A ⊗ alg B as the w ∗ -dense *-subalgebra of ( A ⊗ B ) ′′ . Note that h a i ⊗ , a i ⊗ i = h a i , a i i ⊗ , hence sup i k a i ⊗ k = sup i k a i k < ∞ . Given an elementary tensor x ⊗ y ∈ A ¯ ⊗ B we have, in the w ∗ − topology:lim i h a i ⊗ , ( x ⊗ y ) ^ α t ⊗ id( a i ⊗ i = lim i h a i , x ˜ α t ( a i ) i ⊗ y = x ⊗ y. This completes the proof. (cid:3)
Corollary 7.9.
Let α be an AD-amenable action of the group G on the C ∗ -algebra A. If A is exact, then so it is A ⋊ G = A ⋊ r G. Proof.
It suffices to use the Lemma above and the very same ideas of the proof of[28, Proposition 7.1]. (cid:3)
Corollary 7.10. If B is a Fell bundle (over G ) with the WAP and B e is exact,then so is C ∗ r ( B ) = C ∗ ( B ) . Proof.
We know, by Proposition 7.5, that k ( B ) is exact and the canonical action on k ( B ) is AD-amenable (Corollaries 5.26 and 6.10). By Corollary 7.9, k ( B ) ⋊ r G = k ( B ) ⋊ G is exact. Since this algebra is Morita equivalent to C ∗ r ( B ) = C ∗ ( B ) , C ∗ r ( B )is exact. (cid:3) In [24, Definition 21.19] Exel introduces the notion of conditional expectation forFell bundles: if A is a Fell subbundle of B , a conditional expectation from B to A is a map P : B → A which restricts to bounded surjective idempotent linear maps P g : B g ։ A g ⊆ B g such that P e : B e ։ A e is an ordinary conditional expectationand P g ( b ) ∗ = P g − ( b ∗ ) and P gh ( ba ) = P g ( b ) a for all b ∈ B g and a ∈ A h , g, h ∈ G . Theorem 7.11.
Let M be a W ∗ -Fell bundle over G, N a W ∗ -Fell subbundle of M and P : M → N a (not necessarily w ∗ − continuous) conditional expectation. Thenthere exists a conditional expectation P k : k w ∗ ( M ) → k w ∗ ( N ) which is equivariantwith respect to the canonical W ∗ -actions. Moreover, the restriction of P k to k ( M ) is a conditional expectation onto k ( N ) .Proof. By Corollary 5.31 we can think of k w ∗ ( N ) as a W ∗ -subalgebra of k w ∗ ( M ) . The matrix algebras M F ( M ) (for F ⊂ G finite) of Remark 6.7 form an upwarddirected set of C ∗ -subalgebras of k w ∗ ( M ) with norm closure equal to k ( M ) . More-over, M F ( M ) is hereditary in k ( M ) and in the proof of Theorem 6.8 we showedthat M F ( M ) is in fact a W ∗ -subalgebra of k w ∗ ( M ) . Hence the family { M F ( M ) } F is an upward directed family of hereditary W ∗ -subalgebras of k w ∗ ( M ) whose unionis w ∗ − dense in k w ∗ ( M ) . We denote 1 F the unit of M F ( M ). Then 1 F may or maynot be equal to the unit of M F ( N ) , which we denote 1 ′ F . Define, for each finite subset F ⊂ G, the map P F : M F ( M ) → M F ( N ) ⊂ k w ∗ ( N )as the entriwise application of P. We claim that this map is a conditional expecta-tion. Indeed, by Tomiyama’s theorem it suffices to prove it is contractive.Let X F M be the M F ( M ) − Hilbert module obtained by considering on M F ( M )the inner product h X, Y i F M := trace( X ∗ Y ) and the action given by matrix mul-tiplication. Then matrix multiplication on the left gives a faithful representation M F ( M ) → L ( X F M ) . If A ∈ M F ( M ) and X, Y ∈ X F N , then h P F ( A ) X, Y i F N = trace( P F ( X ∗ AY )) = P ( h AX, Y i F M ) . This implies k P F ( A ) k ≤ k A k and P F is contractive.We can extend P F to k w ∗ ( M ) by defining P F : k w ∗ ( M ) → k w ∗ ( N ) as P F ( x ) = P F (1 F x F ) . Then P F is clearly ccp, in fact it is a conditional expectation over M F ( N ) . In this way we get a net of ccp maps { P F } F from k w ∗ ( M ) to k w ∗ ( N ) . Let P k be a pointwise w ∗ − limit of a converging subnet { P F j } j . Clearly P k is ccp. Take x ∈ k w ∗ ( N ) . Since P F ( x ) = 1 ′ F x ′ F , both { P F ( x ) } F and { P F j ( x ) } F j w ∗ − convergeto x. Thus P k ( x ) = x and P k is a conditional expectation.We claim that P k is equivariant with respect to the canonical W ∗ -actions. Take t ∈ G and note that β w ∗ t (1 F ) = 1 F t − and that, given x ∈ M F ( N ) , it follows that P F t − ( β w ∗ t ( x )) = β w ∗ t ( P F ( x )) . Considering w ∗ − limits we have P k ( β w ∗ t ( x )) = lim j P F j (1 F j β w ∗ t ( x )1 F j ) = lim j P F j ( β w ∗ t (1 F j t x F j t ))= lim j β w ∗ t ( P F j t (1 F j t x F j t )) = β w ∗ t (lim j P F j t (1 F j t x F j t ))Thus { P F j t (1 F j t x F j t ) } j actually has a w ∗ − limit (for every x ) and it suffices toshow that this limit is P k ( x ) . Every element v of k w ∗ ( N ) is completely determined by the products uvw, for u, v ∈ k c ( M ) . Then it suffices to prove that lim j uP F j t (1 F j t x F j t ) w = uP k ( x ) w, forall u, w ∈ k c ( M ) . Fix u, w ∈ k c ( M ) and take a finite set K ⊂ G such that K × K contains both the supports of u and w. Since the families { F j } j and { F j t } j arecofinal in the finite subsets of G we havelim j uP F j t (1 F j t x F j t ) w = lim j u ′ K P F j t (1 F j t x F j t )1 ′ K w = lim j uP F j t (1 ′ K F j t x F j t ′ K ) w = uP K (1 ′ K x ′ K ) v = uP k (1 ′ K x ′ K ) v = u ′ K P k ( x )1 ′ K v = uP k ( x ) v. Finally, the last statement is clear from the computations above. In fact if F := { F ⊂ G : F is finite } , then it is easy to see also that the C ∗ -limits of the directsystems { M F ( M ) } F ∈F and { M F ( N ) } F ∈F are k ( M ) and k ( N ) respectively, andthat P k | k ( M ) is the limit of the direct system { M F ( M ) } F ∈F P F → { M F ( N ) } F ∈F . (cid:3) Corollary 7.12.
Let M be a W ∗ -Fell bundle over G, N a W ∗ -Fell subbundle of M and P : M → N a (not necessarily w ∗ − continuous) conditional expectation. If M has the W*AP then so does N . Proof.
Recall that M has the W*AP iff the canonical W ∗ -action on k w ∗ ( M ) isW*AD-amenable. Then everything follows from [7, Proposition 3.8] and the Theo-rem above. (cid:3) Corollary 7.13.
Let A be a Fell subbundle of B . If B has the WAP and the as-sociated inclusion A ′′ ֒ → B ′′ admits conditional expectation, then A has the WAP.In particular, this is the case if A is hereditary in B (i.e. A e B r A e ⊂ A r for all r ∈ G ).Proof. If A is hereditary in B and p is the unit of A ′′ e ⊂ B ′′ e , then the map P : B ′′ →A ′′ , b pbp, is a conditional expectation. Then the proof follows from our lastCorollary. (cid:3) MENABILITY FOR PARTIAL ACTIONS AND FELL BUNDLES 41
Corollary 7.14.
Let A be a Fell subbundle of a Fell bundle B and suppose thereexists a conditional expectation P : B ։ A . If B has the WAP, then so does A .Proof. It suffices to construct a conditional expectation P ′′ : B ′′ → A ′′ . Take a fiber B t and consider it’s linking algebra as L ( B t ) = (cid:18) I t B t B t − I t − (cid:19) , where I t is (the closed linear span of) B t B ∗ t in B e . Then we can form a condi-tional expectation L ( P ) t : L ( B t ) → L ( A t ) by entrywise computation of P. UsingStinespring’s factorization theorem can be extended L ( P ) t w ∗ − continuously to aconditional expectation L ( P ) ′′ t : L ( B t ) ′′ → L ( A t ) ′′ . If we now restrict this last mapto B ′′ t we get a map P ′′ t : B ′′ t → A ′′ t . We leave to the reader the verification of thefact that P ′′ := { P ′′ t } t ∈ G : B ′′ → A ′′ is a conditional expectation. (cid:3) Fell bundles with commutative unit fibre
This section is dedicated to the study of amenability for Fell bundles with com-mutative unit fiber. Let B = ( B t ) t ∈ G be a Fell bundle such that B e = C ( X ) is acommutative C ∗ -algebra. Such a Fell bundle canonically induces a partial action of G on X and hence also on C ( X ). This is because imprimitivity bimodules betweencommutative C ∗ -algebras yield isomorphisms between their spectra. Since each B t can be viewed as an imprimitivity I t - I t − -bimodule, where I t := B t B ∗ t ∼ = C ( A t )for some open subset A t ⊆ X , it yields an isomorphism α t : C ( A t − ) → C ( A t )or, equivalently, to a homeomorphism θ t : A t − → A t . The collection α = ( α t ) t ∈ G (resp. θ = ( θ t ) t ∈ G ) is then the desired partial action of G on C ( X ) (resp. X ); theyare related by the equation α t ( f ) = f ◦ θ t − for all f ∈ C ( A t − ). Definition 8.1.
The partial action θ of G on X or its associated partial action α on C ( X ) = B e defined above will be call the spectral partial action of B .These partial actions are analogous to the central partial actions defined in Sec-tion 5.2. Moreover, they are special cases of the more general partial actions on thespectrum of B e , as in [1] for an arbitrary Fell bundle B ( B e need not be abelianhere). Moreover, we see from the constructions that the central partial action of B ′′ is the double dual partial action α ′′ of G on the W*-algebra C ( X ) ′′ associatedwith α . From this we immediately derive the following result: Corollary 8.2.
A Fell bundle B with commutative unit fiber is AD-amenable (or,equivalently, has the WAP) if and only if its spectral partial action is AD-amenable(or has the WAP).Proof. By definition, α is AD-amenable if and only if its double dual partial action α ′′ is W*AD-amenable. Since α ′′ is the central partial action of B ′′ , the resultfollows from Corollary 5.20, the definition of AD-amenability (Definition 5.21) andthe equivalence between AD-amenability and the WAP, Theorem 6.11. (cid:3) From the spectral partial action α of B , we obtain another Fell bundle B α , the oneassociated to the partial action α . These Fell bundles are not necessarily isomorphicin general because the original Fell bundle may contain some “twist”. One formof twist is given in terms of 2-cocycles for partial actions as defined by Exel in[25]. More precisely, Exel introduces the notion of a twisted partial action . In the commutative case, besides a partial action α of G on C ( X ), this envolvescertain unitary multipliers ω ( s, t ) ∈ U M ( D st ). We do not need to recall the preciseconditions on ω and its relation with α . We only recall that the Fell bundle B α,ω associated with the twisted partial action ( α, ω ) has fibres B α,ω,t := C ( D t ) δ t ∼ = C ( D t ) and multiplications and involutions given by:( f δ s ) · ( gδ t ) := ω ( s, t ) α s ( α − s ( f ) g ) δ st , ( f δ s ) ∗ := ω ( s − , s ) ∗ α s − ( f ∗ ) δ s − for all s, t ∈ G , f ∈ C ( D s ) and g ∈ C ( D t ).The main result in [25] states that every regular Fell bundle is isomorphic toone associated with a twisted partial action. The regularity of B concerns thestructure of the fibres B t as imprimitivity I t - I t − -bimodules. Since the C ∗ -algebras I t = C ( D t ) are commutative, such imprimitivity bimodules are necessarily given as C -sections of a certain (complex) line bundle L t over D t . The C -section C ( L t ) ofsuch a line bundle may be viewed as an imprimitivity C ( D t )- C ( D t − )-bimodule;using the isomorphism α t : C ( D t − ) ∼ −→ C ( D t ) we may also view C ( L t ) as animprimitivity C ( D t )- C ( D t − )-bimodule which is then isomorphic to B t . Theregularity of B t is then equivalent to L t being topologically trivial as a complexline bundle. This is always the case for Fell bundles associated with twisted partialactions but it might be not the case in general, so that our original Fell bundle B is not necessarily isomorphic to B α,ω , not even as Banach bundles. Howeveramenability does not see these diferences. Indeed, again the spectral and centralpartial actions of all three Fell bundles B , B α and B α,ω are isomorphic, so weimmediately obtain the next corollary (an improvement of the previous one): Corollary 8.3.
A Fell bundle with commutative unit fibre B is AD-amenable (orhas the WAP) if and only if so is B α or, equivalently, B α,ω . Notice that by Proposition 7.3 the equivalent conditions in the above corollaryare also equivalent to nuclearity of one of the C ∗ -algebras C ∗ (r) ( B ), C ∗ (r) ( B α ) ∼ = C ( X ) ⋊ α, (r) G or C ∗ (r) ( B α,ω ) = C ( X ) ⋊ α,ω, (r) G . In particular we obtain theinteresting consequence that C ( X ) ⋊ α, (r) G is nuclear if and only if C ( X ) ⋊ α,ω, (r) G is nuclear for every twisted partial action ( α, ω ). The parenthesis around r heremeans that we can take either the full or the reduced crossed product (or cross-sectional C*-algebras) in all equivalent statements. Indeed, any other exotic crossed-product norm between the full and reduced could be used for that matter.The above results can also be interpreted using groupoid descriptions of theassociated C ∗ -algebras. To explain this, let us first recall that a partial action θ of G on X yields a locally compact Hausdorff étale transformation groupoid Γ = X ⋊ θ G (see [3]). As a set it consists of pairs ( x, t ) with t ∈ G and x ∈ D t − . The sourceand range maps are s ( x, t ) := x and r ( x, t ) := t · x := θ t ( x ) and multiplication andinversion are given by( x, s ) · ( y, t ) = ( y, st ) , ( x, t ) − = ( t · x, t − ) for x = t · y. The topology is the one inherited from the product topology on X × G . The domains D t give rise to subsets Γ t := D t ×{ t − } ⊆ Γ that are clopen bisections of Γ (althoughthe domains D t are only assumed to be open in X ). Hence Γ decomposes as adisjoint union Γ = ⊔ t ∈ G Γ t of clopen subsets. In particular the vector space C c (Γ)identifies canonically with the algebraic direct sum ⊕ alg t ∈ G C c (Γ t ), that is, functions ζ ∈ C c (Γ) correspond bijectively to finite sets of functions ζ t ∈ C c (Γ t ), t ∈ G . This MENABILITY FOR PARTIAL ACTIONS AND FELL BUNDLES 43 identification extends to a canonical isomorphism C ∗ (r) (Γ) ∼ = C ( X ) ⋊ α, (r) G , where α is the partial action of G on C ( X ) corresponding to θ .Now, given a Fell bundle B over G with unit fibre B e = C ( X ) as above, let θ beits spectral action and Γ the corresponding transformation groupoid that we call the spectral groupoid of B . As previously, we also identify each fibre B t with the sections C ( L t ) of a line bundle L t over D t . The disjoint union L := ⊔ t ∈ G L t can then beviewed as a line bundle over Γ = ⊔ t ∈ G Γ t . Moreover, with the Fell bundle structureinherited from B , L is indeed a Fell line bundle over Γ; such a Fell bundle is alsousually viewed as a twist over Γ. By construction we get an obvious identification C c (Γ , L ) ∼ = C c ( B ) that extends to an isomorphism C ∗ (r) (Γ , L ) ∼ = C ∗ (r) ( B ). In otherwords, we have described every Fell bundle over a discrete group with commutativeunit fibre in terms of a twisted groupoid. This result can be deduced from theconstructions and results in [17] that describe Fell bundles over inverse semigroupswith commutative fibres over idempotents (also call semi-abelian Fell bundles in[17]) in a similar way via twisted groupoids. The Fell bundles in [17] are assumedto be saturated, but the same constructions can also be done in general for non-saturated ones; alternatively, one can view a non-saturated Fell bundle over G as asaturated Fell bundle over the inverse semigroup S ( G ) constructed by Exel in [23],see [16].Next we relate amenability of B in terms of amenability of its spectral groupoid.Amenable groupoids are defined and studied mainly in [11]. We shall use thecharacterisation from [14, Lemma 5.6.14] that says that an étale groupoid Γ isamenable if and only if there is a net ( ζ i ) ⊆ C c (Γ) with k ζ i k ≤ i and( ζ ∗ i ∗ ζ i )( γ ) → γ in compact subsets of Γ. One of the main resultsin this direction states that Γ is amenable if and only if C ∗ (r) (Γ) is nuclear.Notice that the spectral groupoids of B and B α (and also of B α,ω ) are the same;they are just the transformation groupoid Γ = X ⋊ θ G of the spectral partial actionof B . In particular we can reinterpret our previous results as follows: Corollary 8.4.
A Fell bundle with commutative unit fibre is AD-amenable if andonly if its spectral groupoid is amenable.
Using the description of B in terms of a twisted groupoid (Γ , L ) and that AD-amenability is equivalent to nuclearity of the corresponding C ∗ -algebras, we can alsointerpret the above result as the statement that C ∗ (r) (Γ , L ) is nuclear if and only if C ∗ (r) (Γ) is nuclear. In other words, nuclearity of a twisted groupoid C*-algebra isindependent of the twist. Indeed, in this form this result is already known, see [34].Up to this point we have only looked at AD-amenability or, equivalently, theWAP for Fell bundles with commutative unit fibre. We also want to consider theAP for such Fell bundles. We know already that the AP always implies the WAPbut we do not know whether the converse holds in general. The next result aimsat a partial converse: Theorem 8.5.
Let B be a Fell bundle over G with commutative unit fibre B e = C ( X ) . Let θ be its spectral partial action with associated partial action α on C ( X ) , and let Γ = X ⋊ θ G be its spectral groupoid. Then the following assertionsare equivalent:(i) B has the WAP or is AD-amenable, that is, C ∗ (r) ( B ) is nuclear;(ii) B α has the WAP or is AD-amenable; (iii) C ∗ (r) ( B α ) = C ∗ (r) (Γ) = C ( X ) ⋊ α, (r) G is nuclear;(iv) Γ is amenable;(v) B α has the AP;(vi) for every -cocycle ω for α , the corresponding Fell bundle B α,ω has the AP;(vii) B has the AP.Proof. We already checked the equivalences (i) ⇔ (ii) ⇔ (iii) ⇔ (iv). It remains tocheck the equivalence of these conditions with (v), (vi) and (vii). Assume that Γis amenable and let { ζ i } i ∈ I be a net of functions in C c (Γ) with k ζ i k ≤ i and ζ ∗ i ∗ ζ i ( γ ) → γ in compact subsets of Γ. Define ξ i : G → C ( X )by ξ i ( t ) | x := ζ i ( x, t − ) if x ∈ D t and 0 otherwise. In other words, we just use thecanonical identification C c (Γ) ∼ = ⊕ alg t ∈ G C c ( D t ) to view each ζ i as a finitely supportedfunction ξ i : G → C c ( X ) with ξ i ( t ) ∈ C c ( D t ) for all t ∈ G . We verify that this netyields the AP for B α . The boundedness of ( ζ i ) for the ℓ -norm implies the sameboundedness for ( ξ i ). It remains to check the convergence condition that gives theAP. For this it is enough to check that if f ∈ C c ( D t ), then P s ∈ G ξ i ( ts ) ∗ ( f δ t ) ξ i ( s )converges in norm to f δ t . By definition, we have(8.6) X s ∈ G ξ i ( ts ) ∗ ( f δ t ) ξ i ( s ) = X s ∈ G ξ i ( ts ) ∗ α t ( α − t ( f ) ξ i ( s )) δ t . Computing this sum at some x ∈ D t we get an expression of the form: X s ∈ G ζ i ( x, s − t − ) f ( x ) ζ i ( θ − t ( x ) , s − )where the sum varies over all s ∈ G in a finite subset (depending on the supportof ζ i ) satisfying θ − t ( x ) ∈ D s or, equivalently, x ∈ D ts . The above sum can berewritten as X α ∈ Γ ζ i ( α ) ζ i ( αγ ) ! f ( x ) = ( ζ ∗ i ∗ ζ i )( γ ) f ( x )where γ = ( θ − t ( x ) , t ) and α ∈ Γ varies in a finite subset (depending on the supportof ζ i ) satisfying s ( α ) = r ( γ ); those α are necessarily of the form ( x, s − t − ) with x ∈ D ts . If x varies in the compact support K := supp( f ) ⊆ D t of f , then γ = ( θ − t ( x ) , t ) varies in a compact subset of Γ so that ( ζ ∗ i ∗ ζ i )( γ ) → B α . Hence (iv) ⇒ (v). Moreover, if ω is a 2-cocycle for α , then the computation (8.6)is the same because ω ( t, e ) = 1. Therefore the same argument also yields theimplication (iv) ⇒ (vi). Conversely, if B α or B α,ω has the AP, then it also has theWAP and we already observed that this is equivalent to amenability of Γ. Thisyields the implications (v),(vi),(vii) ⇒ (iv).It remains to check (iv) ⇒ (vii). The proof is essentially the same as before, let usgive more details: let { ζ i } i ∈ I ⊆ C c (Γ) be a net that gives the amenability of Γ andlet { ξ i } i ∈ I be the same net as above defined from { ζ i } i ∈ I that gives the AP for B α .We check that this net also gives the AP for B . For this we identify B t ∼ = C ( L t ) fora line bundle L t as before. The structure of C ( L t ) as a Hilbert C ( D t )- C ( D t − ) isas follows: the left and right inner products are given by l h ξ | η i ( x ) := h η ( x ) | ξ ( x ) i x and h ξ | η i r ( x ) := h ξ ( θ t ( x ) | η ( θ t ( x )) i x . Here we use that L t is a Hermitian complexline bundle and h· | ·i x denotes the inner product on each fibre L t,x ; this innerproduct is assumed to be linear on the second variable and it is continuous sothat the inner products on C ( L t ) are well defined. The left action of C ( D t ) and MENABILITY FOR PARTIAL ACTIONS AND FELL BUNDLES 45 the right action of C ( D t − ) on C ( L t ) are given by ( f · ξ )( x ) := f ( x ) ξ ( x ) and( ξ · g )( x ) := ξ ( x ) g ( θ − t ( x )) for all x ∈ D t .Having this, essentially the same proof as before still works: we take an element η ∈ C c ( L t ) and verify that(8.7) X s ∈ G ξ i ( ts ) ∗ · η · ξ i ( s ) → η with respect to the norm of C ( L t ) ∼ = B t . But the expression above is a section of L t and when we compute at some x ∈ D t we get: X s ∈ G ζ i ( x, s − t − ) η ( x ) ζ i ( θ − t ( x ) , s − )which as before can be rewritten as X α ∈ Γ ζ i ( α ) ζ i ( αγ ) ! η ( x ) = ( ζ ∗ i ∗ ζ i )( γ ) η ( x ) . Using that ( ζ ∗ i ∗ ζ i )( γ ) → (cid:3) Corollary 8.8.
Let B be a Fell bundle which is weakly equivalent to a Fell bundle A with commutative unit fibre. Then B has the AP if and only if it has the WAP(i.e. is AD-amenable).Proof. We showed that both the WAP and the AP are preserved by the weakequivalence of Fell bundles and our last Theorem implies A has the AP if and onlyif it has the WAP. Thus the claim follows. (cid:3) Given a Fell bundle B as in the Corollary above, there may not be a suitablecandidate for the spectral grupoid, mainly because the spectrum of B e may notbe Hausdorff. Consider for example the semidirect product bundle of a Moritaenveloping action of a partial action on a commutative C*-algebra which does nothave an enveloping action [2]. If we want to generalize Theorem 8.5, it is thenreasonable to assume B e is Morita equivalent to a commutative C*-algebra. Theorem 8.9.
Suppose B is a Fell bundle over G, A is a C*-algebra and M isan A - B e -equivalence bimodule. Consider, for each t ∈ G, B t as a left B e -Hilbertmodule and let M ⊗ B t be the B e -inner tensor product. Then there exists a uniqueright Hilbert B -bundle structure [6, Definition 2.1] on X := { M ⊗ B t } t ∈ G such that,for all x, y ∈ M and a, b ∈ B : h x ⊗ a, y ⊗ b i B = a ∗ h x, y i B e b and ( x ⊗ a ) b = x ⊗ ( ab ) . Let also K ( X ) = { K t } t ∈ G be the bundle of generalized compact operators, as in [6, Theorem 3.9] . Then X is a strong K ( X ) − B equivalence and the unit fibre K e is isomorphic to A. Proof.
Uniqueness follows from [6] and the fact that elementary tensor productsspan a dense subspace of the tensor products M ⊗ B t , thus we only need to provethe existence claim. First we show that the action of B on X is defined. Take r, s ∈ G, x , . . . , x n ∈ M,a , . . . , a n ∈ B r and b ∈ B s . Then k n X i =1 x i ⊗ ( a i b ) k = k n X i,j =1 b ∗ a ∗ i h x i , x j i B e a j b k = k b ∗ h n X i =1 x i ⊗ a i , n X i =1 x i ⊗ a i i b k≤ k b k k n X i =1 x i ⊗ a i k . With the inequality above we can easily prove the existence of a bilinear map( M ⊗ B r ) × B s → M ⊗ B rs , ( u, b ) ub, such that ( x ⊗ a ) b = x ⊗ ( ab ) and k ub k ≤ k u kk b k . These maps define the action of B on X . We now construct the B -valued inner product of X . Take r, s ∈ G, x , . . . , x n ∈ M, y , . . . , y n ∈ M, a , . . . , a n ∈ B r and b , . . . , b n ∈ B s . Set u := P ni =1 x i ⊗ a i ∈ M ⊗ B r , v := P ni =1 y i ⊗ b i ∈ M ⊗ B s and w := P ni,j =1 a ∗ i h x i , y j i B e b j ∈ B r − s . Inorder to prove the inner product is defined it suffices to show that(8.10) k w k ≤ k u kk v k , because after this inequality we can set h u, v i B := w. Let [ u, u ] r ∈ K ( M ⊗ B r ) represent the generalized compact operator z u h u, z i , and let ϕ r : A = K ( M ) → K ( M ⊗ B r ) be the unique *-homomorphismsuch that ϕ r ( a )( z ⊗ c ) = ( az ) ⊗ c. It is straightforward to show that [ u, u ] r = ϕ r ( P ni,j =1 A h x i a i b ∗ j , x j i ) . We know ϕ r may not be injective, but it is injective whenrestricted to the ideal span h M B r B ∗ r , M i . Thus(8.11) k u k = k [ u, u ] r k = k n X i,j =1 A h x i a i b ∗ j , x j ik = k n X i,j =1 A h x i , x j b j a ∗ i ik . To prove (8.10) note that w ∗ w = P ni,j,k,l =1 b ∗ j h y j , x i i B e a i a ∗ l h x l , y k i B e b k . We have a := ( a i a ∗ l ) ni,l =1 ∈ M n ( B e ) + and, if d := a / ∈ M n ( B e ) , then a i a ∗ l = P np =1 d i,p d l,p ∗ . This implies w ∗ w = n X i,j,k,l,p =1 b ∗ j h y j , x i i B e d i,p d l,p ∗ h x l , y k i B e b k = n X i,j,k,l,p =1 b ∗ j h A h x l d l,p d ∗ i,p , x i i y j , y k i B e b k = n X j,k =1 b ∗ j h n X i,l =1 A h x l a l a ∗ i , x i i y j , y k i B e b k . Consider the direct sum of n copies of M, ⊕ n M, as a M n ( A ) − B e equivalencebimodule with left and right inner products given by M n ( A ) h ( f , . . . , f n ) , ( g , . . . , g n ) i := ( A h f i , g j i ) ni,j =1 h ( f , . . . , f n ) , ( g , . . . , g n ) i B e := n X i =1 h f i , g i i B e . The action on the right of B e on ⊕ n M is given by entrywise multiplication, while theaction of M n ( A ) on the left is given by matrix multiplication by considering the ele-ments of ⊕ n M as column vectors with entries in M. Let A → M n ( A ) , d diag( d ) , be the inclusion of A in the diagonal. Consider also ⊕ n B s as a M n ( B e ) − B e bimod-ule, with analogous operations. In (8.11) we identified d := P ni,l =1 A h x l a l a ∗ i , x i i MENABILITY FOR PARTIAL ACTIONS AND FELL BUNDLES 47 with [ u, u ] r via ϕ r , thus d ≥ . With these considerations and defining ξ :=( b , . . . , b n ) ∈ ⊕ n B s and η := ( y , . . . , y n ) ∈ ⊕ n M we have w ∗ w = h ξ, M n ( B e ) h η, diag( d ) η i ξ i B e . Viewing the direct sum of adjoints ⊕ n M ∗ as an M n ( B e ) − A Hilber bimodule, wededuce that M n ( B e ) h η, diag( d ) η i ≤ k d k M n ( B e ) h η, η i . By (8.11), in B e we have w ∗ w ≤ k d kh ξ, M n ( B e ) h η, η i ξ i B e = k u k h v, v i B e , and this clearly implies k w k ≤ k u kk v k . At this point we have shown that the inner product of X and the action of B on X are defined, the reader can now check that these operations satisfy the conditionsof [6, Definition 2.1].In the rest of the proof we use the notation of [6, Theorem 3.9]. Recall inparticular that, for u ∈ M ⊗ B r and v ∈ M ⊗ B s , [ u, v ] is the adjointable operatorof order rs − given by X → X , w u h v, w i B . The ajointable operators of order e of X , B e ( X ) , form a C*-algebra and K e = span { [ u, v ] : u, v ∈ M ⊗ B t , t ∈ G } = span { [ x ⊗ b, y ⊗ c ] : x ⊗ b, y ⊗ c ∈ M ⊗ B r , r ∈ G } . There exists a unique *-homomorphism ϕ : A → B e ( X ) such that ϕ ( a )( x ⊗ b ) =( ax ) ⊗ b. Note that ϕ is injective because we can think of the unit fiber M ⊗ B e as an A − B e equivalence bimodule. If x ⊗ b, y ⊗ c ∈ M ⊗ B r , then for all z ⊗ d ∈ M ⊗ B s we have[ x ⊗ b, y ⊗ c ]( z ⊗ d ) = x ⊗ bc ∗ h y, z i B e d = xbc ∗ h y, z i B e ⊗ d = A h xbc ∗ , y i z ⊗ d = ϕ ( A h xbc ∗ , y i ) z ⊗ d. Since the elements [ x ⊗ b, y ⊗ c ] span a dense subset of K e , we conclude that ϕ ( A ) = K e is isomorphic to A. In order to prove that X is a strong equivalence we must show that(8.12) span K t K ∗ t = span[ M ⊗ B t , M ⊗ B t ] , ∀ t ∈ G. Recall that K t is the closure in B t ( X ) of span { [ u, v ] : u ∈ M ⊗ B tr , v ∈ M ⊗ B r , r ∈ G } . Fix t ∈ G and take r , r ∈ G, x i ⊗ a i ∈ B tr i , y i ⊗ b i ∈ M ⊗ B r i , for i = 1 , . Then[ x ⊗ a , y ⊗ b ][ x ⊗ a , y ⊗ b ] ∗ ( z ⊗ c ) = [ x ⊗ a , y ⊗ b ] y ⊗ b a ∗ h x , z i B e c = x ⊗ a b ∗ h y , y i B e b a ∗ | {z } ∈ B e h x , z i B e c = x a b ∗ h y , y i B e b a ∗ h x , z i B e ⊗ c = A h x a b ∗ h y , y i B e b a ∗ , x i z ⊗ c = ϕ ( A h x a b ∗ h y , y i B e b a ∗ , x i ) z ⊗ c = [ x ⊗ a b ∗ h y , y i B e , x ⊗ a b ∗ ]( z ⊗ c ) . This implies the inclusion ⊆ in (8.12). To prove the converse take x ⊗ a, y ⊗ b ∈ M ⊗ B t . We can write a = a b ∗ and b = a b ∗ for some a , a ∈ B t and b , b ∈ B e (by Cohen-Hewitt’s Theorem). We can also approximate a = a b ∗ in norm by sums of elements of the form a b ∗ h y , y i B e . This allows us to approximate, in B e ( X ) , the operator [ x ⊗ a, y ⊗ b ] by sums of operators of the form[ x ⊗ a b ∗ h y , y i B e , x ⊗ a b ∗ ] = [ x ⊗ a , y ⊗ b ][ x ⊗ a , y ⊗ b ] ∗ ∈ K t K ∗ t . Thus the inclusion ⊇ in (8.12) follows. (cid:3) Corollary 8.13.
Suppose B is a Fell bundle over G and that B e is Morita equivalentto a commutative C*-algebra C ( X ) through an equivalence bimodule M. Identify X with the primitive ideal space of B e and let α be the partial action defined by B on C ( X ) . Let also Γ be the grupoid associated to α. Then the following assertionsare equivalent:(i) B has the WAP or is AD-amenable, that is, C ∗ (r) ( B ) is nuclear;(ii) B α has the WAP or is AD-amenable;(iii) C ∗ (r) ( B α ) = C ∗ (r) (Γ) = C ( X ) ⋊ α, (r) G is nuclear;(iv) Γ is amenable;(v) B α has the AP;(vi) for every -cocycle ω for α , the corresponding Fell bundle B α,ω has the AP;(vii) B has the AP.Proof. Let X = { M ⊗ B t } t ∈ G be the equivalence bundle of Theorem 8.9. Since X isa strong equivalence bundle and C ( X ) is the unit fibre of K ( X ) , α is (isomorphicto) the partial action defined by K ( X ) [5]. These facts and Theorem 8.5 imply that: • (ii) to (vi) are equivalent to: (i’) K ( X ) has the WAP or is AD-amenable,that is, C ∗ (r) ( K ( X )) is nuclear; and to (vii’) K ( X ) is amenable. • (i) ⇔ (i’). • (vii) ⇔ (vii’). (cid:3) Example . In [24, Proposition 37.9] Exel provides a partial crossed productdescription for the C*-algebra of every directed graph E = ( s, r : E → E ) withno sinks (i.e. s − ( v ) = ∅ for all v ∈ E ). In other words, we have an isomorphism C ∗ ( E ) ∼ = C ( X ) ⋊ α G for a certain partial action α of the free group G = F n on n = | E | generators(this can be infinite), and X is a certain (totally disconnected) locally compactHausdorff space. The exact description of this space and the partial action isslightly complicated in general but it simplifies under certain regularity conditionson E . For instance, if every vertex v ∈ E is regular in the sense that r − ( v ) isnon-empty and finite, that X is just the infinite path space E ∞ of E .Regardless of how X and the partial action α above are defined, using thatgraph C ∗ -algebras are always nuclear (a well-known fact, see [29, Proposition 2.6]),it follows from our previous theorem that α has the AP. Indeed, Exel gives a moredirect proof of this fact in [24, Theorem 37.10].We shall give more details about the partial action α and its amenability inwhat follows in the case of the graph E that describes the Cuntz algebra O n , thatis, the graph with one vertex and n loops with 2 ≤ n < ∞ . This is a specialand representative case. This is a finite graph that has no sinks or sources. Inthis case, X ∼ = { , . . . , n } ∞ is Cantor space and G = F n is the free group on n generators that we also view as the free group generated by E . The partialaction α is defined as follows: the domains D g for g ∈ F n are defined in terms ofthe cylinders X a = { aµ : µ ∈ X } if g ∈ F n can be written in reduced form as MENABILITY FOR PARTIAL ACTIONS AND FELL BUNDLES 49 g = ab − for a, b ∈ E ∗ , the set of finite paths viewed as elements of F n . In this case D g − = C ( X b ) and D g = C ( X a ) and α g : D g − ∼ −→ D g is given α g ( f ) = f ◦ θ − g ,where θ g : X b ∼ −→ X a is the canonical homeomorphism sending bµ aµ . If g is notof the form ab − , then D g is defined to be the zero ideal (and α g is the zero map).The AP for α means the existence of a net of finitely supported functions ξ i : G → C ( X ) that is uniformly bounded for the ℓ -norm and satisfying(8.15) h ξ i | a ˜ α g ( ξ i ) i := X h ∈ G ξ i ( h ) ∗ α g ( α − g ( aξ i ( g − h ))) → a for all g ∈ G and a ∈ D g . Notice that all the ideals D g are unital here. If 1 g denotes its unit (so that D g = A · g ), then (8.15) is equivalent to X h ∈ F n ξ i ( h ) α g (1 g − ξ i ( g − h )) → g for all g ∈ G . One explicit sequence ξ i : G → C ( X ) that gives the AP for thispartial action can be defined by ξ i ( g ) = √ i g if g ∈ F + n (the positive cone of F n )with length | g | ≤ i and ξ i ( g ) = 0 otherwise. Recall that 1 g denotes the characteristicfunction on the cylinder set X g = { gµ : µ ∈ X = E ∞ } which makes sense because g is positive.The fact that all domain ideals D g are unital also implies that α has an envelop-ing global action and we know from Corollary 6.26 that this global action also hasthe AP or, equivalently, it is AD-amenable. Indeed, a concrete description of theenveloping action for the partial action of F n on X is as follows: instead of con-sidering only words on positive words, we also consider their inverses, that is, weconsider the generators of F n and their inverses, and then look at all infinite reducedwords on this new alphabet. This yields a new space, denoted ¯ X that naturallycontains X as a clopen subspace. Now notice that F n naturally acts (globally) on¯ X by (left) concatenation and the partial action on X is just the restriction of thisglobal action. Moreover, the global action of F n on ¯ X is known to be amenable:this action can be viewed as the action on a certain boundary of F n , and this is anamenable action, see [9, Examples 2.7(4)] and [14, Proposition 5.1.8]. Indeed, thisis the standard way to see that F n is an exact group. Appendix A. W*-bimodules and their representations
Let M be a W ∗ -algebra. A W ∗ -Hilbert M -module is an ordinary C ∗ -Hilbert M -module X which is isometrically isomorphic to a dual Banach space, X ∼ = X ′∗ , andsuch that the M -action and M -inner product are separately w ∗ -continuous. Theseare exactly the self-dual Hilbert modules; this means that every bounded M -linearmap X → M is of the form y
7→ h x | y i M for some (uniquely determined) element x ∈ X . In this case the predual X ∗ is unique up to isomorphism.In a similar fashion one defines left W ∗ -modules and W ∗ -bimodules (requiringboth left and right inner products and actions to be separately w ∗ -continuous).Specially, we want to emphasise the W ∗ -equivalence bimodules: Definition A.1.
Given two W ∗ -algebras, M and N , a W ∗ -equivalence bimodule is a W ∗ -Hilbert M - N -bimodule X such that the left and right inner products spanw ∗ -dense ideals of M and N .Here is an elementary concrete example: for Hilbert spaces H , K , the space L ( H, K ) is a W ∗ -equivalence L ( K )- L ( H )-bimodule with respect to the obvious operations given by composition and adjunction of operators. For example, theright inner product is given by h S | T i L ( H ) := S ∗ T . The predual in this case canbe identified with K ( H, K ). On bounded subsets the w ∗ -topology coincides withthe weak topology, that is, T i → T with respect to the w ∗ -topology if and only if h u | T i v i → h u | T v i whenever { T i } i ∈ I is a norm-bounded net.Every W ∗ -equivalent bimodule can be faithfully represented into some concretebimodule of the form L ( H, K ) as above. We explain in what follows how this canbe done.Let X be a W ∗ -equivalence M - N -bimodule. Using the notation of [35], we view X as a ternary W ∗ -ring with the ternary operation( x, y, z ) := x h y, z i N = M h x, y i z. We now indicate how to translate the fundamental results of Zettl [35] to repre-sent W ∗ -equivalence bimodules on Hilbert spaces.For example, Zettl shows that the adjointable operators of X N , L ( X ) , form aW*-algebra. We indicate how to represent this algebra W*-faithfully. Lemma A.2.
Let X be a W*-Hilbert right M -module. Consider a unital andfaithful W*-representation M ⊂ L ( H ) , and let K be the Hilbert space X ⊗ M H. Then the representation ρ : L ( X ) → L ( K ) , such that ρ ( T )( x ⊗ h ) = T x ⊗ h, is aunital and faithful W*-representation. Moreover, a bounded net { T i } i ∈ I ⊂ L ( X ) w ∗ -converges to T if and only if {h y, T i x i} i ∈ I w ∗ -converges to h y, T x i , for all x, y ∈ X. Proof.
Clearly ρ is an injective and unital *-homomorphism. To show that theimage of ρ, M, is a concrete W*-algebra it suffices to prove that its closed unit ball M is wot closed.Take a net { ρ ( T i ) } i ∈ I ⊂ M that weakly converges to R ∈ L ( K ) . If X isthe closed unit ball of X with the w ∗ -topology, then X is compact and so it is Y := Π x ∈ X ( X × X ) . Let h : L ( X ) → Y be such that h ( T ) x = ( T x, T ∗ x ) . Then { h ( T i ) } i ∈ I has a converging subnet { h ( T i j ) } j ∈ J . This implies the existence of twolinear maps
U, V : X → X such that U x = lim j T i j x and V x = lim j T ∗ i j x, in thew ∗ -topology, for all x ∈ X. Hence, for all x, y ∈ X, h U x, y i = lim j h T i j x, y i = lim j h x, T ∗ i j y i = h x, V y i , where the limits are taken with respect to the w ∗ -topology. Then U ∈ L ( X ) andour construction implies k U k ≤ . We have ρ ( U ) = R because, for all x, y ∈ X and h, k ∈ H, h x ⊗ h, ρ ( U )( y ⊗ k ) i = lim j h h, h x, T i j y i k i = lim j h x ⊗ h, ρ ( T i j )( y ⊗ k ) i = h x ⊗ h, R ( y ⊗ k ) i . This shows that M is wot closed, hence M is a concrete W*-algebra.On bounded sets of L ( K ) the wot topology (and hence the w ∗ -topology) isdetermined by the functionals R
7→ h x ⊗ h, R ( y ⊗ k ) i . When translated to L ( X )this means that on bounded sets of L ( X ) the w ∗ -topology is determined by thefunctionals L ( X ) → X, T
7→ h x, T y i , considering on X the w ∗ -topology. (cid:3) Definition A.3.
Let X be a W ∗ -equivalence M - N -bimodule. A representation of X is a linear and w ∗ -continuous map π : X → L ( H, K ), where H and K areHilbert spaces and π ( x h y, z i N ) = π ( x ) π ( y ) ∗ π ( z ) for all x, y, z ∈ X . We say that π is nondegenerate if K = span π ( X ) H and H = span π ( X ) ∗ K . MENABILITY FOR PARTIAL ACTIONS AND FELL BUNDLES 51
Proposition A.4 ([35]) . Every W ∗ -equivalence bimodule admits a nondegeneratefaithful representation.Proof. Every W ∗ -equivalence M - N -bimodule X is a ternary W ∗ -ring, and by [35]has a faithful representation π : X → L ( H, K ). Let H := span π ( X ) ∗ K and K = span π ( X ) H . Clearly, span π ( X ) H ⊆ K and span π ( X ) ∗ K ⊆ H . Weclaim that π ( X )( H ⊥ ) = 0. If h ∈ H ⊥ and x ∈ X , then π ( x ) ∗ π ( x ) h ∈ H and k π ( x ) h k = h π ( x ) h, π ( x ) h i = h π ( x ) ∗ π ( x ) h, h i = 0 . In a similar way it can be shown that π ( X ) ∗ ( K ⊥ ) = 0. Thus we may consider therepresentation π : X → L ( H , K ) given by π ( x ) h = π ( x ) h .It suffices to show that π is nondegenerate. We show that span π ( X ) H = K ;the proof of span π ( X ) ∗ K = H is analogous. Take x ∈ X and h ∈ H . Wecan approximate x by sums of elements of the form u h v, w i , thus π ( x ) h lies in theclosed linear span of π ( X ) π ( X ) ∗ π ( X ) h ⊆ π ( X )( H ) = π ( X ) H . Hence π ( x ) h ∈ span π ( X ) H . (cid:3) Definition A.5.
Given a representation π of a W ∗ -equivalence bimodule, the rep-resentation π constructed in the proof above is called de essential part of π . Proposition A.6.
Let X be a W ∗ -equivalence M - N -bimodule and π : X → L ( H, K ) a nondegenerate representation. Then there exists a unique unital and normal rep-resentation π l : M → L ( K ) such that π l ( M h x, y i ) = π ( x ) π ( y ) ∗ for all x, y ∈ X . If π is faithful then so it is π l .Proof. Let M X be the norm closure of span M h X, X i in M . By [2, Proposition 4.1]there exists a unique *-homomorphism ρ l : M X → L ( K ) such that ρ l ( M h x, y i ) = π ( x ) π ( y ) ∗ for all x, y ∈ X . We claim this representation can be extended in aunique way to a normal representation π l of M .Let ρ r : N X → L ( H ) be the *-homomorphism such that ρ r ( h x, y i N ) = π ( x ) ∗ π ( y ).Since π is nondegenerate, there exists a unique unitary U : X ⊗ ρ r H → H such that U ( x ⊗ h ) = π ( x ) h . We also have a representation µ : M → L ( X ⊗ π H ) = L ( K ) suchthat µ ( a ) π ( x ) h = π ( ax ) h . We claim µ is w ∗ -continuous, to show this it suffices toprove µ is w ∗ -continuous on the closed unit ball M . Since µ ( M ) is bounded, itsuffices to prove that given a net { a λ } λ ⊆ M that w ∗ -converges to a ∈ M , thenlim λ h µ ( a λ )( x ⊗ π ( y ) ∗ k ) , ( x ⊗ π ( y ) ∗ k ) i = h µ ( a )( x ⊗ π ( y ) ∗ k ) , ( x ⊗ π ( y ) ∗ k ) i , for every x, y ∈ X and k ∈ K . Butlim λ h µ ( a λ )( x ⊗ π ( y ) ∗ k ) , ( x ⊗ π ( y ) ∗ k ) i = lim λ h h, π ( y h a λ x, x i ) π ( y ) ∗ h i = h h, π ( y h ax, x i ) π ( y ) ∗ h i = h µ ( a )( x ⊗ π ( y ) ∗ k ) , ( x ⊗ π ( y ) ∗ k ) i . It is straightforward to show that the restriction of µ to M X is ρ l , thus π l := µ is the unique w ∗ -continuous extension of ρ l .If π is faithful and π l ( a ) = 0 then π ( ax ) h = π l ( a ) π ( x ) h = 0 for every x ∈ X and h ∈ H . Hence ax = 0 for every x ∈ X and this implies a = 0. (cid:3) Definition A.7.
Let X be a W ∗ -equivalence A - B -bimodule and let Y be a W ∗ -equiv-alence M - N -bimodule. A map π : X → Y is a W ∗ -homomorphism if it is linear,w ∗ -continuous and π ( x h y, z i N ) = π ( x ) h π ( y ) , π ( z ) i B for every x, y, z ∈ X . Proposition A.8 ([2, Proposition 4.1]) . Let X and Y be W ∗ -equivalence M − N and A - B -bimodules, respectively, and π : X → Y a w ∗ -continuous linear mapsuch that π ( x h y, z i N ) = π ( x ) h π ( y ) , π ( z ) i B for every x, y, z ∈ X . Then thereexist unique w ∗ -continuous homomorphisms π l : M → A and π r : N → B suchthat π l ( h x, y i N ) = h π ( y ) , π ( z ) i B and π ( M h y, z i ) = A h π ( x ) , π ( y ) i and π ( h y, z i N ) = h π ( x ) , π ( y ) i B for every x, y ∈ X . If π is an isomorphism then so are π r and π l .Proof. We construct π r , the map π l can be constructed considering the adjointmodule of X . Take a nondegenerate and faithful representation ρ : Y → L ( H, K ).In this situation ρ l : M → L ( K ) is a faithful unital W ∗ -representation. Then ρ ◦ π : X → L ( H, K ) is a representation, that may or may not be nondegenerate.In any case, the essential part ( ρ ◦ π ) is nondegenerate and we may think of ρ ◦ π as the null extension of ( ρ ◦ π ) (from H to H ⊕ ( H ⊥ )).We know ( ρ ◦ π ) r : A → L ( K ) is a W ∗ -homomorphism. Define ( ρ ◦ π ) r : A →L ( K ) as the null extension of ( ρ ◦ π ) r . We claim ( ρ ◦ π ) r ( A ) ⊆ ρ r ( M ). Indeed, note( ρ ◦ π ) r ( A ) is the w ∗ -closure of ( ρ ◦ π ) r ( A Y ). Considering only the C*-structre Y andusing the map π r : M X → A Y of [2, Proposition 4.1], we get that ( ρ ◦ π ) r ( A h x, y i ) = ρ r ( M h π r ( x ) , π r ( y ) i ) ∈ ρ r ( M ). Then the map π r : M → A we are looking for isthe unique w ∗ -continuous extension of π r : M X → A Y and can be computed as( ρ r ) − ◦ ( ρ ◦ π ) r . In case π is an isomorphism ( π − ) r is the inverse of π r . (cid:3) Let X be a W ∗ -equivalence M - N -bimodule. The W ∗ -linking algebra of X is theBanach space formed by all all the matrices (cid:18) a x ˜ y b (cid:19) , where ˜ Y is the module conjugate to X . To give L a W ∗ -algebra structure take afaithful and nondegenerate representation π : X → L ( H, K ). Then ρ : L → L ( H ⊕ K ) = (cid:18) L ( H ) L ( H, K ) L ( K, H ) L ( H, K ) (cid:19) ; ρ (cid:18) a x ˜ y b (cid:19) = (cid:18) π l ( a ) π ( x ) π ( y ) π r ( b ) (cid:19) , is a faithful representation of *-algebras and ρ induces a C*-algebra structure on L . Moreover, ρ ( L ) is a unital subalgebra closed with respect to the weak operatortopology (wot) because convergence in L ( H ⊕ K ) in the wot is just entrywise wot-convergence. This implies that M, N and X are w ∗ -closed subspaces of L . Inparticular M and N are hereditary W ∗ -subalgebras of L . Proposition A.9.
Let X be a W ∗ -equivalence M - N -bimodule. Then there existsa unique W ∗ -isomorphism π X : Z ( N ) → Z ( M ) such that xa = π ( a ) x for all a ∈ Z ( M ) and x ∈ X .Proof. By the definition of the centre of an algebra we have (cid:18) a x ˜ y b (cid:19) ∈ Z ( L ) ⇔ x = y = 0 , a ∈ Z ( M ) , b ∈ Z ( N ) , az = zb ∀ z ∈ X. Note a and b completely determine each other, thus we have an injective w ∗ -continuous *-homomorphism π : Z ( L ) → Z ( N ) , π (cid:18) a b (cid:19) = b. MENABILITY FOR PARTIAL ACTIONS AND FELL BUNDLES 53
We claim π is surjective and hence an isomorphism. Note the image of π isa W ∗ -algebra because π is w ∗ -continuous. Hence it suffices to show that Im ( π )contains every projection of Z ( N ).Let p ∈ Z ( N ) be a projection and take a unital and normal representation on aHilbert space ρ : N → L ( H ) with ker( ρ ) = (1 − p ) N . Consider the representationof M induced by ρ through X , Ind ( ρ ), and let q ∈ M be projection such thatker( Ind ( ρ )) = (1 − q ) M . Then for all x, y ∈ X and h, k ∈ H : h k, π ( y ∗ ( qx − xp )) k i = h k, π ( y ∗ qx ) k i − h k, π ( y ∗ x ) k i = h Ind ( ρ )( q )( y ⊗ ρ k ) , ( x ⊗ ρ h ) i − h ( y ⊗ ρ k ) , ( x ⊗ ρ h ) i = 0We conclude y ∗ ( qx − xp ) p = 0 for all x, y ∈ X and this implies qxp = xp for all x ∈ X . By symmetry (or double induction) we get qx = xp for all x ∈ X . Then p = π (cid:0) q p (cid:1) and π is surjective. (cid:3) It may look strange to consider the above isomorphism π X as a homomorphism Z ( N ) → Z ( M ) and not the opposite. Implicitly we have chosen this conventionbecause we view X as a “generalised” morphism from N to M . This choice alsomakes it easier to see that the constructions we perform in Section 5.2 are exactlythe W ∗ -counterparts of that in [1]. Another motivation for our notation is therelation between the composition and W ∗ -tensor products in Remark B.4. Corollary A.10.
Let X and Y be two W ∗ -equivalence M - N -bimodules and ρ : X → Y an isomorphism such that π l and π r are the identities on M and N , respectively.Then the isomorphisms π X and π Y of Proposition A.9 are equal.Proof. Take a ∈ Z ( N ). For all x ∈ X we have xa = π x ( a ) x , hence ρ ( x ) a = ρ ( x ) ρ r ( a ) = ρ ( ax ) = ρ ( π X ( a ) x ) = ρ l ( π X ( a )) ρ ( x ) = π X ( a ) ρ ( x ) . Since ρ is surjective we deduce that π X ( a ) = π Y ( a ). (cid:3) Appendix B. Induction of representations and tensor products
Proposition B.1.
Let X be an W ∗ -equivalence M - N -bimodule and π : N → L ( H ) a unital W ∗ -representation. If K := X ⊗ π H , then there exists a unique nonde-generate representation ˆ π : X → L ( H, K ) such that ˆ π ( x ) h = x ⊗ h . Moreover, if ρ : X → L ( H, K ) is a nondegenerate representation then there exists a unique uni-tary U : X ⊗ ρ r H → K such that U ( x ⊗ h ) = ρ ( x ) h and we have ρ ( x ) = U ◦ b ρ r ( x ) for all x ∈ X . In particular, ρ is faithful if and only if ρ r is faithful.Proof. Regarding ˆ π , we only have to show it is w ∗ -continuous. It suffices to showthat given a bounded net { x i } i ∈ I ⊆ X that w ∗ -converges to x ∈ X , we havelim i h ˆ π ( x i ) h, y ⊗ k i = h π ( x i ) h, y ⊗ k i for all y ∈ X and h, k ∈ H . But the separatew ∗ continuity of the inner products implieslim i h ˆ π ( x i ) h, y ⊗ k i = lim i h h, π ( h x i , y i N ) k i = h h, π ( h x, y i N ) k i = h ˆ π ( x ) h, y ⊗ k i Now consider a nondegenerate representation ρ as in the statement. There existsa unique linear isometry U : X ⊗ ρ r H → K such that U ( x ⊗ h ) = ρ ( x ) h because h ρ ( x ) h, ρ ( y ) k i = h h, ρ r ( h x, y i N ) k i . This isometry is in fact surjective because ρ is nondegenerate. Then U ◦ b ρ r ( x ) h = U ( x ⊗ h ) = π ( x ) h . We know ρ r is faithful whenever ρ is. Assume ρ r is faithful and ρ ( x ) = 0.Then, for every h ∈ H , h h, ρ r ( h x, x i N ) h i = k ρ ( x ) h k = 0. Thus h x, x i N = 0 and x = 0. (cid:3) Definition B.2.
Given a W ∗ -equivalence M - N -bimodule X and a unital W ∗ -rep-resentation π : N → L ( H ), the representation induced by π through X , denotedInd X ( π ), is b π l .Let X and Y be W ∗ -equivalence M - N and N - P -bimodules, respectively. Wewant to construct a tensor product X ⊗ w ∗ N Y , with a natural M − P W ∗ -equivalencebimodule structure.Take faithful and unital W ∗ -representation π : P → L ( H ) and define H π :=( X ⊗ N Y ) ⊗ π H , where X ⊗ N Y is the usual tensor product of Hilbert modules. Wehave a natural representation ˆ π : X ⊗ N Y → L ( H, H π ) such that ˆ π ( x ⊗ y ) h = x ⊗ y ⊗ h .We define X ⊗ πN Y as the wot-closure of π ( X ⊗ N Y ). Note π ( P ) = span wot { π ( u ) ∗ π ( v ) : u, v ∈ X ⊗ N Y } Ind Y (Ind X ( π ))( M ) = span wot { π ( u ) π ( v ) ∗ : u, v ∈ X ⊗ N Y } Then we may think of X ⊗ πN Y as a W ∗ -equivalence M - P -bimodule.Let ρ : P → L ( H ) be another unital and faithful W ∗ -representation. We haverepresented X ⊗ N Y faithfully, as a ternary C*-ring, in L ( H, H π ) and in L ( K, K ρ ).With some abuse of notation we denote these representations ˆ π : X ⊗ N Y →L ( H, H π ) and ˆ ρ . There exists a unique isomorphism of ternary C*-rings µ : ˆ π ( X ⊗ N Y ) → ˆ ρ ( X ⊗ N Y ) such that µ (ˆ π ( x ⊗ y )) = ˆ ρ ( x ⊗ y ).We claim that µ is continuous on bounded sets with respect to the wot topologies.Indeed, let { u i } i ∈ I ⊆ X ⊗ N Y be a bounded net and u ∈ X ⊗ N Y such that { ˆ π ( u i ) } i ∈ I wot-converges to ˆ π ( u ). Then, for every h, k ∈ H and v ∈ X ⊗ N Y , wehave lim i h h, π ( h u i , v i M ) k i = lim i h ˆ π ( u i ) h, ˆ π ( v ) k i = h ˆ π ( u ) h, ˆ π ( v ) k i = h h, π ( h u, v i ) k i . In fact we can conclude that the wot-convergence of { ˆ π ( u i ) } i ∈ I to ˆ π ( u ) is equivalentto the w ∗ convergence of {h u i , v i M ) } i ∈ I to h u, v i M for every v ∈ X ⊗ N Y , whichin turn is equivalent to the wot-convergence of { ˆ ρ ( u i ) } i ∈ I to ˆ ρ ( u ). Then µ has aunique extension to a W ∗ -isomorphism µ : X ⊗ πN Y → X ⊗ ρN Y . Definition B.3.
The W ∗ -tensor product X ⊗ w ∗ N Y is the W ∗ -isomorphism class ofthe modules X ⊗ πN Y . As usual we abuse the notation and view X ⊗ w ∗ N Y as anyof its representatives. Remark
B.4 . (1) Almost by construction we have, for any unital W ∗ -representation π : P → L ( H ), that Ind X ⊗ w ∗ N Y ( π ) is unitarly equivalent to Ind Y (Ind X ( π )).(2) If π X : Z ( N ) → Z ( M ) and π Y : Z ( P ) → Z ( N ) are the isomorphisms ofProposition A.9, then π X ◦ π Y = π X ⊗ w ∗ N Y . Appendix C. Biduals of Hilbert bimodules
Proposition C.1.
Let X be a Hilbert A - B -bimodule. Then there exists a uniqueHilbert A ′′ - B ′′ -bimodule structure on X ′′ extending that of X and with (left andright) inner products and actions of A ′′ and B ′′ separately w ∗ -continuous. Moreover,if X is an equivalence A - B -bimodule, that is, if the left and right inner products on MENABILITY FOR PARTIAL ACTIONS AND FELL BUNDLES 55 X generate A and B as C ∗ -algebras, then the inner products on X ′′ generate A ′′ and B ′′ as W ∗ -algebras, that is, X ′′ is a W ∗ -equivalence A ′′ - B ′′ -bimodule.Proof. Uniqueness follows immediately because X and A and B are w ∗ -dense in X ′′ , A ′′ and B ′′ , respectively. Let L be the linking algebra of X . Since A and B are C ∗ -subalgebras of L , we may view A ′′ and B ′′ as W ∗ -subalgebras of L ′′ . Moreover,we also view X as a closed subspace of L and identify X ′′ with the w ∗ -closure of X in L ′′ . Note that AXB ⊆ X , XX ∗ ⊆ A and X ∗ X ⊆ B imply A ′′ X ′′ B ′′ ⊆ X ′′ , X ′′ X ′′∗ ⊆ A ′′ and X ′′∗ X ′′ ⊆ B ′′ because the multiplication of L ′′ is separatelyw ∗ -continuous and the involution of L ′′ is w ∗ -continuous. The rest follows directlybecause the Hilbert module operations of X ′′ are defined in terms of the W ∗ -algebrastructure of L ′′ . (cid:3) The reader should note that X ′′ is usually not an A ′′ - B ′′ -equivalence bimodule inthe C ∗ -sense because the images of the inner products on X ′′ might be not linearly norm dense in A ′′ or B ′′ (only w ∗ -dense). Proposition C.2.
Let A and B be C ∗ -algebras and X an A - B -equivalence bimod-ule. Given a nondegenerate representation π : B → L ( H ) , write π ′′ : B ′′ → L ( H ) for its unique w ∗ -continuous extension, and Ind AX π : A → L ( X ⊗ π H ) for the rep-resentation induced by π through X . Then (Ind AX π ) ′′ is faithful if and only if π ′′ isfaithful.Proof. This result is certainly well-known, but we could not find it explicitly in theliterature, so we give a proof here. A quick way to prove the statement is to noticethat the induction process of representations via an equivalence bimodule preservesquasi-equivalence of representations which is, in turn, determined by their centralcover projections, see [32, Section 3.8]. And π ′′ is faithful if and only if its centralcover is zero.A more elementary way to prove the result is as follows: for a ∈ A ′′ , ξ , ξ ∈ X and v , v ∈ H , we have h ξ ⊗ v | Ind AX ( π ) ′′ ( a )( ξ ⊗ v ) i = h v | π ′′ ( h ξ | a · ξ i ) v i where a · ξ ∈ X ′′ means the left action of A ′′ on X ′′ . The above equation holdsbecause it does for a ∈ A and all the operations involved are w ∗ -continuous. Now,if Ind AX ( π ) ′′ ( a ) = 0, then h ξ | a · ξ i = 0 for all ξ , ξ ∈ X from which it follows that a = 0. The converse (faithfulness of π from Ind AX ( π )) follows by symmetry since π can be seen as the induced representation of Ind AX ( π ) through the dual equivalencebimodule X ∗ . (cid:3) Corollary C.3.
Let X be a Hilbert A -module and π : A → L ( H ) a nondegeneraterepresentation. Then the representation π X : X → L ( H, X ⊗ π H ) , π X ( x ) h = x ⊗ h ,has a unique w ∗ -continuous extension π ′′ X : X ′′ → L ( H, X ⊗ π H ) to a representationof ternary W ∗ -rings [35] . Moreover, if π ′′ is faithful then so is π ′′ X .Proof. We view X , A and the algebra of generalized compact operators of X , B , assubspaces of the linking algebra L of X . Then LA = X ⊕ A and the representationInd LX ⊕ A π of L induced by π through X ⊕ A can be seen asInd LX ⊕ A π : L → L (( X ⊕ A ) ⊗ π H ) ∼ = L (( X ⊗ π H ) ⊕ H ) , Ind LX ⊕ A π (cid:18) T x ˜ y a (cid:19) = (cid:18) Ind BX π ( T ) π X ( x ) π X ( y ) ∗ π ( a ) (cid:19) . Since Ind LX ⊕ A π is nondegenerate, we have a canonical extension(Ind LX ⊕ A π ) ′′ : L ′′ → L (( X ⊕ A ) ⊗ π H ) . In the proof of Proposition C.1 we have identified X ′′ with the w ∗ -closure of X in L ′′ . Thus the restriction of (Ind LX ⊕ A π ) ′′ to X ′′ is a w ∗ -continuous extensionof (Ind LX ⊕ A π ) | X . The image of (Ind LX ⊕ A π ) | X consists entirely of operators of theform (cid:0) y (cid:1) . Under the identification y = (cid:0) y (cid:1) we may view (Ind LX ⊕ A π ) ′′ | X ′′ asthe unique w ∗ -extension of π X .In case π ′′ is faithful then so is (Ind LX ⊕ A π ) ′′ , so π ′′ X is faithful because it is arestriction of a faithful map. (cid:3) Corollary C.4.
Let X be a Hilbert A -module. For a bounded net { x i } i ∈ I ⊆ X ′′ and x ∈ X ′′ , the following assertions are equivalent:(i) { x i } i ∈ I ⊆ X ′′ w ∗ -converges to x .(ii) For every y ∈ X , {h x i , y i A ′′ } i ∈ I w ∗ -converges to h x, y i A ′′ .(iii) For every y ∈ X , {h y, x i i A ′′ } i ∈ I w ∗ -converges to h y, x i A ′′ .Proof. Let π : A → L ( H ) be the universal representation and consider π ′′ X : X ′′ →L ( H, X ⊗ π H ). On bounded sets the w ∗ -topology on X ′′ coincides with the weakoperator topology it inherits from L ( H, X ⊗ π H ) via π ′′ X and it is determined by thefunctionals of the form X → C , z
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