aa r X i v : . [ m a t h . OA ] S e p FIXED POINT ALGEBRAS FORWEAKLY PROPER FELL BUNDLES
DAMIÁN FERRARO
Abstract.
We define weakly proper Fell bundles and construct exotic fixedpoint algebras for such bundles. Three alternative constructions of such alge-bras are given. Under a kind of freeness condition, one of our constructionsimplies that every exotic cross sectional C*-algebra of a weakly proper Fellbundle is Morita equivalent to an exotic fixed point algebras. The other con-structions are used to show that ours generalizes that of Buss and Echterhoffon weakly proper actions on C*-algebras. We also generalize to Fell bundlesthe fact that every C*-action which is proper in Kasparov’s sense is amenable.
Contents
Introduction 11. Proper partial actions on LCH spaces 32. Weakly proper Fell bundles 62.1. Kasparov proper Fell bundles and amenability 113. Fixed point algebras 133.1. Basic examples 133.2. General weakly proper Fell bundles 153.3. The module E B as a tensor product 223.4. Bra-ket operators and the fixed point algebra 24References 28 Introduction
Green’s Symmetric Imprimitivity Theorem [14] implies that given a free andproper action σ of a locally compact and Hausdorff (LCH) group G on a LCH space X ; the full crossed product C ( X ) ⋊ σ G is strongly Morita equivalent to C ( X/G ) ,X/G being the space of σ − orbits. The “large fixed point algebra” of σ, C b ( X ) σ , isthe set of fixed points of the the canonical action σ of G on C b ( X ) determined by σ,σ t ( f )( x ) = f ( σ t − ( x )) . Note C b ( X ) σ is C*-isomorphic to C b ( X/G ) , hence C ( X/G )is C*-isomorphic to a C*-subalgebra of the large fixed point algebra of σ. That iswhy C ( X/G ) is called a “fixed point algebra”.Green’s Theorem may be used to show that σ is amenable in the sense that C ( X ) ⋊ σ G agrees with the reduced cross product C ( X ) ⋊ r σ G. Indeed, let I bethe kernel of the regular representation C ( X ) ⋊ σ G → C ( X ) ⋊ r σ G. This ideal isinduced from some ideal of C ( X/G ) , i.e. from a σ − invariant open set U ⊂ X. But C ( U ) is a σ − invariant ideal of C ( X ) , so using Green’s Theorem once again wededuce that I = C ( U ) ⋊ σ | U G. The only way of having C ( U ) ⋊ σ | U G contained inthe kernel of the regular representation is if U = ∅ , meaning that I = { } . Date : September 3, 2020.2010
Mathematics Subject Classification.
Key words and phrases.
Fell bundles, partial actions, proper actions, fixed point algebras.
In [10] Kasparov used proper actions on LCH spaces in order to construct fixedpoint algebras for actions on C ( X ) − algebras. The same year Rieffel tried to givea definition of proper action on a C*-algebra without using proper actions on LCHspaces [15]. Rieffel starts with an action α of a LCH group G on a C*-algebra A and seeks for a definition of proper action that allows him to establish a strongMorita equivalence between A ⋊ α G and a C*-subalgebra of the (large) fixed pointalgebra M ( A ) α := { T ∈ M ( A ) : ˜ α t ( T ) = T, ∀ t ∈ G } , where ˜ α is the natural extension of α to the multiplier algebra M ( A ) . Rieffel notices that certain C c ( G, A ) − valued inner products are positive definitein the reduced crossed product, but not in the full crossed product in general. Kas-parov does not have this problem because his proper actions are always amenable(the full and reduced crossed products agree).Buss and Echterhoff introduced in [5] the concept of the weakly proper action.Every Kasparov’s proper actions is weakly proper and any weakly proper action isproper in Rieffel’s sense. Eventhoug not every weakly proper actions is amenable,they are nice enough as to be able to prove that the C c ( G, A ) − valued inner productsare positive in the full crossed product. Moreover, one even has Symmetric Imprim-itivity Theorems for weakly proper actions [6] (generalizing Raeburn’s SymmetricImprimitivity Theorem [13]).An interesting feature of weakly proper actions is that for every exotic crossedproduct ⋊ µ lying between the full ⋊ u and reduced ⋊ r , there exists a µ -fixed pointalgebra A αµ strongly Morita equivalent to A ⋊ µα G. So a weakly proper action α isamenable ( ⋊ u = ⋊ r ) if and only if it has only one exotic fixed point algebra.There have been other attempts to define proper actions on C*-algebras that havelead to study integrable (or square integrable) actions, see for example [6, 12, 16]and the references therein. Our work can be seen as a generalization of Buss andEchterhoff’s one to Fell bundles, which in turn generalizes Kasparov’s.With some notational effort one can extend Buss and Echterhoff’s work to twistedactions and to partial actions, separately. But a natural preservation instinct shouldprevent everybody to try give a definition of “weakly proper twisted partial action”.Twisted partial action are defined in [7], where it is shown that under certainhypotheses a Fell bundles can be described as the semidirect product bundle of atwisted partial action. This is of importance to us because it says that there isa kind of natural action associated to every Fell bundle. Then (at least in somecases) it should be possible to determine whether or not an action on a C*-algebrais weakly proper using only the semidirect product bundle of the action. A stepfurther in this direction would be to state the definition of weakly proper actionitself using the semidirect product bundle. After this one should obtain somethingvery close to a definition of weakly proper Fell bundle. This is what we have done,and we show our results in this work.Quite often authors working with Fell bundles assume the bundles are saturated.This is even true in some parts of [9], where Fell bundles are called C*-algebraicbundles. But we do not want to assume our Fell bundles are saturated because thesemidirect product bundle of a C*-partial action (as considered in [7]) is saturatedif and only if the partial action is global (i.e. just an action).The “test case” that motivated this work was the semidirect product bundle of apartial action on a commutative C*-algebra or, equivalently, a C*-partial action ona commutative C*-algebra. This turned out to be quite important in the generaltheory, so we dedicate the first section of this work to study this test case.In Section 2 we give the definition of a general weakly proper and Kasparovproper Fell bundles. We also show that every Kasparov proper Fell bundles is IXED POINT ALGEBRAS FOR FELL BUNDLES 3 amenable. Finally, in the last section, we construct the fixed point algebra of a Fellbundle and give three different ways of constructing these algebras.1.
Proper partial actions on LCH spaces
Let’s start by recalling some equivalent definitions of proper actions on LCHspaces. Suppose σ is an action of the LCH group G on the LCH space X. Then σ is proper if satisfies any (and hence all) the equivalent conditions:(1) For every pair of compact sets, L, M ⊂ X, the set(( L, M )) := { t ∈ G : σ t ( L ) ∩ M = ∅} has compact closure.(2) For every pair of compact sets, L, M ⊂ X, the set (( L, M )) is compact.(3) Them map G × X → X × X, ( t, x ) ( σ t ( x ) , x ) , is proper (the preimage ofevery compact set is compact).Let’s translate each one of the conditions above to partial actions and comparethe respective candidates for the definition of proper partial action. Assume σ :=( { X t } t ∈ G , { σ t } t ∈ G ) is a LCH partial action, that is, σ is a topological partial actionin the sense of [2, Definition 1.1] with both G and X being LCH. The domain andgraph of σ are, respectively,Γ σ := { ( t, x ) ∈ G × X : x ∈ X t − } Gr( σ ) := { ( t, x, y ) ∈ G × X × X : x ∈ X t − , y = σ t ( x ) } . The natural translation of conditions (1-3) above are(P1) For every pair of compact sets,
L, M ⊂ X, the set(( L, M )) := { t ∈ G : σ t ( L ∩ X t − ) ∩ M = ∅} has compact closure.(P2) For every pair of compact sets, L, M ⊂ X, the set (( L, M )) is compact.(P3) The map F σ : Γ σ → X × X, ( t, x ) ( σ t ( x ) , x ) , is proper.The following examples show the conditions above are not equivalent. Example . Let ϕ be the flow of the vector field f : R \{ (0 , } → R , f ( x, y ) = (1 , . Then ϕ defines a partial action of R on X := R \ { (0 ,
0) that we now describe. Given t ∈ R , let X t be X \ { ( st,
0) : s ∈ [0 , } . Then ϕ t : X − t → X t is given by ϕ t ( x, y ) = ( x + t, y ) . We leave to the reader to verifythat ϕ satisfies (P1). But ϕ does not satisfy (P2) because for the closed segments L := [( − , , ( − , M = [(1 , , (1 , , (( L, M )) = (2 ,
3] is not compact.
Example . Let G := Z act partially on X = [0 ,
1] by thepartial action such that σ = id X and σ = id [0 , / . Since every subset of G iscompact, σ satisfies (P2). But σ does not satisfy (P3) because Γ σ = F − σ ( X × X )is not compact.With the previous examples and some extra work one can show that Proposition 1.3.
For every LCH partial action σ it follows that (P3) ⇒ (P2) ⇒ (P1)but none of the converses holds in general (as the previous examples show).Proof. Assume σ satisfies (P3) and take two compacts sets, L, M ⊂ X. Take a net { t i } i ∈ I ⊂ (( L, M )) . Then for all i ∈ I there exists x i ∈ L ∩ X t − i such that σ t i ( x i ) ∈ M. Thus { ( t i , x i ) } i ∈ I ⊂ F − σ ( L × M ) and there exists a sub net { ( t i j , x i j ) } j ∈ J converging to some ( t, x ) ∈ F − σ ( L × M ) . Thus, x ∈ X t − , x = lim j x i j ∈ L,σ t ( x ) = lim j σ t ij ( x i j ) ∈ M and this implies lim j t i,j = t ∈ (( L, M )) . Hence (Pefitem:F is proper) ⇒ (P2).The implication (P2) ⇒ (P2) is trivial because G is Hausdorff. (cid:3) DAMIÁN FERRARO
A key feature of proper actions is that one can construct fixed point algebraswith them. In the topological context this means that the orbit space is a LCHspace. So lets define the orbit space of a topological partial action an lets try tosee if any of the conditions (P1-P3) guarantees a LCH orbit space.
Definition 1.4.
Let τ = ( { τ t } t ∈ H , { Y t } t ∈ H ) be a topological partial action. Theorbit of a set U ⊂ Y is defined as [ U ] τ := ∪ t ∈ H τ t ( U ∩ Y t − ) and the orbit of a point y ∈ Y is [ y ] τ := [ { y } ] τ . A subset U ⊂ Y is said to be invariant (or τ − invariant) if[ U ] τ ⊂ U (or, equivalently, U = [ U ] τ ).Note that U ⊂ [ U ] τ because for t = e, τ t ( U ∩ X t − ) = U. Besides, [ U ] τ is invariantbecause the properties of set theoretic partial actions imply[[ U ] τ ] τ = [ t,s ∈ H σ t ( σ s ( X s − ∩ U ) ∩ X t − )= [ t,s ∈ H σ t ( σ s ( X s − ∩ U ) ∩ σ s ( X s − ∩ X t − ))= [ t,s ∈ H σ t ( σ s ( X s − ∩ X s − t − ∩ U )) = [ t,s ∈ H σ ts ( X s − ∩ X s − t − ∩ U ) ⊂ [ t,s ∈ H σ ts ( X s − t − ∩ U ) ⊂ [ U ] τ ⊂ [[ U ] τ ] τ . Remark . If U ⊂ V ⊂ Y then [ U ] τ ⊂ [ V ] τ . This implies [ U ] τ is the smallestinvariant set containing U. Note also that [ U ] τ is open if U is. Remark . The whole space f Y is the disjoint union of the orbits of it’s point.Indeed, it is clear that y ∈ [ y ] τ for all y ∈ Y. Assume y, z ∈ Y are such that[ y ] τ ∩ [ z ] τ = ∅ . Then there exists r, s ∈ H such that y ∈ X r − , z ∈ X s − and τ r ( y ) = τ s ( z ) . Hence, by de definition of partial action, y ∈ X r − ∩ X r − s and τ s − r ( y ) = τ s − ( τ r ( y )) = z. Thus z ∈ [ y ] τ and this implies [ z ] τ ⊂ [ y ] τ . By symmetrywe get that [ z ] τ = [ y ] τ . It is evident that the relation y ∼ z ⇔ [ y ] τ = [ z ] τ is an equivalence relation.This relation is open in the sense that[ U ] τ = { z ∈ Y : [ z ] σ ∩ U = ∅} is open if U is. Definition 1.7.
The orbit space of τ is Y /τ := { [ y ] τ : y ∈ Y } , the canonical projec-tion is π : Y → Y /τ, y [ y ] τ , and the topology of Y /τ is { U ⊂ Y /τ : π − ( U ) is open } . The canonical projection is continuous, open and surjective. Hence the orbitspace of a topological partial action on a locally compact space is always locallycompact. The next example shows that condition (P2) does not guarantee a Haus-dorff orbit space.
Example . Let G = Z act partially on X = [ − ,
2] by the partial action σ = ( { σ , σ } , { X, ( − , } ) with σ := id X and σ : ( − , → ( − ,
1) given by σ ( x ) = − x. Then σ satisfies (P2) and [1] σ = [ − σ , but every open invariant sub-set containing 1 intersects every open invariant subset containing − . Thus
X/σ isnot Hausdorff, but it is locally compact.Now we relate the orbit space of a topological partial action with the orbit spaceof it’s topological enveloping action, as defined in [2, Definition 1.2]. The topologicalenveloping action of τ is, up to isomorphism of actions, a topological (global) action τ e of H on a topological space Y e such that: • Y is an open subset of Y e . IXED POINT ALGEBRAS FOR FELL BUNDLES 5 • For all t ∈ H, Y t = Y ∩ τ et ( Y ) . • For all t ∈ H and y ∈ Y t − , τ t ( y ) = τ et ( y ) . • Y e = S t ∈ H σ et ( Y ) or, equivalently, Y e = [ Y ] τ e . Proposition 1.9.
Let τ be a topological partial action of H on Y and τ e theenveloping action of τ, acting on the enveloping space Y e . Then the map
Y /τ → Y e /τ e , [ y ] τ [ y ] τ e , is defined and is a homeomorphism.Proof. We claim that [ y ] τ e ∩ Y = [ y ] τ , for all y ∈ Y. Indeed, it is clear that [ y ] τ ⊂ [ y ] τ e ∩ Y. Conversely, if z ∈ [ y ] τ e ∩ Y then there exists t ∈ H such that τ et ( y ) = z ∈ Y. Thus y ∈ Y ∩ τ et − ( Y ) = Y t − and τ t ( y ) = τ et ( y ) = z. Hence z ∈ [ y ] τ . The function Y → Y e /τ e , y [ y ] τ e , is continuous and constant in the τ − orbits.Moreover, it is surjective because [ Y ] τ e = Y. Then there exists a unique continuousand surjective map h : Y /τ → Y e /τ e , h ([ y ] τ ) = [ y ] τ e . Note h is injective because,if h ([ y ] τ ) = h ([ z ] τ ) , then [ y ] τ = h ([ y ] τ ) ∩ Y = h ([ z ] τ ) ∩ Y = [ z ] τ . We also have that h is open because if U ⊂ Y is open, then h ([ U ] τ ) = [ U ] τ e is open. (cid:3) The next result (together with the previous one) is our main reason to adoptcondition (P3) as the definition of proper partial action.
Proposition 1.10.
Let σ be a LCH partial action of G on X and let σ e be it’senveloping action, acting on the enveloping space X e . Then the following are equiv-alent:(1) σ e is a LCH and proper action.(2) Given a net { ( t i , x i ) } i ∈ I ⊂ Γ σ such that { ( σ t i ( x i ) , x i ) } i ∈ I ⊂ X × X con-verges (to a point of X × X ), there exists a sub net of { ( t i , x i ) } i ∈ I convergingto a point of Γ σ . (3) σ satisfies condition (P3), that is: F σ : Γ σ → X × X, F σ ( t, x ) = ( σ t ( x ) , x ) , is proper.Proof. Assume (1) and take a net { ( t i , x i ) } i ∈ I ⊂ Γ σ such that ( σ t i ( x i ) , x i ) → ( y, x ) ∈ X × X. Take compact neighbourhoods of x and y, U and V, respectively.Then there exists i ∈ I such that { ( t i , x i ) } i ≥ i ∈ F − σ e ( U × V ) . Hence there ex-ists a sub net { ( t i j , x i j ) } j ∈ J converging to a point ( t, z ) ∈ G × X. We then have z = lim j x i j = lim i x i = x because X is Hausdorff and σ et ( x ) = lim j σ et ij ( x i j ) =lim j σ t ij ( x i j ) = y ∈ X. Thus x ∈ X ∩ σ et − ( X ) = X t − , meaning that ( t, x ) ∈ Γ σ . Now assume (2) holds and take a compact set L ⊂ X × X and a net { ( t i , x i ) } i ∈ I ⊂ F − σ ( L ) . Then { ( σ t i ( x i ) , x i ) } i ∈ I ⊂ L has a converging sub net and, by passing toa sub net again and relabelling, we get a sub net { ( t i j , x i j ) } j ∈ J converging to apoint ( t, x ) ∈ Γ σ and such that { ( σ t i ( x i ) , x i ) } i ∈ I converges to some ( y, z ) ∈ L. We then have z = x and, by continuity, σ t ( x ) = lim j σ t ij ( x i j ) = y. This implies( t, x ) ∈ F − σ ( L ) . Now (3) follows because the net { ( t i j , x i j ) } j ∈ J ⊂ F − σ ( L ) convergesto ( t, x ) ∈ F − σ ( L ) . Suppose σ satisfies (3). Note that X e = S t ∈ G σ et ( X ) is locally compact becauseit is the union of open locally compact subsets.To show that X e is Hausdorff it suffices, by [2, Proposition 1.2], to show Gr( σ )is closed in G × X × X. Take a net { ( t i , x i , y i ) } i ∈ I ⊂ Gr( σ ) converging to ( t, x, y ) . Then { ( t i , x i ) } i ∈ I ⊂ Γ σ and { ( σ t i ( x i ) , x i ) } i ∈ I = { ( y i , x i ) } i ∈ I converges to ( y, x ) . By taking a compact neighbourhood of ( y, x ) , L, and considering F − σ ( L ) we geta sub net { ( t i j , x i j ) } j ∈ J converging to some ( s, z ) ∈ F − σ ( L ) ⊂ Γ σ . Since G × X is Hausdorff, ( t, x ) = ( s, z ) ∈ Γ σ and by continuity we get σ t ( x ) = lim i σ t i ( x i ) =lim i y i = y. This means that ( t, x, y ) ∈ Gr( σ ) . Hence Gr( σ ) is closed and X e isHausdorff.Take a compact set L ⊂ X e × X e and a net { ( t i , x i ) } i ∈ I ∈ F − σ e ( L ) . Then thereexists a sub net { ( t i j , x i j ) } J ∈ J such that { ( σ et ij ( x i j ) , x i j ) } j ∈ J ⊂ L converges to a DAMIÁN FERRARO point ( y, x ) ∈ L. Take r, s ∈ G such that σ es ( y ) , σ er ( x ) ∈ X. Note thatlim j σ er ( x i j ) = σ er ( x ) ∈ X lim j σ est ij r − ( σ er ( x i j )) = σ es ( y ) ∈ X. Since X is open in X e there exists j ∈ J such that, for all j ≥ j , σ er ( x i j ) ∈ X and σ est ij r − ( σ er ( x i j )) ∈ X. Let U ⊂ X × X be a compact neighbourhood of ( σ es ( y ) , σ er ( y )) . Then the net { ( st i j r − , σ er ( x i j )) } j ≥ j is contained in F − σ ( U ) and, by passing to a sub net andrelabelling, we can assume { ( st i j r − , σ er ( x i j )) } j ∈ J converges. In particular we getthat { t i j } j ∈ J converges to some t ∈ G. This implies { ( t i j , x i j ) } J ∈ J converges to( t, x ) and ( t, x ) ∈ F − σ e ( L ) because ( σ et ( x ) , x ) = lim j ( σ et ij ( x i j ) , x i j ) ∈ L. (cid:3) Definition 1.11.
A LCH partial action σ is proper if it satisfies the equivalentconditions of Proposition 1.10.At this point Proposition 1.10 becomes a machine to construct every possibleLCH proper partial action: just take a common LCH proper action and restrict itto an ideal. For future reference we give an Example. Example . Let G be a discrete group and denote τ the action of G on G bymultiplication. The restriction τ of σ to the singleton { e } is a proper partial actionbecause τ e = σ and σ is proper. Note that for t ∈ G \ { e } the homeomorphism τ t is the empty function. Remark . The orbit space of every LCH proper partial action σ is LCH. Indeed,this in known to hold for global actions, in particular for σ e . Hence the same holdsfor σ by Proposition 1.9.We hope to have exhibited enough reasons to adopt our definition of properpartial actions. 2. Weakly proper Fell bundles
We start this section by constructing the basic examples of weakly proper Fellbundles. Consider a LCH partial action σ = ( { X t } t ∈ G , { σ t } t ∈ G ) of G on X. The nat-ural partial action of G on C ( X ) defined by σ, θ = θ ( σ ) := ( { C ( X t ) } t ∈ G , { θ t } t ∈ G ) , is given by θ t ( f )( x ) = f ( σ t − ( x )) for all f ∈ C ( X t − ) , x ∈ X t . In general, by a C*-partial action we mean a partial action on a C*-algebraas in [2, Definition 2.2]. The correspondence σ θ ( σ ) is bijective between LCHpartial actions and C*-partial actions on commutative C*-algebras.Given a C*-partial action β = ( { B t } t ∈ G , { β t } t ∈ G ) of G on B, the semidirectproduct bundle of β, B β , is the Banach subbundle { ( t, a ) : a ∈ B t , t ∈ G } of thetrivial Banach bundle B × G over G together with the product and involution givenby ( s, a ) ∗ = ( s − , β s − ( a ∗ )) ( s, a )( t, b ) = ( st, β t ( β t − ( a ) b )) . Exel defines (in [7]) semidirect product bundles even for twisted partial actions.Although our theory will cover this kind of bundles, we will not have to deal withthem explicitly. Following Exel we will write aδ s instead of ( s, a ) . So B s δ s is ac-tually the fibre ( s, B s ) of B β over s ∈ G. This notation is convenient because wecanonically identify the fibre B e δ e = Bδ e (over the group’s unit e ) with B. Underthis identification B s is a C*-ideal of Bδ e , not the fibre B s δ s . In case B = C ( X ) and β = θ ( σ ) , we write A σ instead of B θ ( σ ) . We call A σ thesemidirect product bundle of σ. Definition 2.1.
The basic examples of weakly proper Fell bundle are the semidirectproduct bundles of LCH and proper partial actions.
IXED POINT ALGEBRAS FOR FELL BUNDLES 7
For convenience we introduce some notation to be used quite frequently in therest of the work.
Notation 2.2.
Given sets U and V, a topological vector space W, a binary operation U × V → W, ( u, v ) u · v, and subsets A ⊂ U and B ⊂ V, we denote A · B theclosed linear span of { a · b : a ∈ A, b ∈ B } . Recall from [1] (and from [3] for non discrete groups) that every Fell bundle B = { B t } t ∈ G defines a topological partial action ˆ α of G on the spectrum ˆ B e of theunit fibre B e . Given t ∈ G, the set I B t := B t B ∗ t is an ideal of B e , so the spectrumˆ I B t is an open subset of ˆ B e . The homeomorphism ˆ α t : d I B t − → c I B t is the Rieffelhomeomorphism associated to the I B t − − I B t − equivalence bimodule B t ; consideredwith the left and right inner products ( a, b ) ab ∗ and ( a, b ) a ∗ b, respectively.The partial action ˆ α described before is compatible with the natural quotientmap q : ˆ B e → Prim( B e ) (from the spectrum to the primitive ideal space) givenby q ([ π ]) = ker( π ) , where [ π ] is the unitary equivalence class of the irreduciblerepresentation π. This means that there exists a unique topological partial action˜ α of G on Prim( B e ) such that ˜ α t maps O t − := Prim( I B t − ) bijectively to O t =Prim( I B t ) sending ker( π ) to ker(ˆ α t ( π )) . For a basic example B = A σ we have ˆ α = σ, so the identity A σ = A τ implies σ = τ and there is no possible ambiguity in our last definition. It also follows that,for an arbitrary LCH partial action σ, A σ is a basic example of a weakly properFell bundle if and only if σ is proper.In order to motivate our definition of weakly proper Fell bundle, and to justifythe term “weakly proper” we are using, let’s put Buss and Echterhoff’s weaklyproper actions [6] in the context of Fell bundles.A C*-action β of G on B is weakly proper if there exists a proper LCH action τ of G on Y together with a *-homomorphism φ : C ( Y ) → M ( B ) such that, with θ = θ ( σ ) , (1) B = φ ( C ( Y )) B. By Cohen-Hewitt’s Theorem this is equivalent to sayevery b ∈ B admits a factorization b = φ ( f ) c. (2) For all f ∈ C ( Y ) , b ∈ B and t ∈ G, φ ( θ t ( f )) β t ( b ) = β t ( φ ( f ) b ) . In the situation above one can construct the function A τ × B β → B β , ( f δ s , bδ t ) φ ( f ) β s ( b ) δ st , which we interpret as an action of A τ on B β . The properties satisfied by this actionmotivate the following.
Definition 2.3.
Let A = { A t } t ∈ G and B = { B t } t ∈ G be Fell bundles over G. Wesay a function
A × B → B , ( a, b ) a · b, is an action of A on B by adjointablemaps if(1) For all a ∈ A s , b ∈ B t and s, t ∈ G, a · b ∈ B st . (2) For all s, t ∈ G and b ∈ B t the function A s → B st , a a · b is linear.(3) For all a, c ∈ A and b ∈ B , a · ( c · b ) = ( ac ) · b. (4) For all a ∈ A and b, d ∈ B , ( a · b ) ∗ d = b ∗ ( a ∗ · d ) . (5) For all b ∈ B e the function A → B , a a · b, is continuous.We say the action is non degenerate if for all t ∈ G, ( A t A t − ) · B e = B t B t − (recallNotation 2.2).In the conditions of the Definition above there exists a unique *-homomorphism φ : A e → M ( B e ) such that φ ( a ) b = a · b. If we were working with semidirect prod-uct bundles B = B β and A = A σ , then this map φ would be exactly the map DAMIÁN FERRARO φ : C ( X ) → M ( B ) (after the identification A e = C ( X ) and B e = B ). The differ-ence between weakly proper actions and Kasparov proper actions is that the latterassume φ is central (i.e. the image of φ is contained in the centre ZM ( B e )). Definition 2.4.
A Fell bundle B over G is weakly proper if there exists a basicexample of a weakly proper Fell bundle over G, A , and a non degenerate action of A on B by adjointable maps. If, in addition, the map φ : A e → M ( B e ) describedabove is central, we say B is Kasparov proper. Example . Every basic example of a weakly proper Fell bundle, say A , is weaklyproper. Indeed, one just needs to consider the multiplication of A as an action of A on A . We will show later (in Corollary 2.14) that a semidirect product bundle A σ of aLCH partial action σ is weakly proper if and only if σ is proper. So the Exampleabove produces all weakly proper Fell bundles coming from semidirect productbundles of LCH partial actions.The non degeneracy requirement in Definition 2.4 is motivated by condition (1)in the example below (and to exclude the zero action). Example . Consider a C*-partial action β =( { B t } t ∈ G , { β t } t ∈ G ) of G on B for which there exists a proper LCH partial action σ = ( { X t } t ∈ G , { σ t } t ∈ G ) of G on X and a *-homomorphism φ : C ( X ) → M ( B )such that, with θ = θ ( σ ):(1) For all t ∈ G, B t = φ ( C ( X t )) B. By Cohen-Hewitt’s factorization Theoremthis implies for every b ∈ B t and t ∈ G there exits f ∈ C ( X t ) and c ∈ B t such that b = φ ( f ) c. (2) For all t ∈ G, f ∈ C ( X t − ) and b ∈ B t − , φ ( θ t ( f )) β t ( b ) = β t ( φ ( f ) b ) . Then the semidirect product bundle B β is weakly proper with respect to the action(2.7) A σ × B β → B β , ( f δ s , bδ t ) β t ( φ ( θ t − ( f )) b ) δ st . Note that the map above is defined because φ ( C ( X t − )) B s ∈ B t − ∩ B s , for all s, t ∈ G. Thus β t ( φ ( C ( X t − )) B s ) ⊂ B t ∩ B st . One can not define the action (2.7) ifin (1) one requires, for example, B t = φ ( C ( X t )) B t . The reader may explore otheralternatives, but the author must say (1) is the only satisfactory condition he hasbeen able to find.Is is not straightforward to verify (2.7) is an action by adjointable maps. Aftersome attempts to do this one feels that the computations needed are quite similarto those necessary to show the semidirect product bundle of a C*-partial action isa Fell bundle. We leave to the reader the adaptation of Exel’s method to do this,see [7] (fortunately there are no twists here).The action of a Fell bundle on another has some extra properties that we sum-marize below.
Proposition 2.8. If A × B → B , ( a, b ) a · b, is an action of the Fell bundle A = { A t } t ∈ G on the Fell bundle B = { B t } t ∈ G by adjointable maps then:(1) For all s, t ∈ G the map A s × B t B st , ( a, b ) a · b, is bilinear.(2) For all a ∈ A and b, c ∈ B , a · ( bc ) = ( a · b ) c. (3) For all a ∈ A and b ∈ B , k ab k ≤ k a kk b k and ( a · b ) ∗ ( a · b ) ≤ k a k b ∗ b. (4) The action A × B → B , ( a, b ) a · b, is continuous.(5) For all t ∈ G, A t · B e = A e · B t = B t . Proof.
Clearly the map from (1) is linear in the first variable, to show it is linearin the second variable take a ∈ A s , b, c ∈ B t and λ ∈ C . Then, with z := a · ( b + IXED POINT ALGEBRAS FOR FELL BUNDLES 9 λc ) − ( a · b + λ [ a · c ]) , k z k = k z ∗ z k = k ( b + λc ) ∗ ( a ∗ · z ) − b ∗ ( a ∗ · z ) − λ [ b ∗ ( a ∗ · c )] k = 0 . This implies z = 0 , that is a · ( b + λc ) = a · b + λ [ a · c ] . The proof of (2) is quite similar to the previous one. If z = a · ( bc ) − ( a · b ) c, then k z k = k z ∗ z k = k ( bc ) ∗ ( a ∗ · z ) − c ∗ b ∗ ( a ∗ · z ) k = 0 . Hence (2) follows.To prove (3) take a ∈ A s and b ∈ B t and note that ( a · b ) ∗ ( a · b ) = b ∗ ( a ∗ a · b ) . Forevery c ∈ A e there exists a unique φ c ∈ M ( B e ) such that φ c x = c · x. In fact the map φ : A e → M ( B e ) , c φ c , is a *-homomorphism. Considering the fibre B t as a right B e − Hilbert module (with inner product h x, y i = x ∗ y ) we have a non degenerate*-homomorphism ϕ : B e → B ( B t ) such that ϕ ( x ) y = xy. If ϕ : M ( B e ) → B ( B t ) isthe unique extension of ϕ, then since every *-homomorphism between C*-algebrasis contractive we have( a · b ) ∗ ( a · b ) = b ∗ ( a ∗ a · b ) = h b, ϕ ( φ a ∗ a ) b i ≤ k ϕ ( φ a ∗ a ) kh b, b i ≤ k a ∗ a k b ∗ b = k a k b ∗ b, where we have used that a ∗ a ≥ A e . Then k a · b k = k ( a · b ) ∗ ( a · b ) k / ≤ k a kk b k . To prove (4) take a net { ( a i , b i ) } i ∈ I ⊂ A × B converging to ( a, b ) ∈ A × B . Let { ( s i , t i ) } i ∈ I ⊂ G × G and ( s, t ) ∈ G × G be such that a i ∈ A s i , b i ∈ B t i , a ∈ A s and b ∈ B t . Then s i → s, t i → t, a i · b i ∈ B s i t i and s i t i → st. Fix ε > . Then, since B has approximate units, there exists u ε ∈ B e suchthat k b − u ε b k < ε (1 + k a k ) − . Take i ε ∈ I such that k b i − u ε b i k < ε (1 + k a k ) − and k a i k < k a k + 1 for all i ≥ i ε . Our construction implies that, for all i ≥ i ε , k a i · b i − a i · ( u ε b i ) k = k a i · ( b i − u ε b i ) k < ε. Now the definition of action byadjointable maps and claim (2) implylim i a i · ( u i b i ) = lim i ( a i · u ε ) b i = ( a · u ε ) b = a · ( u ε b ) . Besides, k a · b − a · ( u ε b ) k < ε. Now we can use [9, II 13.12] to deduce that lim i a i · b i = a · b, thus the proof of (4) is complete.Regarding (5), by considering B t as a right B e − Hilbert module we deduce that B t = B t B ∗ t B t = A t A ∗ t · B t ⊂ A t · ( A t − · B t ) ⊂ A t · B e ⊂ B t and also that B t = B e B ∗ e B t = ( A e A ∗ e · B e ) B t = A e · ( A e · B t ) ⊂ A e · B t ⊂ B t . Now the proof is complete. (cid:3)
The next Lemma may look harmless or even unnecessary, but it is of key impor-tance to us because it relates a LCH partial action σ with the action of A σ on aFell bundle. Lemma 2.9.
Let σ be a LCH partial action of G on X, B a Fell bundle over G, A σ × B → B , ( a, b ) a · b, a non degenerate action by adjointable maps and set θ := θ ( σ ) . If the map φ : A e → M ( B e ) , φ ( x ) y := x · y, is central, then θ t ( f ) δ e · bc = b ( f δ e · c ) , ∀ t ∈ G, f ∈ C ( X t − ) , b ∈ B t , c ∈ B . Proof.
Fix s, t ∈ G and f ∈ C ( X t − ) . The maps B t × B s → B ts given by ( b, c ) θ t ( f ) δ e · bc and ( b, c ) b ( f δ e · c ) are continuous and bilinear. Besides B t = C ( X t ) δ t · B e and B s = B e B s , thus we may assume b = uδ t · v and c = zw for some u ∈ C ( X t ) , v ∈ B e , z ∈ B e and w ∈ B s . By considering B s and B t as a right B e − B e − Hilbert bimodules we deduce that b ( f δ e · c ) = ( uδ t · v )([ f δ e · z ] w ) = uδ t · ([ vφ ( f ) z ] w ) = uδ t · ([ φ ( f ) vz ] w )= uδ t f δ e · ( vc ) = θ t ( θ t − ( u ) f ) δ t · ( vc ) = θ t ( f ) uδ t · ( vc )= θ t ( f ) δ e uδ t · ( vc ) = θ t ( f ) δ e · ( uδ t · ( vc )) = θ t ( f ) δ e · bc. (cid:3) In the C*-algebraic context, if A acts on B and B on C by non degenerate actionsby adjointable maps, then one can define a non degenerate action of A on C byadjointable maps. This is also true for Fell bundles. Proposition 2.10.
Let A = { A t } t ∈ G , B = { B t } t ∈ G and C = { C t } t ∈ G be Fellbundles over G and A × B → B , ( a, b ) a · b, and B × C → C , ( b, c ) b ⋄ c, non degenerate actions by adjointable maps. Then there exists a unique action byadjointable maps A × C → C , ( a, c ) a ⋆ c, such that a ⋆ ( b ⋄ c ) = ( a · b ) ⋄ c for all ( a, b, c ) ∈ A × B × C . Moreover, ⋆ is non degenerate.Proof. Fix s, t ∈ G. In order to show there exists a unique bounded bilinear map F s,t : A s × C t → C st such that F s,t ( a, b ⋄ c ) = ( a · b ) ⋄ c for all ( a, c ) ∈ A s × C t and b ∈ B e , it suffices to show that k P nj =1 ( a · b j ) ⋄ c j k ≤ k a kk P nj =1 b j ⋄ c j k , for all n ∈ N , b , . . . , b n ∈ B e and c , . . . , c n ∈ C t . Take b , . . . , b n ∈ B e and c , . . . , c n ∈ C t . Define u := P nj =1 b j ⋄ c j and v := P nj =1 ( a · b j ) ⋄ c j . Consider the map φ : A e → M ( B e ) given by φ ( x ) y = x · y and let φ : M ( A e ) → M ( B e ) be the unique extension of φ. If w is the positive square rootof k a k − a ∗ a in M ( A e ) , then k a k u ∗ u − v ∗ v = n X i,j =1 k a k c ∗ i ( b ∗ i b j ⋄ c j ) − c ∗ i ([ b ∗ i ( a ∗ a · b j )] ⋄ c j )= n X i,j =1 c ∗ i ([ b ∗ i φ ( w ∗ w ) b j ] ⋄ c j ) = n X i,j =1 ( φ ( w ) b i ⋄ c i ) ∗ ( φ ( w ) b j ⋄ c j ) ≥ . This implies k P nj =1 ( a · b j ) ⋄ c j k ≤ k a kk P nj =1 b j ⋄ c j k (i.e. k v k ≤ k a kk u k ).Now we define A × C → C , ( a, c ) a ⋆ c, in such a way that for a ∈ A s and c ∈ C t , a ⋆ c = F s,t ( a, c ) . Clearly, ⋆ satisfies conditions (1) and (2) from Definition2.3.Take ( a, b, c ) ∈ A × B × C and ( r, s, t ) ∈ G such that a ∈ A r , b ∈ B s and c ∈ C t . In order to compute a ⋆ ( b ⋄ c ) we take an approximate unit of B , { u j } j ∈ J ⊂ B e . Then the continuity of the actions and of the maps F p,q ( p, q ∈ G ) implies a ⋆ ( b ⋄ c ) = lim j a ⋆ ( u j ⋄ b ⋄ c ) = lim j ( a · u j ) ⋄ ( b ⋄ c ) = lim j ( a · u j b ) ⋄ c = ( a · b ) ⋄ c. Now we show condition (3) from Definition 2.3 using the identity we have justproved. For all a, d ∈ A and c ∈ C we have a ⋆ ( d ⋆ c ) = lim j a ⋆ ( d ⋆ [ u j ⋄ c ]) = lim j a ⋆ (( d · u j ) ⋄ c ) = lim j ( a · ( d · u j )) ⋄ c = lim j ( ad · u j ) ⋄ c = lim j ( ad ) ⋆ ( u j ⋄ c ) = ( ad ) ⋆ c. IXED POINT ALGEBRAS FOR FELL BUNDLES 11
To prove (4) from Definition 2.3 take a ∈ A and c, f ∈ C . Then( a ⋆ c ) ∗ f = lim j ( a ⋆ ( u j ⋄ c )) ∗ f = lim j (( a · u j ) ⋄ c ) ∗ f = lim j c ∗ (( a · u j ) ∗ ⋄ f )= lim j lim k c ∗ (( a · u j ) ∗ u k ⋄ f ) = lim j lim k c ∗ ( u ∗ j ( a ∗ · u k ) ⋄ f )= lim j lim k c ∗ ( u ∗ j ⋄ ( a ∗ · u k ) ⋄ f ) = lim j lim k ( u j ⋄ c ) ∗ (( a ∗ · u k ) ⋄ f )= lim j lim k ( u j ⋄ c ) ∗ ( a ⋆ ( u k ⋄ f )) = lim j ( u j ⋄ c ) ∗ ( a ⋆ f ) = c ∗ ( a ⋆ f ) . Lets show (5) from Definition 2.3. Note that by construction we get k a ⋆ c k ≤k a kk c k , for all ( a, c ) ∈ A × C . Fix c ∈ C e and take a net { a i } i ∈ I ⊂ A converging to a ∈ A t . Given ε > u ε ∈ B e (for example one of the terms of { u j } j ∈ J ) suchthat k c − u ε ⋄ c k < ε (1 + k a k ) − . Thenlim i a i ⋆ ( u ε ⋄ c ) = lim i a i ⋆ ( u ε ⋄ c ) = lim i ( a i · u ε ) ⋄ c = ( a · u ε ) ⋄ c = a ⋆ ( u ε ⋄ c ) . and k a ⋆ ( u ε ⋄ c ) − a ⋆ c k < ε. Besides, taking i ε ∈ I such that k a i k < k a k + 1 forall i ≥ i ε , we get that k a i ⋆ ( u ε ⋄ c ) − a i ⋆ c k ≤ k a kk u ε ⋄ c − c k < ε for all i ≥ i ε . Now [9, II 13.12] implies lim i a i ⋆ c = a ⋆ c. Finally, ⋆ is non degenerate because for all t ∈ GA t A t − ⋆ C e = A t A t − ⋆ ( B e · C e ) = ( A t A t − · B e ) ⋄ C e = B t B t − ⋄ C e = C t C t − . Now the proof is complete. (cid:3)
We have defined weakly proper Fell bundles using actions of basic examples ofweakly proper Fell bundles. Then one might define “weakly weakly proper Fell bun-dles” using actions of weakly proper Fell bundles and so on. Fortunately “weakly n proper Fell bundles” = “weakly proper Fell bundles”. Corollary 2.11. If B is a weakly proper Fell bundle over G and C is a Fell bundleover G admitting a non degenerate action by adjointable maps of B , then C is weaklyproper.Proof. Let σ be a LCH and proper partial action such that A σ acts on B by a nondegenerate action by adjointable maps. Proposition 2.10 gives a non degenerateaction by adjointable maps of A σ on C , thus C is weakly proper. (cid:3) Kasparov proper Fell bundles and amenability.
The main result in this(sub)section states that every Kasparov proper Fell bundle is amenable in the sensethat it’s full and reduced cross sectional C*-algebras agree.
Theorem 2.12.
Let σ be a LCH partial action of G on X, B a Fell bundle over G and A σ × B → B , ( a, b ) a · b, a non degenerate action by adjointable maps.Denote ˆ α the topological partial action G on Prim( B e ) defined by B (described inSection 2). If the map φ : C ( X ) → M ( B e ) given by φ ( f ) b = ( f δ e ) · b is central,then there exists a continuous function h : Prim( B e ) → X such that(1) For all t ∈ G, h − ( X t ) = O t (recall O t := { P ∈ Prim( B e ) : B t B ∗ t * P } ).(2) For all t ∈ G and P ∈ O t − , h (˜ α t ( P )) = ˜ α t ( h ( P )) . Proof.
Recall that the fibre over t ∈ G of A σ is A t := C ( X t ) δ e . As done beforewe denote θ (instead of θ ( σ )) the C*-partial action defined by σ on C ( X ) . Weidentify C ( X ) with A e canonically and think of C ( X t ) as an ideal of A e . A directcomputation shows that A t A ∗ t = C ( X t ) . The map φ is non degenerate because φ ( C ( X )) B e = ( C ( X ) δ e C ( X ) δ e ) · B e = B e . Then the Dauns-Hoffman Theorem gives a continuous map h : Prim( B e ) → X such that, for all a ∈ C ( X ) , b ∈ B e and P ∈ Prim( B e ) , π P ( aδ e · b ) = a ( h ( P )) π P ( b );where π P : B e → B e /P is the canonical quotient map. Fix t ∈ G. Given P ∈ h − ( X t ) , take a ∈ C ( X t ) such that a ( h ( P )) = 1 and b ∈ B e such that π P ( b ) = 0 . Then π P ( aδ e · c ) = a ( h ( P )) π P ( c ) = 0 and we concludethat C ( X t ) δ e · B e = A t A ∗ t · B e = B t B ∗ t is not contained in P, meaning that P ∈ O t . Assume, conversely, that we have P ∈ O t . Since C ( X t ) δ e · B e = B t B ∗ t * P, thereexists a ∈ C ( X t ) and b ∈ B e such that 0 = π P ( aδ e · b ) = a ( h ( P )) π P ( b ) . Hence a ( h ( P )) = 0 and this implies P ∈ X t . It is now time to prove claim (2). Take t ∈ G and P ∈ O t − . By construction(see [3]) the representation ρ : B e → B ( B t ⊗ π P ( B e /P )) , ρ ( b )( c ⊗ π P ( d )) = bc ⊗ π P ( d ) , has kernel ˜ α t ( P ) . Then, for all a ∈ C ( X t − ) and c ∈ B e , we have ρ ( aδ e · c ) = a ( h (˜ α t ( P ))) ρ ( c ) . Take z ⊗ π P ( w ) , z ′ ⊗ π P ( w ′ ) ∈ B t ⊗ π P ( B e /P ) . Recalling our innerproducts are linear in the second variable and using Lemma 2.9 we get a ( h (˜ α t ( P ))) π P ( w ∗ z ∗ c ∗ z ′ w ′ ) = a ( h (˜ α t ( P ))) h c ( z ⊗ π P ( w )) , z ′ ⊗ π P ( w ′ ) i = h ρ ( a ∗ δ e · c )( z ⊗ π P ( w )) , z ′ ⊗ π P ( w ′ ) i = π P ( w ∗ ( a ∗ δ e · cz ) ∗ z ′ w ′ )= π P ( w ∗ z ∗ c ∗ | {z } ∈ B t − aδ e · z ′ w ′ )= π P ( θ t − ( a ) δ e · w ∗ z ∗ c ∗ z ′ w ′ | {z } ∈ B e )= θ t − ( a )( h ( P )) π P ( w ∗ z ∗ c ∗ z ′ w ′ ) . Since π P ( w ∗ z ∗ c ∗ z ′ w ′ ) can not be null for all z ⊗ π P ( w ) , z ′ ⊗ π P ( w ′ ) ∈ B t ⊗ π P B e /P, we conclude that a ( h (˜ α t ( P ))) = a ( σ t ( h ( P ))) ∀ a ∈ C ( X t ) . Noticing that h (˜ α t ( P )) , σ t ( h ( P )) ∈ X t and recalling that C ( X t ) separates thepoints of X t we deduce that h (˜ α t ( P )) = σ t ( h ( P )) . (cid:3) Corollary 2.13.
Every Kasparov proper Fell bundle B = { B t } t ∈ G is amenable (i.e.the canonical map C ∗ ( B ) → C ∗ r ( B ) is injective). In case Prim( B e ) is Hausdorff,the partial action ˜ α of G on Prim( B e ) is proper.Proof. We can use the notation and hypotheses of Theorem 2.12, with the addi-tional assumption that σ is proper. The enveloping action of σ , σ e , is amenablebecause it is LCH and proper [5]. If X e is the enveloping space for σ, then wecan think of h : Prim( B e ) → X as a map from Prim( B e ) to X e which is ˜ α − σ e equivariant. Then B is amenable by [3, Theorem 6.3].Now assume Prim( B e ) is Hausdorff (hence LCH). We will use condition (2)of Proposition 1.10 to show ˜ α is proper. Take a net { ( t i , P i ) } i ∈ I ⊂ Γ ˜ α suchthat { (˜ α t i ( P i ) , P i ) } i ∈ I converges to a point ( Q, R ) ∈ Prim( B e ) × Prim( B e ) . Then { ( t i , h ( P i )) } i ∈ I ⊂ Γ σ and { ( σ t i ( h ( P i )) , h ( P i )) } i ∈ I converges to ( h ( Q ) , h ( R )) . Thereexists a subnet { ( t i j , P i j ) } j ∈ J such that { ( t i j , h ( P i j )) } i ∈ I ⊂ Γ σ converges to a point( t, x ) ∈ Γ σ . By construction h ( R ) = lim j h ( P i j ) = x ∈ X t − . Since R ∈ h − ( X t − ) = O t − , we obtain ( t, R ) ∈ Γ ˜ α is the limit of { ( t i j , P i j ) } j ∈ J . (cid:3) Now we show there is not such a thing like a LCH partial action which is notproper but it’s associated Fell bundle is weakly proper, what is quite comforting.
Corollary 2.14.
Let σ be a LCH partial action of G on C ( X ) . Then A σ is aweakly proper Fell bundle if and only if σ is proper.Proof. The multiplier algebra of C ( X ) δ e is (C*-isomorphic to) C b ( X ) and hencecommutative. Thus the direct implication follows from Corollary 2.13 because˜ α = σ . For the converse just consider A σ acting on A σ by multiplication. (cid:3) IXED POINT ALGEBRAS FOR FELL BUNDLES 13 Fixed point algebras
As we mentioned in the Introduction, the analysis of the basic examples of weaklyproper Fell bundles plays an important role in the general theory.3.1.
Basic examples.
Let σ be a LCH and proper partial action of G on X anddenote θ the respective partial action on C ( X ) . The enveloping action and en-veloping spaces of σ will be denoted σ e and X e , respectively. We view C ( X ) as anideal of C ( X e ) and θ e := θ ( σ e ) as the enveloping action of θ (in the C*-algebraicsense [2]).The partial crossed product C ( X ) ⋊ σ G is, by definition, the cross sectional C*-algebra of A σ , C ∗ ( A σ ) . The bundle A σ is an hereditary subbundle of A σ e and thuswe can view C ( X ) ⋊ σ G as a full hereditary C*-subalgebra of C ( X e ) ⋊ σ e G [4].Recall from 2.13 that the full and reduced cross sectional C*-algebras are the samein the present situation.The set of compactly supported continuous cross sections of A σ , C c ( A σ ) , isformed by functions of the form f † : G → A σ , t f ( t ) δ t , with f ∈ C σc ( G, C ( X )) := { g ∈ C c ( G, C ( X )) : g ( t ) ∈ C ( X t ) , ∀ t ∈ G } . Let’s recall the construction in [14] of a C ( X e /σ e ) − C ( X e ) ⋊ σ e G − bimodule E σ e . We are not assuming σ e is free, then we will not be able to guarantee E σ e isan equivalence bimodule.The σ e − orbit of a point x ∈ X e will be denoted [ x ] . Recall from Proposition 1.9that we may think
X/σ = X e /σ e by identifying [ x ] = [ x ] σ e with [ x ] σ , for all x ∈ X. Consider E σ e := C c ( X e ) with the pre C ( X/σ ) − left Hilbert module structuregiven by f g ( x ) = f ([ x ]) g ( x ) h g, h i ([ x ]) := Z G ( gh )( σ et − ( x )) dt, for all f ∈ C ( X/σ ) , g, h ∈ C c ( X e ) and x ∈ X. Routine computations show that the operations above are compatible and thatone can complete C c ( X e ) to get a left C ( X/σ ) − Hilbert module E σ e . In fact onecan use Stone-Weierstrass Theorem to show the idealspan {h g, h i : g, h ∈ C c ( X ) } is dense in C c ( X/σ ) in the inductive limit topology. Thus E σ e is in fact a full leftHilbert module.Full and reduced crossed products agree here, then one may appeal to [15] todescribe the right C ( X e ) ⋊ σ e G structure of E σ e (even if σ e is not free). For g, h ∈ C c ( X e ) and k † ∈ C c ( A σ e ) (with k ∈ C c ( G, C ( X e ))) the action and innerproducts are given by f k † ( x ) = Z G f ( σ et ( x )) k ( t )( σ et ( x ))∆( t ) − / dt ∀ x ∈ X e . (3.1) hh f, g ii σ e ( t ) = ∆( t ) − / f ∗ θ et ( g ) δ t ∀ t ∈ G. (3.2)In case σ e is free we have E σ e is a C ( X/σ ) − C ( X e ) ⋊ σ e G − equivalence bimod-ule.Our goal now is to show the closure of C c ( X ) in E σ e , henceforth denoted E σ , inherits a C ( X/σ ) − C ( X ) ⋊ σ G − bimodule from E σ e . Proposition 3.3. E σ is the closure of E σ := C c ( X e ) ∩ C ( X ) in E σ e . Proof.
It suffices to show that every f ∈ C c ( X e ) ∩ C ( X ) can be approximated in E σ e by elements of C c ( X ) . Fix f ∈ C c ( X e ) ∩ C ( X ) and take an approximate unit { g i } i ∈ I of C ( X ) contained in C c ( X ) . Since the projection of supp( f ) into X e /σ e is compact, there exists a compact set K ⊂ X e such that f ( x ) = 0 if [ x ] ∩ K = ∅ . Then, for all i ∈ I, k f − g i f k E σe = sup x ∈ K Z G | f − g i f | ( σ et − ( x )) dt. The trick now is to restrict the integral over G to a compact subset of G. To dothis note that if x ∈ K and | f − g i f | ( σ et − ( x )) = 0 , then σ et − ( x ) ∈ supp( f ) . Thus t ∈ L := { s ∈ G : K ∩ σ et (supp( f )) = ∅} and L is compact because σ e is LCH andproper. If µ ( L ) is the measure of L, then for all i ∈ I k f − g i f k E σe ≤ µ ( L ) k f − g i f k ∞ The proof now follows directly by taking limit in i. (cid:3) Continuing our discussion note that C ( X/σ ) E σ ⊂ E σ because E σ is an ideal of C c ( X e ) . Thus E σ has a natural C ( X/σ ) − left Hilbert module structure inheritedfrom E σ e . Note also that when we showed E σ e is left full we actually showed E σ isleft full.Given f ∈ E σ , k † ∈ C c ( A σ ) and x ∈ X e \ X we have, by (3.1), f k † ( x ) = 0 . Thisproves E σ C c ( A σ ) ⊂ E σ . Besides, for all f, g ∈ E σ and t ∈ G we have f ∗ θ et ( g ) ∈ C ( X ) ∩ θ et ( C ( X )) = C ( X t ) . This implies, by (3.2), that hh f, g ii σ e ∈ C c ( A σ ) . Then we conclude that E σ has a natural C ( X/σ ) − C ( X ) ⋊ σ G − bimodule structure(inherited from E σ e ).The natural choice for the fixed point algebra of σ, or A σ , is C ( X/G ) . To ensureit is strongly Morita equivalent to C ∗ ( A σ ) = C ( X ) ⋊ σ G one needs to show theideal generated by the C ( X ) ⋊ σ G − valued inner products span a dense subset of C ( X ) ⋊ σ G. As for global actions this can be done by assuming the partial actionsis free.
Definition 3.4.
A topological partial action τ of H on Y is free if for all t ∈ G \{ e } , { y ∈ Y t − : τ t ( y ) = y } = ∅ . In terms of topological freeness for partial actions, as defined in [8], a topologicalpartial action is free if it is free considered as an action of a discrete group on adiscrete space.
Proposition 3.5.
A topological partial action is free if and only if it’s envelopingaction is free.Proof.
Assume τ is a free topological partial action of H on Y and let τ e be it’senveloping action, with enveloping space Y e . Take y ∈ Y e and t ∈ H such that σ et ( y ) = y. There exists r ∈ H such that x := σ er ( y ) ∈ Y. Then σ ertr − ( x ) = x ∈ Y, this implies x ∈ Y rt − r − and σ rtr − ( x ) = x, thus rtr − = e and we get t = e. Theconverse is trivial because σ is a restriction of σ e . (cid:3) The Morita equivalence between the fixed point algebra and the cross sectionalC*-algebra is now available, at least for the basic examples of weakly proper Fellbundles coming from free partial actions.
Theorem 3.6.
Let σ be a LCH free and proper partial action of G on X. Then thebimodule E σ described before in this section is a C ( X/σ ) − C ( X ) ⋊ σ G − equivalencebimodule.Proof. All we need to do is to show the ideal generated by the C ( X ) ⋊ σ G − valuedinner products is dense in C ( X ) ⋊ σ G. We know, by Propositions 3.5 and 1.10,that σ e is free and proper. Thus E σ e is a C ( X/σ ) − C ( X e ) ⋊ σ e G − equivalencebimodule (see for example [14]). IXED POINT ALGEBRAS FOR FELL BUNDLES 15
Using (3.1) it is straightforward to prove that C c ( X e ) C c ( A σ ) ⊂ E σ . Recallingthat C ( X ) ⋊ σ G is a full hereditary C*-subalgebra of C ( X e ) ⋊ σ e G we get that C ( X ) ⋊ σ G = span C ( X ) ⋊ σ G hh C c ( X e ) , C c ( X e ) ii σ e C ( X ) ⋊ σ G = span hh E σ , E σ ii σ e ⊂ C ( X ) ⋊ σ G. (cid:3) Notation 3.7.
The C ( X ) ⋊ σ G − valued inner product of E σ will be denoted hh , ii σ . By construction hh f, g ii σ = hh f, g ii σ e for all f, g ∈ E σ . General weakly proper Fell bundles.
Take now a Fell bundle B = { B t } t ∈ G which is weakly proper with respect to the action A σ × B → B , ( a, b ) a · b, where σ = ( { X t } t ∈ G , { σ t } t ∈ G ) is a LCH and proper partial action of G on X. As usualwe set θ := θ ( σ ) , σ e is the enveloping action of σ, X e the enveloping space and theenveloping action of θ, θ e , is the action on C ( X e ) defined by σ e . To avoid repetition, whenever we write σ, B , θ (and any other mathematical sym-bol appearing in the paragraph above) we will be implicitly assuming the situationis the one we described before. The same will happen for objects constructed outof σ, B , θ, etc; like the space E B or the fixed point algebras we will construct somelines below.Unfortunately, the construction of the fixed point algebra for B depends on σ, but this is no surprise because something similar happens for weakly proper actionson C*-algebras [5].The map φ : C ( X ) → M ( B e ) , φ ( f ) b = f δ e · b, is a non degenerate *-homomor-phism and, since C ( X ) is a C*-ideal of C ( X e ) , there exists a unique extension φ e of φ to C ( X e ) . Motivated by Proposition 3.3 we define E B := { φ e ( f ) b : f ∈ C c ( X e ) , b ∈ B e } . For future reference we set the following.
Lemma 3.8. E B is a subspace of B e and for all b ∈ E B there exists f ∈ E σ = C c ( X e ) ∩ C ( X ) and b ′ ∈ B e such that b = f δ e · b ′ . Proof.
Given b, c ∈ E B and λ ∈ C take b ′ , c ′ ∈ B e and f, g ∈ C c ( X e ) such that b = φ e ( f ) b ′ and c = φ e ( g ) c ′ . Now take h ∈ C c ( X e ) such that hf = f and hg = g. Then b + λc = φ e ( h ) b + λφ e ( h ) c = φ e ( h )( b + λc ) ∈ E B . By Cohen-Hewitt’s factorization Theorem there exists k ∈ C ( X ) and b ′′ ∈ B e such that b ′ = kδ e · b ′ . Then b = φ e ( h ) b ′ = φ e ( h ) kδ e · b ′′ = ( hk ) δ e · b ′′ , and hk ∈ E σ . (cid:3) Now we construct the C c ( B ) valued inner product of E B using the C c ( A σ ) − valuedinner product of E σ . Proposition 3.9.
There exists a unique function hh , ii B : E B × E B → C c ( B ) , ( a, b )
7→ hh a, b ii B , such that for all f, g ∈ E σ , a, b ∈ B e and t ∈ G, (3.10) hh f δ e · a, gδ e · b ii B ( t ) = a ∗ ( hh f, g ii σ ( t ) · b ) . Moreover,(1) hh , ii B is linear in the second variable.(2) For all a, b ∈ E B , hh a, b ii ∗B = hh b, a ii B . (3) Given f, g ∈ C ( X e ) and a net { ( a j , b j ) } j ∈ J ⊂ B e × B e converging to ( a, b ) ∈ B e × B e , the net {hh φ e ( f ) a j , φ e ( g ) b j ii B } j ∈ J converges to hh φ e ( f ) a, φ e ( f ) b ii B in the inductive limit topology of C c ( B ) . Proof.
To show existence take f, g, h, k ∈ E σ and a, b, c, d ∈ B e such that f δ e · a = hδ e · c and gδ e · b = kδ e · d. Fix t ∈ G and take an approximate unit of C ( X t ) , { u i } i ∈ I . Then, a ∗ ( hh f, g ii σ ( t ) · b ) = ∆( t ) a ∗ ( f ∗ θ et ( g ) δ t · b ) = lim i ∆( t ) a ∗ (( u i f ) ∗ θ et ( θ t − ( u i ) g ) δ t · b )= lim i ∆( t ) a ∗ (( f δ e ) ∗ · ( u i δ e ) ∗ · ( u i δ t ) · ( gδ e ) · b )= lim i ∆( t )( f δ e · a ) ∗ (( u i δ t ) · ( gδ e ) · b )= lim i ∆( t )( hδ e · c ) ∗ (( u i δ t ) · ( kδ e ) · d ) = c ∗ ( hh h, k ii σ ( t ) · d ) . The identities above imply formula (3.10) can actually be used as a definition andcan also be used to show that hh , ii B is linear in the second variable.The following identities prove claim (2): hh f δ e · a, gδ e · b ii ∗B ( t ) = ∆( t ) − hh f δ e · a, gδ e · b ii B ( t − ) ∗ = ∆( t ) − [ a ∗ ( hh f, g ii σ ( t − ) · b )] ∗ = ∆( t ) − ( hh f, g ii σ ( t − ) · b ) ∗ a = b ∗ (∆( t ) − hh f, g ii σ ( t − ) ∗ · a )= b ∗ ( hh g, f ii σ ( t ) · a ) = hh gδ e · b, f δ e · a ii B ( t ) . Before proving claim (3) we develop an alternative way of computing hh x, y ii B ( t ) , for x, y ∈ E B and t ∈ G. Take an approximate unit of C ( X ) , { u i } i ∈ I , and factor-izations x = f · a and y = g · b with f, g ∈ E σ and a, b ∈ B e . Then(3.11) lim i hh u i · x, u i · y ii B ( t ) = lim i hh u i f · a, u i g · b ii B ( t )= lim i ∆( t ) − a ∗ ( hh u i f, u i g ii σ ( t − ) · b )= lim i ∆( t ) − a ∗ ( f ∗ θ et ( g ) u i θ et ( u i ) δ t · b ) = hh x, y ii B ( t ) , where the last identity holds because { u i θ et ( u i ) } i ∈ I is an approximate unit to C ( X t )and f ∗ θ et ( g ) ∈ C ( X t ) . Now take a net { ( a j , b j ) } j ∈ J ⊂ B e × B e and f, g ∈ C ( X e ) as in claim (3). Using(3.11) we deduce that, for all j ∈ J, supp hh f δ e · a j , gδ e · b j ii B ⊂ supp hh f, g ii σ e . Thus itsuffices to prove {hh f δ e · a j , gδ e · b j ii B } j ∈ J converges uniformly to hh f δ e · a j , gδ e · b j ii B . Again by (3.11) we have, for all t ∈ G, khh f δ e · a j , gδ e · b j ii B ( t ) − hh f δ e · a, gδ e · b ii B ( t ) k ≤≤ khh f δ e · ( a j − a ) , gδ e · b j ii B ( t ) k + khh f δ e · a, gδ e · ( b j − b ) ii B ( t ) k≤ k a j − a kk b j kkhh f, g ii σ e k ∞ + k a kk b j − b kkhh f, g ii σ e k ∞ . It is then straightforward to show thatlim j khh f δ e · a j , gδ e · b j ii B − hh f δ e · a, gδ e · b ii B k ∞ = 0 (cid:3) Our intention is to use hh , ii B as a C ∗ ( B ) − valued inner product and constructa Hilbert module with it. To do so we will need to show hh , ii B is positive. Lemma 3.12.
Consider two Fell bundles over G, C = { C t } t ∈ G and D = { D t } t ∈ G , and an non degenerate action by adjointable maps C × D → D , ( c, d ) c · d. Thenfor every non degenerate *-representation T : D → B ( V ) there exists a unique *-representation ˆ T : C → B ( V ) such that ˆ T c T d ξ = T c · d ξ, for all ( c, d, ξ ) ∈ C × D × V. Moreover, ˆ T is non degenerate. IXED POINT ALGEBRAS FOR FELL BUNDLES 17
Proof.
Fix c ∈ C , d , . . . , d n ∈ D e and ξ , . . . , ξ n ∈ V. Let w be a square root of k c k − c ∗ c ∈ M ( C e ) . Using the arguments of the proof of Proposition 2.10 we get k c k k n X i =1 T d i ξ i k − k n X i =1 T c · d i ξ i k = n X i,j =1 k c k h T d i ξ i , T d j ξ j i − h T c · d i ξ i , T c · d j ξ j i = n X i,j =1 h ξ i , T k c k d ∗ j d j − d ∗ j cc ∗ d j ξ j i = n X i,j =1 h ξ i , T d ∗ j w ∗ wd j ξ j i = k n X i =1 T wd i ξ i k ≥ . Since the restriction T | D e is non degenerate, the inequalities above imply thereexists a unique operator ˆ T c ∈ B ( V ) such that ˆ T c T d ξ = T c · d ξ, for all d ∈ D e and ξ ∈ V. Given any d ∈ D , taking an approximate unit { d j } j ∈ J of D e , it follows thatˆ T c T d ξ = lim j ˆ T c T d j T d ξ = lim j T c · d j · d ξ = T c · d ξ. Having defined the operators ˆ T c (for all a ∈ C ) we leave the rest of the proof tothe reader. (cid:3) Lemma 3.13.
For all x ∈ E B , hh x, x ii B ≥ in C ∗ ( B ) . Moreover, hh x, x ii B = 0 ifand only if x = 0 . Proof.
Take a faithful and non degenerate *-representation T : B → B ( V ) withfaithful integrated form ˜ T : C ∗ ( B ) → B ( V ) . Let ˆ T : A σ → B ( V ) be *-representationgiven by Lemma 3.12 and take f ∈ E σ and b ∈ B e such that x = f δ e · b. Then, forall ξ ∈ V, h ˜ T hh x,x ii B ξ, ξ i = Z G h T hh x,x ii B ( t ) ξ, ξ i dt = Z G h T b ∗ ( hh f,f ii σ ( t ) · b ) ξ, ξ i dt = Z G h ˆ T hh f,f ii σ ( t ) T b ξ, T b ξ i dt = h ˜ˆ T hh f,f ii σ T b ξ, T b ξ i ≥ , (3.14)where ˜ˆ T is the integrated form of ˆ T and the last inequality holds because hh f, f ii σ ≥ C ∗ ( A σ ) . In case hh x, x ii B = 0 , x ∗ x = hh x, x ii B ( e ) = 0 and this implies x = 0 . The converseis immediate. (cid:3)
Now we define an action ⋄ of C c ( B ) on E B on the right. Lemma 3.15.
For each x ∈ E B and f ∈ C c ( B ) there exists a unique function x ⊳ f ∈ C c ( G, B e ) such that given any approximate unit { u i } i ∈ I of C σ ( G, X ) := { f ∈ C ( G, C ( X )) : f ( t ) ∈ C ( X t ) , ∀ t ∈ G } , the net { r u i ( r − ) δ r − · xf ( r ) } i ∈ I ⊂ C c ( G, B e ) converges to x ⊳ f in the inductivelimit topology. Besides, (3.16) x ⋄ f := Z G ∆( r ) − / x ⊳ f ( r ) dr ∈ E B . Proof.
The set of continuous sections of B vanishing at ∞ , C ( B ) , is a Banach spacewith the norm k k ∞ . The function C σ ( G, X ) × C ( B ) → C ( B ) , ( f, g ) f ⋆ g, with f ⋆ g ( r ) = f ( r ) δ e · g ( r ) , is a linear action such that k f ⋆ g k ∞ ≤ k f k ∞ k g k ∞ . We claim that ⋆ in non degenerate in the sense that C σ ( G, X ) ⋆ C ( B ) = C ( B ) . Indeed, let S := span { f ⋆ g : f ∈ C θc ( G, X ) , g ∈ C c ( B ) } . It is clear that { vf : v ∈ C c ( G ) , v ∈ S } ⊂ S and, for all t ∈ G, { f ( t ) : f ∈ S } = span { u · v : u ∈ C ( X r ) δ e , v ∈ B r } . The non degeneracy condition of the action of A σ on B implies B r = B r B ∗ r B e B r = C ( X r ) δ e · B e B r ⊂ C ( X r ) δ e · B r ⊂ B r . Hence { f ( t ) : f ∈ S } is dense in B r . By [9, II 14.6] the conditions above imply S isdense in C c ( B ) in the inductive limit topology.For all f ∈ S and approximate unit { u i } i ∈ I of C σ ( G, X ) we have lim i k u i ⋆ f − f k ∞ = 0 and this implies the same holds for all f ∈ C ( B ) . Now the Cohen-HewittTheorem implies for all f ∈ C ( B ) there exists g ∈ C σ ( G, X ) and f ′ ∈ C ( B ) suchthat f = g ⋆ f ′ . Fix x ∈ E B and f ∈ C c ( B ) . The function xf ∈ C c ( B ) , given by ( xf )( r ) = xf ( r ) , admits a factorization g ⋆ h with g ∈ C σc ( G, X ) and h ∈ C c ( B ) . Consideran approximate unit { u i } i ∈ I of C σ ( G, X ) and define, for each i ∈ I, the function[ xf ] i ∈ C c ( G, B e ) by [ xf ] i ( r ) := u i ( r − ) δ r − · xf ( r ) . Clearly, supp[ xf ] i ⊂ supp f. Thus to show { [ xf ] i } i ∈ I converges in the inductive limit topology it suffices to proveit converges uniformly.Define k : G → B e by k ( r ) := θ r − ( g ( r )) δ r − · h ( r ) . Then k ∈ C c ( G, B e ) and, forall r ∈ G, k [ xf ] i ( r ) − k ( r ) k = k u i ( r − ) δ r − · xf ( r ) − θ r − ( g ( r )) δ r − · h ( r ) k = k u i ( r − ) δ r − · g ( r ) δ e · h ( r ) − θ r − ( g ( r )) δ r − · h ( r ) k≤ k u i ( r − ) δ r − g ( r ) δ e − θ r − ( g ( r )) δ r − kk h k ∞ ≤ k θ r ( u i ( r − )) g ( r ) − g ( r ) kk h k ∞ The function µ : C σ ( G, X ) → C σ ( G, X ) given by µ ( z )( r ) = θ r ( z ( r − )) is anisomorphism of C*-algebras. Then { µ ( u i ) } i ∈ I is an approximate unit of C σ ( G, X )and the inequalities above imply k [ xf ] i − k k ≤ k µ ( u i ) g − g kk h k ∞ . Thus { [ xf ] i } i ∈ I converges to k in the inductive limit topology.We set, by definition, k := x ⊳ f. In order to prove (3.16) choose w ∈ C c ( X e )such that wδ e · x = x. Then ( wg ) ⋆ h ( r ) = wg ( r ) δ r · h ( r ) = wδ e · g ( r ) δ r · h ( r ) = wδ e · xf ( r ) = xf ( r ) . Performing the construction of x ⊳ f using the factorization xf = ( wg ) ⋆ h we obtain x ⊳ f ( r ) = θ r − ( wg ( r )) δ r − · h ( r ) . For every t ∈ supp( h ) we have supp( θ r − ( wg ( r ))) ⊂ σ er − (supp( w )) . Since σ e is proper there exist a compact subset of the enveloping space X e containing S { supp( θ r − ( wg ( r ))) : t ∈ supp( h ) } . Thus we may find z ∈ C c ( X e ) such that zθ r − ( wg ( r )) = θ r − ( wg ( r )) , for all r ∈ G. This construction of z guarantees that zδ e · ( x ⊳ f ( r )) = x ⊳ f ( r ) , for all r ∈ G. Then we have x ⋄ f = Z G ∆( r ) − / zδ e · ( x ⊳ f ( r )) dr = zδ e · ( x ⋄ f ) ∈ E B . (cid:3) We want to construct a right C c ( B ) − module with inner product out of E B . Forthis we need to show the following.
Lemma 3.17.
For all x, y ∈ E B and f, g ∈ C c ( B ) , the identities hh x, y ⋄ f ii B = hh x, y ii B ∗ f ( x ⋄ f ) ⋄ g = x ⋄ ( f ∗ g ) obtain, where ∗ is the convolution product in C c ( B ) . Proof.
Without loss of generality we can replace x and y with gδ e · x and hδ e · y, with f, g ∈ E σ . Fix t ∈ G and let { u i } i ∈ I and { v j } j ∈ J be approximate units of IXED POINT ALGEBRAS FOR FELL BUNDLES 19 C ( X ) and C σ ( G, X ) , respectively. The construction of hh , ii B described in theproof of Proposition 3.9 together with Lemma 3.15 imply(3.18) hh gδ e · x, ( hδ e · y ) ⋄ f ii B ( t )= lim i ∆( t ) − / ( gδ e · x ) ∗ ( u i θ et ( u i ) δ t · [( hδ e · y ) ⋄ f ])= lim i Z G lim j ∆( ts ) − / ( gδ e · x ) ∗ ( u i θ et ( u i ) δ t · v j ( s − ) δ s − · ( hδ e · y ) f ( s )) ds = lim i Z G lim j ∆( ts ) − / x ∗ ( g ∗ u i θ et ( u i v j ( s − ) θ es − ( h )) δ ts − · yf ( s )) ds = lim i Z G ∆( ts ) − / x ∗ ( g ∗ u i θ et ( u i θ es − ( h )) δ ts − · yf ( s )) ds = lim i Z G ∆( ts − ) − / ∆( s − ) x ∗ ( g ∗ u i θ et ( u i θ es ( h )) δ ts · yf ( s − )) ds = lim i Z G ∆( s ) − / x ∗ ( g ∗ u i θ et ( u i θ et − s ( h )) δ s · yf ( s − t )) ds = lim i Z G ∆( s ) − / x ∗ ( u i θ et ( u i ) δ e · g ∗ θ es ( h ) δ s · yf ( s − t )) ds. Note that g ∗ θ es ( h ) δ s · yf ( s − t ) ∈ B t for all s ∈ G. Let F i , F ∈ C c ( G, B t ) bedefined as F i ( s ) : = u i θ et ( u i ) δ e · g ∗ θ es ( h ) δ s · yf ( s − t ) F ( s ) : = g ∗ θ es ( h ) δ s · yf ( s − t )The supports of both F i and F are contained in t supp( f ) − , thus the net { F i } i ∈ I converges to F in the inductive limit topology if and only if it converges uniformly.To show uniform convergence it suffices to prove that given a net { s i } λ ∈ Λ ⊂ G converging to s ∈ G, it follows that lim λ k F i ( s i ) − F ( s ) k = 0 . Since F ( s ) ∈ B t = B t B ∗ t B e B t = C ( X t ) δ e · B t , there exist m ∈ C c ( X t ) and z ∈ B t such that F ( s ) = mδ e · z. Then0 ≤ lim i k F i ( s i ) − F ( s ) k ≤ lim i k F i ( s i ) − F i ( s ) k + k F i ( s ) − F ( s ) k≤ lim i k u i θ et ( u i ) kk F ( s i ) − F ( s ) k + k F i ( s ) − F ( s ) k≤ lim i k F i ( s ) − F ( s ) k = lim i k u i θ et ( u i ) δ e · F ( s ) − F ( s ) k = lim i k u i θ et ( u i ) m − m kk z k = 0 , where the last identity holds because { u i θ et ( u i ) } i ∈ I is an approximate unit of C ( X t ) . Now we can continue the computations (3.18) to get hh gδ e · x, ( hδ e · y ) ⋄ f ii B ( t ) = lim i Z G F i ( s ) ds = Z G F ( s ) ds = Z G ∆( s ) − / x ∗ ( g ∗ θ es ( h ) δ s · y ) f ( s − t ) ds = Z G hh gδ e · x, hδ e · y ii ( s ) f ( s − t ) ds = hh gδ e · x, hδ e · y ii ∗ f ( t ) . This proves hh gδ e · x, ( hδ e · y ) ⋄ f ii B = hh gδ e · x, hδ e · y ii ∗ f. To show that ( x ⋄ f ) ⋄ g = x ⋄ ( f ∗ g ) (for x ∈ E B and f, g ∈ C c ( B )) it suffices toshow, by Lemma 3.13, that for y := ( x ⋄ f ) ⋄ g − x ⋄ ( f ∗ g ) one has hh y, y ii B = 0 . But this is so because hh y, y ii B = ( hh y, x ii B ∗ f ) ∗ g − hh y, x ii B ∗ ( f ∗ g ) = 0 . (cid:3) By a C*-norm of a *-algebra A we mean a norm k k of A such that k ab k ≤ k a kk b k and k a ∗ a k = k a k , for all a, b ∈ A. Then a C*-algebra is a *-algebra which has aC*-norm k k such that ( A, k k ) is a Banach space.An exotic C*-norm of C c ( B ) is (for us) any C*-norm k k of C c ( B ) such that k k r ≤ k k ≤ k k u , where k k r and k k u are the reduced and universal C*-norms (bythe universal C*-norm we mean the largest one dominated by k k ). Definition 3.19.
Let µ be an exotic C*-norm of C c ( B ) and denote C µ ( B ) theC*-algebra obtained by completing C c ( B ) with respect to µ. Then we define E µ B as the completion of E B with respect to the norm k k µ : E B → [0 , + ∞ ) , x µ ( hh x, x ii B ) / , and regard E µ B as a right C µ ( B ) − Hilbert module. The µ − fixedpoint algebra for B , F µ B , is the C*-algebra of generalized compact operators of E µ B , K ( E µ B ) . For future use we give a bound on k k µ , for every exotic C*-norm µ. Remark . For every f ∈ E σ and x ∈ B e , k f δ e · x k µ ≤ k f k σ k x k , where k k σ isthe norm of E σ . Indeed, it suffices to consider µ as the universal C*-norm. Thenthe bound follows from (3.14).The fixed point algebras F µ B have a natural C ( X/σ ) − algebra structure, as weshow below. Proposition 3.21.
Let C b ( X ) = M ( C ( X )) act on B e by extending the action of C ( X ) = C ( X ) δ e on B e . Consider C ( X/σ ) as a C*-subalgebra of C b ( X ) and let C ( X/σ ) act on B e through the action of C b ( X ) . Then C ( X/σ ) E B ⊂ E B and thisgives an action C ( X/σ ) × E B → E B , ( f, x ) f x. Moreover, for every exotic C*-norm µ of C c ( B ) there exits a unique *-homomorphism φ µ : C ( X/σ ) → B ( E µ B ) = M ( F µ B ) such that φ µ ( f ) x = f x, for all f ∈ C ( X/σ ) and x ∈ E µ . Besides, φ µ isnon degenerate.Proof. Take f ∈ C ( X/σ ) and x ∈ E B . Consider a factorization x = gδ e · y with g ∈ E σ and y ∈ B e . Then, by construction, f x = f ( gδ e · y ) = ( f g ) δ e · y ∈ E B . Now let µ be an exotic C*-norm of C c ( B ) . Take a non degenerate *-representation T : B → B ( V ) such that the integrated form ˜ T : C ∗ ( B ) → B ( V ) factors through afaithful representation of C ∗ µ ( B ) . Let ˆ T : A σ → B ( V ) be the *-representation de-scribed in Lemma 3.12. Given f ∈ C ( X/σ ) and x ∈ E σ take a factorization x = gδ e · y as explained in the last paragraph. We know (see Section 3.1) that hh f g, f g ii σ ≤ k f k hh g, g ii σ in C ∗ ( A σ ) . Using the computations in (3.14) one ob-tains, for all ξ ∈ V, that h ˜ T k f k hh x,x ii B −hh fx,fx ii B ξ, ξ i = h ˜ˆ T k f k hh g,g ii σ −hh fg,fg ii σ T y ξ, T y ξ i ≥ . Thus hh f x, f x ii B ≤ k f k hh x, x ii B in C ∗ γ ( B ) . This implies that for all f ∈ C ( X/σ )there exists a unique bounded operator φ µ ( f ) : E µ B → E µ B such that φ µ ( f ) x = f x, for all x ∈ E µ B . Moreover, k φ µ ( f ) x k µ ≤ k f kk x k µ . The operator φ µ ( f ) is adjointable with adjoint φ µ ( f ∗ ) because, for all x, y ∈ B e ,g, h ∈ E σ and t ∈ G, hh φ µ ( f )( gδ e · x ) , hδ e · y ii B ( t ) = x ∗ ( hh f g, h ii σ ( t ) · y ) = x ∗ ( hh g, f ∗ h ii σ ( t ) · y )= hh gδ e · x, φ µ ( f ∗ )( hδ e · y ) ii B ( t ) . IXED POINT ALGEBRAS FOR FELL BUNDLES 21
Now that we know the map φ µ : C ( X/σ ) → M ( F µ B ) is defined and preserves theinvolution, we leave to the reader the verification of the fact that φ µ is linear andmultiplicative.In order to show that φ µ is non degenerate it suffices to show that given anapproximate unit { f i } i ∈ I of C ( X/σ ) , g ∈ E σ and x ∈ B e , we have thatlim i k φ µ ( f i ) gδ e · x − gδ e · x k µ = 0 . By Remark 3.20 we have k φ µ ( f i ) gδ e · x − gδ e · x k µ ≤ k f i g − g k σ k x k . The construction of E σ in Section 3.1 implies lim i k f i g − g k σ = 0 . Thus φ µ is nondegenerate. (cid:3) The next result implies there are as many exotic fixed point algebras (for a givenFell bundle) as exotic C*-norms.In the proof below we consider Hilbert modules as ternary C*-rings (C*-trings)[17]. More precisely, given a right A − Hilbert module Y we consider on Y theternary operation ( x, y, z ) Y := x h y, z i A . An homomorphism of C*-trings is a linearmap φ : E → F such that φ ( x, y, z ) = ( φ ( x ) , φ ( y ) , φ ( z )) , for all x, y, z ∈ E. Proposition 3.22.
Given two exotic C*-norms of C c ( B ) , µ and ν with µ ≤ ν, there exists a unique homomorphism of C*-trings κ µν : : E ν B → E µ B extending thenatural identity map of E B . Moreover, κ µν is surjective. In case the inner products hh ii B span a dense subset of C ∗ ν ( B ) the following are equivalent:(1) κ µν is injective (and hence an isomorphism).(2) κ µν is isometric (and hence an isomorphism).(3) µ = ν. Proof.
For all x ∈ E B we have k x k µ = µ ( hh x, x ii B ) / ≤ ν ( hh x, x ii B ) / = k x k ν . Thus the identity map of E B admits a unique linear and continuous extension κ µν . This extension is a homomorphism because the identity κ µν ( x, y, z ) = ( κ µν ( x ) , κ µν ( y ) , κ µν ( z ))holds for all x, y, z ∈ E B and hence, by continuity, for all x, y, z ∈ E ν B . The range of κ µν is closed by [4, Corollary 4.8]. Clearly (3) implies (1) and (2)and these last two conditions are equivalent by [4, Proposition 3.11].Assume (2) holds. Since the inner products span a dense subset of C ν ( B ) , theyalso span a dense subset of C µ ( B ) . Regarding E ν B (respectively, E µ B ) as full right C ∗ ν ( B ) − Hilbert module (respectively, C ∗ ν ( B ) − Hilbert module) we obtain, for all f ∈ C c ( B ) , ν ( f ) = sup {k x ⋄ f k ν : x ∈ E B , k x k ν ≤ } = µ ( f ) . This completes the proof. (cid:3)
As explained in [17] one can recover the µ − fixed point algebra out of the C*-tring structure of E µ B . In fact the maps κ µν : E ν B → E µ B induce surjective morphismof C*-algebras κ µν r : F ν B → F µ B [2, Proposition 4.1] and the equivalence in our lastProposition also holds for these maps.The main result of this section is the following one, in which we prove a Moritaequivalence between exotic crossed products and exotic fixed point algebras. Theorem 3.23. If σ is free then I B := span {hh x, y ii B : x, y ∈ E B } is dense in C c ( B ) in the inductive limit topology. In particular, for every exoticC*-norm µ of C c ( B ) the bimodule E µ B is a F µ B − C ∗ µ ( B ) − equivalence bimodule.Proof. It suffices to work with the universal C*-norm k k u of C c ( B ) . The proofof Green’s Symmetric Imprimitivity Theorem presented in [14], used here for theenveloping action σ e , implies that I σ e := span {hh f, g ii σ e : f, g ∈ E σ e } ⊂ C c ( A σ e )is dense in the inductive limit topology in C c ( A σ e ) . Moreover, as shown in [14], forevery k ∈ C c ( A σ e ) there exists a compact set L ⊂ G and a net { k i } i ∈ I ∈ I σ e suchthat supp( k i ) ⊂ L for all i ∈ I and k k i − k k ∞ → . Since C c ( A σ ) is hereditary in C c ( A σ e ) , by using the approximate units constructed in [9, VIII 16.4] we get thatfor all k ∈ C c ( A σ ) there exists a compact set L ⊂ G and a net { k i } i ∈ I ⊂ span { u ∗ hh f, g ii σ e ∗ v : f, g ∈ C c ( X e ) , u, v ∈ C c ( A σ ) } such that supp( k i ) ⊂ L, for all i ∈ I, and k k i − k k ∞ → . But since C c ( X e ) C c ( A σ ) ⊂ E σ , the last approximate unit { k i } i ∈ I is included in I σ := span {hh f, g ii σ : f, g ∈ E σ } ⊂ C c ( A σ ) . Now define I B := span {hh x, y ii : x, y ∈ E B } ⊂ C c ( B )and let I B be the closure of I B in the inductive limit topology of C c ( B ) . For I B tobe equal to C c ( B ) we just need to show, by [9, II 14.6],(i) C c ( G ) I B ⊂ I B . (ii) For all t ∈ G, I B ( t ) := { z ( t ) : z ∈ I B } is dense in B t . Given f ∈ C c ( G ) and k ∈ I B take a net { k i } i ∈ I ⊂ I B converging to k in theinductive limit topology (and hence uniformly over compact sets). Thus { f k i } i ∈ I converges to f k in the inductive limit topology and to show f k ∈ I B it suffices toshow that f k i ∈ I B , for all i ∈ I. In other words, we can assume from the beginningthat k ∈ I B . Moreover, by linearity we may assume k = hh gδ e · x, hδ e · y ii B with g, h ∈ E σ and x, y ∈ B e . For all t ∈ G we have( f k )( t ) = x ∗ ( f ( t ) hh g, h ii σ ( t ) · y ) . Now take a compact set L and a net { m j } j ∈ J ⊂ I σ ∩ C L ( A σ ) converging uniformlyto the function t f ( t ) hh g, h ii σ ( t ) . Then the net { t x ∗ ( m j ( t ) · y ) } j ∈ J is containedin I B and converges to f k in the inductive limit topology. Thus f k ∈ I B . To prove (ii) take t ∈ G and b ∈ B t . By the Cohen-Hewitt factorization Theoremand the non degeneracy of the action of A σ on B there exists c, d ∈ B e and g ∈ C ( X t ) such that b = c ∗ ( gδ t · d ) . Since there exists an element of C c ( A σ ) taking thevalue gδ t at t, there exists a net { m j } j ∈ J ⊂ I σ such that k m j ( t ) − gδ t k → . Thenthe net { s c ∗ ( m ( s ) · d ) } j ∈ J lies in I B and, after evaluation at t, converges to b. Thus b ∈ I B ( t ) . The rest of the proof is straightforward because the inductive limit topology isstronger that any topology coming from an exotic C*-norm. (cid:3)
The module E B as a tensor product. For this section we need exactly thesame setting we used in the first paragraph of Section 3.2.In Section 3.1 we constructed the C ∗ ( A σ ) module E σ , there are no exotic C*-norm to be considered for σ because A σ is amenable. The action of A σ on B passesto an action of C ∗ ( A σ ) on C ∗ µ ( B ) , for any exotic C*-norm µ. Proposition 3.24.
For every exotic C*-norm µ of C c ( B ) there exists a unique*-representation S µ : A σ → M ( C ∗ µ ( B )) such that: • For all a ∈ A σ , S µa C c ( B ) ⊂ C c ( B ) . IXED POINT ALGEBRAS FOR FELL BUNDLES 23 • For all s, t ∈ G, a ∈ C ( X t ) δ t and f ∈ C c ( B ) , S µa f ( s ) = a · f ( t − s ) . Moreover, S µ is non degenerate and f S µ : C ∗ ( A σ ) → M ( C ∗ µ ( B )) is the unique*-homomorphism satisfying the following • For all f ∈ C c ( A σ ) , f S µf C c ( B ) ⊂ C c ( B ) . • For all f ∈ C c ( A σ ) , g ∈ C c ( B ) and t ∈ G, f S µf g ( t ) = R G f ( s ) · g ( s − t ) dt. Proof.
Uniqueness claims are immediate, we will only prove the existence. Forconvenience we write A t instead of C ( X t ) δ t . Let T : B → B ( V ) be a non degenerate *-representation on a Hilbert spacewhose integrated form e T : C ∗ ( B ) → B ( V ) factors through a faithful representationof C ∗ µ ( B ) . Thus we can actually think of e T as a non degenerate and faithful *-representation of C ∗ µ ( B ) . We will denote D the image of e T .
The canonical extensionof e T to M ( C ∗ µ ( B )) will be denoted T .
This extension is injective and it’s image is
M D := { R ∈ B ( V ) : RD ∪ DR ⊂ D } . Given a ∈ A t and f ∈ C c ( B ) we define the function a · f ∈ C c ( B ) by a · f ( s ) := a · f ( t − s ) . Let ˆ T : A σ → B ( V ) be the *-representation given by Lemma 3.12. Then for all a ∈ A t , f ∈ C c ( B ) and ξ ∈ V :ˆ T a e T f ξ = Z G T a · f ( t ) ξ dt = Z G T a · f ( s − t ) ξ dt = e T a · f ξ. This implies ˆ T a e T ( C c ( B )) ⊂ D and by continuity we get ˆ T a D ⊂ D. Now define g ∈ C c ( B ) by g ( t ) := ( a ∗ · f ( t ) ∗ ) ∗ and take a factorization ξ = T b η, with b ∈ B e and η ∈ V. Then e T f ˆ T a ξ = Z G T f ( t )( a · b ) η dt = Z G T ( a ∗ · f ( t ) ∗ ) ∗ b η dt = Z G T ( a ∗ · f ( t ) ∗ ) ∗ ξ dt = e T g ξ. This implies D ˆ T a ⊂ D and we conclude that ˆ T a ∈ M D.
By thinking of T as in isomorphism between M ( C ∗ µ ( B )) and M D one just needsto set S µ := T − ◦ ˆ T .
The computations above show S µ satisfies the desired prop-erties.Under the isomorphism M ( C ∗ µ ( B )) ≈ M D, S µ is identified with ˆ T (that is thewhole point of the proof). Then we can think of the integrated form of ˆ T as theintegrated form of S µ . Take f ∈ C c ( A σ ) and g ∈ C c ( B ) . Define F : G → C c ( B ) , F ( s )( t ) = f ( s ) · g ( s − t ) . Note supp( F ( s )) ⊂ supp( f )supp( g ) , for all s ∈ G, and F is k k ∞ − continuous andhas compact support. Then F can be integrated with respect to the inductive limittopology and, since evaluation at s ∈ G is continuous with respect to this topology, R G F ( s ) ds ( t ) = R G F ( s )( t ) ds = R G f ( s ) · g ( s − t ) ds. We define f · g := R G F ( s ) ds. For all f ∈ C c ( A σ ) , g ∈ C c ( B ) and ξ ∈ V we have e ˆ T f e T g ξ = Z G e ˆ T f T g ( t ) ξ dt = Z G Z G ˆ T f ( s ) T g ( t ) ξ dsdt = Z G Z G T f ( s ) · g ( t ) ξ dsdt = Z G Z G T f ( s ) · g ( s − t ) ξ dtds = e T f · g ξ. Then we must have S µf g = f · g, and this identity completes the proof. (cid:3) Theorem 3.25.
For every exotic C*-norm µ of C c ( B ) the right C ∗ µ ( B ) − Hilbertmodule E µ B is unitarily equivalent to E σ ⊗ f S µ C ∗ µ ( B ) , where f S µ : C ∗ ( A σ ) → M ( C ∗ µ ( B )) is the integrated form given by Proposition 3.24.Proof. Let E σ ⊗ C c ( B ) be the subspace of E σ ⊗ f S µ C ∗ µ ( B ) spanned by the elementarytensor product f ⊗ g with f ∈ E σ and g ∈ C c ( B ) . Take f , . . . , f n ∈ E σ and g , . . . , g n ∈ C c ( B ) . By considering the action of B e on C ( B ) by multiplication we can get a factorizations g i = b i h i with b i ∈ B e and h i ∈ C c ( B ) , for i = 1 , . . . , n. We claim that(3.26) k n X i =1 ( f i · b i ) ⋄ h i k = k n X i =1 f i ⊗ b i h i k . To show this it suffices to prove that(3.27) hh ( f i · b i ) ⋄ h i , ( f j · b j ) ⋄ h j ii B = ( b i h i ) ∗ ( f S µ hh f i ,f j ii σ ( b j h j )) , ∀ i, j = 1 , . . . , n. For any i, j = 1 , . . . , n and t ∈ G we have(3.28) hh ( f i · b i ) ⋄ h i , ( f j · b j ) ⋄ h j ii B ( t ) = h ∗ i ∗ hh f i · b i , f j · b j ii B ∗ h j ( t )= Z G Z G h ∗ i ( r ) hh f i · b i , f j · b j ii B ( s ) h j ]( s − r − t ) dsdr = Z G Z G h ∗ i ( r ) b ∗ i [ hh f i , f j ii σ ( s ) · b j ] h j ( s − r − t ) dsdr = Z G Z G ( b i h i ) ∗ ( r )[ hh f i , f j ii σ ( s ) · ( b j h j ( s − r − t ))] dsdr = Z G ( b i h i ) ∗ ( r )[ f S µ hh f i ,f j ii σ ( b j h j )( r − t )] dr = ( b i h i ) ∗ ( f S µ hh f i ,f j ii σ ( b j h j ))( t ) . Now that we know (3.26) holds we can construct a unique bounded linear op-erator U : E σ ⊗ f S µ C ∗ µ ( B ) → E µ B such that U ( f ⊗ bh ) = ( f · b ) ⋄ h, for all f ∈ E σ ,b ∈ B e and h ∈ C c ( B ) . Moreover, U is an isometry with dense range, thus it isan isometric isomorphism of Banach spaces. But now (3.27) says U preserves theinner products, thus it is a unitary operator. (cid:3) After the Theorem above Proposition 3.21 should be completely natural. Weleave to the reader the verification of the fact that the unitary constructed in ourlast proof intertwines the action constructed in Proposition 3.21 with the naturalaction of C ( X/σ ) on E σ ⊗ f S µ C ∗ µ ( B ) . Theorem 3.25 can also be used to give an alternative proof of Theorem 3.23.Indeed, in case σ is free then E σ is full on the right, and since f S µ is non de-generate we conclude that E µσ = E σ ⊗ f S µ C ∗ µ ( B ) is full on the right and hence a F µ B − C ∗ µ ( B ) − equivalence bimodule.Our last Theorem also implies our exotic fixed point algebras (an even the mod-ules used to construct them) are generalizations of those constructed in [5] forweakly proper actions (see the discussion preceding Definition 2.3 and Example2.6).3.4. Bra-ket operators and the fixed point algebra.
In [11, 12] Meyer de-fines square integrable actions, which are a generalization of proper actions onC*-algebras (or even of weakly proper actions). One can extend Meyer’s definitionto partial action on C*-algebras, but we will not pursue this goal here. We aremore interested in the so called bra-ket operators in the context of Fell bundles.
IXED POINT ALGEBRAS FOR FELL BUNDLES 25
Assume α is an action of G on the C*-algebra A and assume there exists a densesubset A of A such that for all a, b ∈ A the function hh a, b ii : G → A, t α t ( a ) ∗ b, has compact support. Then the element a ∈ A is said to be square integrable ifthe bra-operator hh a | : A → C c ( G, A ) , b
7→ hh a, b ii , is the restriction of some adjointable operator from A to L ( G, A ) . If such anextension exits, it is unique and it is denoted hh a | . The ket-operator is | a ii := hh a | ∗ and it should satisfy | a ii ( f ) = Z G α t ( a ) f ( t ) dt, ∀ f ∈ C c ( G, A ) . If a, b ∈ A are square integrable then hh a | ◦ | b ii ∈ B ( L ( G, A )) and in case α isweakly proper one gets that hh a | ◦ | b ii ∈ C c ( G, A ) ⊂ A ⋊ r α G ⊂ B ( L ( G, A )) . In order to translate the previous construction to weakly proper Fell bundles onemust first note that L ( G, A ) is not equal to L ( B α ) , but it is unitary equivalent.This explains the absence of the modular function in the formula hh a, b ii ( t ) = α t ( a ) ∗ b. The inclusion A ⋊ r α G ⊂ B ( L ( G, A )) , as given in [11, Section 3], takesthis equivalence into account. All we will do here will be compatible with thatidentification.Take a Fell bundle B = { B t } t ∈ G which is weakly proper with respect to the LCHand proper partial action σ of G on X. As usual we denote θ the partial action of G on C ( X ) defined by σ. The action of A σ on B will be denoted A σ ×B → B , ( a, b ) a · b. The space E B ⊂ B e is that of Section 3.2. Theorem 3.29.
For every x ∈ E B there exists a unique adjointable operator hh x | : B e → L ( B ) such that hh x | y = hh x, y ii B for all y ∈ E B . The adjoint | x ii := hh x | ∗ is the unique linear operator from L ( B ) to B e such that | x ii f = x ⋄ f, for all f ∈ C c ( B ) . Moreover, if f Λ B : C ∗ ( B ) → B ( L ( B )) is the regular representation, thenfor all x, y, z ∈ E B and f ∈ C c ( B ) we have f Λ Bhh x,y ii B = hh x | ◦ | y ii ; | x iihh y | z = x ⋄ hh y, z ii B ; | x ⋄ f ii = | x ii ◦ f Λ B f . Proof.
Fix x ∈ E B and define the functions P : E B → C c ( B ) , y
7→ hh x, y ii B ,Q : C c ( B ) → E B , f x ⋄ f. Note both P and Q are linear. If P and Q are to be extended to adjointableoperators, then we must have, for all y ∈ E B and f ∈ C c ( B ) , (3.30) h f, hh x, y ii i L ( B ) = h f, P y i L ( B ) = h Qf, y i B e = ( Qf ) ∗ y = ( x ⋄ f ) ∗ y. To prove the identities above it suffices to show the first term equals the last one.Take f ∈ C c ( B ) and y ∈ E B and consider factorizations x = gδ e · u and y = hδ e · v with g, h ∈ E σ and u, v ∈ B e . Using an approximate unit { u i } i ∈ I of C σ ( G, X ) as the one in Lemma 3.15 we deduce that( x ⋄ f ) ∗ y = Z G lim i ∆( t ) − / ( u i ( t − ) δ t − · xf ( t )) ∗ y dt = Z G lim i ∆( t ) − / ( u i ( t − ) δ t − · gδ e · uf ( t )) ∗ ( h · δ e v ) dt = Z G lim i ∆( t ) − / ( h ∗ δ e u i ( t − ) δ t − gδ e · uf ( t )) ∗ v dt = Z G lim i ∆( t ) − / ( u i ( t − ) h ∗ θ et − ( g ) δ t − · uf ( t )) ∗ v dt = Z G ∆( t ) − / ( h ∗ θ et − ( g ) δ t − · uf ( t )) ∗ v dt = Z G f ( t ) ∗ ( u ∗ ∆( t ) − / g ∗ θ et ( h ) δ t · v ) dt = Z G f ( t ) ∗ hh x, y ii ( t ) dt = h f, hh x, y ii i L ( B ) This completes the proof of (3.30).By taking y = x ⋄ f in (3.30) and recalling that f Λ B g h = g ∗ h for all g, h ∈ C c ( B )we get ( Qf ) ∗ ( Qf ) = ( x ⋄ f ) ∗ ( x ⋄ f ) = h f, hh x, x ⋄ f ii B i L ( B ) = h f, hh x, x ii B ∗ f i L ( B ) ≤ khh x, x ii B k C ∗ r ( B ) h f, f i L ( B ) (3.31)Then Q is bounded and k Q k ≤ khh x, x ii B k C ∗ r ( B ) . Using (3.30) and that Q is bounded we deduce that k P y k = sup {kh f, P y i L ( B ) k : f ∈ C c ( B ) , k f k L ( B ) ≤ } = sup {kh Qf, y i L ( B ) k : f ∈ C c ( B ) , k f k L ( B ) ≤ }≤ k Q kk y k . Hence P is also bounded.Let hh x | : B e → L ( B ) and | x ii : L ( B ) → B e be the unique continuous extensionsof P and Q, respectively. By (3.30) hh x | is adjointable with adjoint | x ii . Now (3.31) can be used to deduce that f Λ Bhh x,x ii B = hh x |◦| x ii for all x ∈ E B . Thenthe polarization identity implies that f Λ Bhh x,y ii B = hh x |◦| y ii for all x, y ∈ E B . Finally,for all x, y, z ∈ E B and f, g ∈ C c ( B ) one has | x iihh y | z = x ⋄ ( hh y | z ) = x ⋄ hh y, z ii B and | x ⋄ f ii g = ( x ⋄ f ) ⋄ g = x ⋄ ( f ∗ g ) = x ⋄ ( f Λ B f g ) = (cid:16) | x ii ◦ f Λ B f (cid:17) g. Then the last identity holds for all g ∈ L ( B ) and the proof is complete. (cid:3) Consider the subspace F B := span {| x iihh y | : x, y ∈ E B } ⊂ M ( B e ) , which is in fact a *-subalgebra of M ( B e ) because | x iihh y || z iihh w | = | x ii f Λ Bhh y,z ii B hh w | = | x ⋄ hh y, z ii B iihh w | ∈ F B . Given an exotic C*-seminorm µ of C c ( B ) , the generalized compact operator[ x, y ] ∈ F µ B = B ( E µ B ) corresponding to the x, y ∈ E B is given by [ x, y ]( z ) = x ⋄hh y, z ii B = | x ii ◦ hh y | z, for all z ∈ E B . Thus one gets a unique morphism of *-algebras π µ : F B → F µ B ⊂ B ( E µ B ) , such that π µ ( T ) x = T x ∀ T ∈ F B , x ∈ E B . In fact π µ is injective and has dense range. Then the exotic fixed point algebra F µ B is a C*-completion of F B ⊂ M ( B e ) . IXED POINT ALGEBRAS FOR FELL BUNDLES 27
We need a Lemma to determine the (exotic) fixed point algebra correspondingto the closure (completion) of F B in M ( B e ) . Lemma 3.32.
For all x ∈ E B , x is contained in the image under | x ii of the closedunit ball of L ( B ) . In particular k x k B e ≤ k| x iik . Proof.
The thesis follows immediately if x = 0 , otherwise we may assume k x k B e = 1without loss of generality.Given ε > b ∈ B e such that k x − xb k B e < ε and k b k < . Now take f ∈ C c ( B ) such that f ( e ) = b and set g := x ⊳ f ∈ C c ( G, B e ) as in Lemma 3.15. Byconstruction g ( e ) = xf ( e ) = xb, thus there exists a compact neighbourhood V of e ∈ G such that: (a) it’s measure µ ( V ) is less than 1; (b) k x − ∆( r ) − / g ( r ) k < ε and k f ( r ) k < , for all r ∈ V. Take a ∈ C c ( G ) + with support contained in V and such that R G a ( r ) dr = 1 . Then k af k L ( B ) = k Z G a ( r ) f ( r ) ∗ f ( r ) dr k ≤ Z G a ( r ) k f ( r ) k , dr ≤ x ⊳ ( af ) = a ( x ⊳ f ) = ag. Thus k x − | x ii ( af ) k B e = k x − Z G ∆( r ) − / a ( r ) g ( r ) dr k ≤ Z V a ( r ) k x − ∆( r ) − / g ( r ) k dr ≤ (cid:18)Z V a ( r ) dr (cid:19) / (cid:18)Z V k x − ∆( r ) − / g ( r ) k dr (cid:19) / ≤ εµ ( V ) / < ε. The proof is complete because we have been able to find, for every ε > , afunction h = ag ∈ L ( B ) such that k h k L ( B ) ≤ k x − | x ii h k B e < ε. (cid:3) Proposition 3.33.
For every T ∈ F B and x ∈ E B one has T x ∈ E B and | T x ii = T | x ii . Besides, the completion of F B in M ( B e ) is the fixed point algebracorresponding to the reduced crossed sectional C*-algebra norm on C c ( B ) . Proof. If T = P ni =1 | y i iihh z i | , then T x = P ni =1 y i ⋄ hh z i , x ii B ∈ E B . Besides, for all f ∈ C c ( B ) | T x ii f = ( T x ) ⋄ f = n X i =1 ( y i ⋄ hh z i , x ii B ) ⋄ f = n X i =1 y i ⋄ hh z i , x ⋄ f ii B = n X i =1 | y i iihh z i | ( x ⋄ f ii B ) = ( T | x ii ) f. Hence | T x ii = T | x ii . The norm of T in the reduced fixed point algebra, F r B , satisfies k T k F r B = sup {khh T x, T x ii B k r : x ∈ E B , khh x, x ii B k r ≤ } = sup {k| T x iik : x ∈ E B , khh x, x ii B k r ≤ } = sup {k T | x iik : x ∈ E B , khh x, x ii B k r ≤ }≤ sup {k T k M ( B e ) k| x iik : x ∈ E B , khh x, x ii B k r ≤ }≤ k T k M ( B e ) . Hence k T k F r B ≤ k T k M ( B e ) . Let D be the completion of F B in M ( B e ) . Then the conclusion of the last para-graph implies the existence of a unique surjective *-homomorphism π : D → F r B extending the identity operator of F B . The proof will be completed if we can show π is injective, because in that case it is isometric. Suppose T ∈ D satisfies µ ( T ) = 0 and take a sequence { T n } n ≥ ∈ F B ⊂ D approximating T. Then by Theorem 3.29 and Lemma 3.32, for all x ∈ E B we have k T x k B e = lim n k T n x k B e ≤ lim sup n k| T n x iik = lim sup n khh T n x, T n x ii B k / ≤ lim sup n k T n x k E r B = lim sup n k π ( T n ) x k E r B = k π ( T ) x k E r B = 0 . This shows T ∈ M ( B e ) vanishes in the dense set E B ⊂ B e , thus T = 0 and π isinjective. (cid:3) The results presented above for bra-ket operators are generalizations, to Fellbundles, of those presented in [11, 12]. In a forthcoming article we will prove animprimitivity theorem for exotic crossed sectional C*-algebras of Fell bundles usingthe exotic fixed point algebras constructed in this article.
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