An equivariant PPV theorem and Paschke-Higson duality
aa r X i v : . [ m a t h . K T ] J a n AN EQUIVARIANT PPV THEOREM ANDPASCHKE-HIGSON DUALITY
MOULAY-TAHAR BENAMEUR AND INDRAVA ROY
Abstract.
We prove an equivariant localized and norm-controlled version of the Pimsner-Popa-Voiculescutheorem. As an application, we deduce a proof of the Paschke-Higson duality for transformation groupoids.
Contents
Introduction 11. Statement of the main theorems 31.1. An extended PPV theorem 31.2. Equivariant Paschke-Higson duality 52. Proof of the extended PPV theorem 62.1. The support-localized PPV theorem 62.2. The general case 143. Application to equivariant Paschke duality 17Appendix A. Localized operators on uniformly proper Γ-spaces 19Appendix B. The norm-controlled PPV theorem 21References 25
Introduction
The classical Paschke-Higson duality expresses the K -homology of a compact space Y as the K -theoryof some dual C ∗ -algebra Q ( Y, H ) which can be taken to be the commutant of C ( Y ) in the Calkin alge-bra of any ample representation of C ( Y ) in a separable Hilbert space H , see [P:81, Hig:95, HPR:97] orthe more recent book [HR:00]. This duality, or more precisly its equivariant version including a properand cocompact action of a countable discrete group, plays a significant part in the study of secondaryinvariants of Dirac-type operators. It allowed for instance Higson and Roe to express the classical Baum-Connes map [BC:00] as a boundary map in K -theory associated with an “elementarily” defined short exactsequence of C ∗ -algebras. They could thereby define a K -theory receptacle for some secondary eta invari-ants. Hence, one can understand the Paschke-Higson duality as a bridge between equivariant K -homologyand the K -theory of appropriate coarse algebras associated with proper, co-compact group actions on non-compact spaces. It allows to conceptualize the coarse geometric approach so as to extend the Baum-Connesframework even to the non-cocompact setting, and yields to the reformulation and generalization of manyclassical results, see for instance [Roe:16]. Further development of the coarse geometric approach enabledas well Higson and Roe to give in [HR:10] an elegant proof of the Keswani vanishing result of reducedeta invariants in the presence of positive scalar curvature metrics. Other results were obtained follow-ing the same line of ideas, in relation with the Novikov and Gromov-Lawson conjectures, see for instance[HR1:05, HR2:05, HR3:05, BM:15, BR:15, BEKW:18, PS:13, XY:14, Z:19, Zen:16] as well as some slightlydifferent approaches in [HPS:15, STY:02, Yu1:97, Yu2:00]. Recall that the original Paschke-Higson duality relied on the so-called Voiculescu theorem [V:76] whichallowed to canonically identify the dual algebras Q ( Y, H ) as far as the representations H are chosen to beample and hence to avoid some set theory complications. The Voiculescu theorem is indeed a precise andindependently important statement which provides a crucial step in the proof of this duality, and althoughit embodies a large class of C ∗ -algebras, the commutative case was more important in the Paschke-Higsonduality used for the Higson-Roe exact sequence in [BR1:20, BR2:20]. It is worthpointing out, from a his-torical perspective, that the Voiculescu theorem answered then as byproducts some then open questions inoperator theory [H:70], and also implied, and in fact was motivated by a noncommutative version of, theclassical Brown-Douglas-Fillmore theorem about the existence of the trivial element of the Ext( Y ) group[BDF:73]. This latter theorem was in turn itself a far-reaching generalization of the classical Weyl-von Neu-mann classification theorem for self-adjoint, and even normal, operators [W:09, vN:35, B:71].The first goal of the present paper was to provide a rigorous proof of an equivariant family version ofthe Paschke-Higson duality theorem, which contains as a special case the equivariant version alluded toabove. Exactly as the (non-equivariant) non-family version was deduced from the Voiculescu theorem, wewere naturally led to an equivariant family version of this Voiculescu theorem. In the non-equivariant case,this family result can be deduced, as we shall see from a classical theorem due to Pimsner-Popa-Voiculescu(PPV) [PPV:79], see also [Ka:81]. Indeed, in the PPV work, the “covariant” variable is a commutative,unital C ∗ -algebra while the “contravariant” variable is a noncommutative (unital) C ∗ -algebra as in the workof Voiculescu, and a bivariant Ext theory was then proposed and expanded later on in relation with bivari-ant K -theory. The PPV results can hence also be understood as extensions of the fundamental Weyl-vonNeumann theorem in operator theory as explained in [PPV:79].In order to prove the equivariant version of the family Paschke-Higson duality, one needs to work withRoe algebras and to keep track of the finite propagation properties of the intertwining unitaries appearingin the PPV work. More precisely, we have added here the action of a countable discrete group and we showthat one can indeed ensure the localization of the support of the involved intertwining unitaries. In thecocompact and metric case, we thus obtain the finite propagation needed for this duality to hold. Moreover,we also state a norm-controlled version ensuring estimates of the involved defect compact operators, whichwere crucial for the operator theory applications in the original work of Voiculescu. Again the commutativecase is all we really needed for the family Paschke-Higson theorem in [BR2:20] but we believe that the moregeneral version given here will have a wider field of applications.In summary, our strategy here is to prove an equivariant, and since this is needed in the non-compact case,support-localized version of the PPV theorem, ensuring in the cocompact case the needed finite propagationof the intertwining unitaries. The equivariant family Paschke-Higson theorem is then deduced as a byprod-uct by some standard arguments. The main application of this Paschke-Higson theorem is to the deductionof a Higson-Roe exact sequence incorporating now the Baum-Connes map for the transformation groupoid X ⋊ Γ and hence yielding to potential applications with generalized eta invariants [BP:09], this was themain result proved in the companion paper [BR2:20] which relied on the previous paper [BR1:20]. When X is reduced to the point, our Paschke-Higson duality reduces to the classical one, which was the startingpoint for proving many invariance properties of reduced eta invariants [HR:10, BR:15]. When the groupis trivial, all involved spaces are compact and we are in the context of the PPV work, but notice that thePaschke result that we obtain then, is essentially already stated in a different form by Connes and Skandalisin [CS:84]. Our equivariant family version of the Paschke-Higson duality will certainly gain importance inthe generalization of eta invariants for laminations as constructed via suspensions, such as some principalsolenoidal tori [CC:00] especially used in [BM:20].In order to prove our main Theorem 1.3 below, some new ideas were needed. Recall that the original PPVconstruction yields intertwining unitaries between fiberwise ample representations, but does not address the QUIVARIANT PPV AND PASCHKE DUALITY 3 support localization property (finite propagation in the cocompact case) of these unitaries, a crucial condi-tion to be able to deduce the Paschke-Higson theorem. The equivariance property on the other hand neededus to proceed with some standard averaging procedures, but a PPV-unitary which does not have appropriatelocalized support produces an operator which is a priori only well-defined in the strong topology (fibrewise)and violates the desired intertwining up to compacts property, and this technical point was precisely missingin the literature.We finally point out that Kasparov studied in the early eighties representations of unital, nuclear C ∗ -algebras on Hilbert C ∗ -modules, and gave a Voiculescu theorem in this more general setting which playedthen a crucial role in establishing his powerful KK -theory, see [Ka:80, Ka:81]. Another important contri-bution to the Paschke duality is the non-commutative version proved by Valette in [V:83] and which alsoallows, modulo the identification of the involved bivariant Ext groups with corresponding Kasparov’ bivari-ant groups, to deduce a non-equivariant family version of the Paschke-Higson duality. As explained above,we have chosen to extend the original approach of Voiculescu and Pimsner-Popa-Voiculescu so as to get aresult which is independent of the Paschke-Higson duality but which implies it. Acknowledgements.
The authors wish to thank A. Carey, T. Fack, J. Heitsch, N. Higson, H. Oyono-Oyono,V. Nistor, S. Paycha, M. Puschnigg, A. Rennie, G. Skandalis and A. Valette for many helpful discussions.MB thanks the French National Research Agency for the financial support via the ANR-14-CE25-0012-01 (SINGSTAR). IR thanks the Homi Bhabha National Institute and the Indian Science and EngineeringResearch Board via MATRICS project MTR/2017/000835 for support.1.
Statement of the main theorems
All the spaces considered in the present paper are assumed second-countable. We devote this preliminarysection to the detailed statement of the main results. Let X be a compact metrizable space of finite dimension,and let Γ be a discrete infinite countable group acting by homeomorphisms on X . Consider a separable Γ-algebra A which is a Γ-proper C ∗ -algebra over a locally compact Hausdorff space Z . Recall that this meansthat Z is a proper Γ-space in the usual sense and that there exists a Γ-equivariant morphism C ( Z ) → ZM ( A )from C ( Z ) to the center ZM ( A ) of the multiplier algebra M ( A ) of A such that C ( Z ) A is dense in A . Wedenote for simplicity by f a ∈ A the resulting action of f ∈ C ( Z ) on a ∈ A . The first example of suchalgebra A is C ( Z ) itself but given such A for any extra separable Γ-algebra B , the Γ-algebra A ⊗ B isthen again Γ-proper over Z . Since we are mainly interested in examples like C ( Z, B ) where B is a givenseparable unital Γ-algebra, we shall always assume that C ( Z ) maps to the center ZA of the C ∗ -algebra A itself. Notice that this can be ensured by replacing A by A + C ( Z ) where C ( Z ) is meant as its range inthe multiplier algebra M ( A ).We shall need furthermore that, when the Γ-space Z is cocompact (say Z/ Γ is compact) then the space Z is endowed with a chosen proper Γ-invariant distance d , so that its closed balls are compact subspaces of Z .The diagonal action of Γ on X × Z then endows X × Z with a proper action. Let G denote the transformationgroupoid X ⋊ Γ. If a Hilbert space H is endowed with a unitary action of Γ, then a given C ( X )-representation b π : C ( X, A ) → L C ( X ) ( C ( X ) ⊗ H ) is a G -equivariant representation if the corresponding field ( π x ) x ∈ X ofrepresentations of A is Γ-equivariant. The same comment applies to a G -operator from L C ( X ) ( C ( X ) ⊗ H )which then corresponds to a Γ-equivariant ∗ -strongly continuous field of operators in H .1.1. An extended PPV theorem.
We fix the proper Γ-algebra A over Z as above. Recall again thatwe have assumed that C ( Z ) maps inside the center ZA of A . The C ∗ -algebra C ( X, A ) of continuousfunctions from X to A is naturally equipped with a C ( X )-algebra structure and the action of Γ endows itwith the structure of a G -algebra, see [LeGall:99, BR1:20]. Suppose that E is a countably-generated Hilbert C ( X )-module. We shall denote abusively by L C ( X ) ( E ) the C ∗ -algebra of adjointable operators in E and by K C ( X ) ( E ) its ideal of C ( X )-compact operators [Ka:80]. M-T. BENAMEUR AND I. ROY
A given representation b π : C ( X, A ) → L C ( X ) ( E ) is called a C ( X )-representation if the action of C ( X )on C ( X, A ) is compatible with the right C ( X )-module structure on E . Such a C ( X )-representation thencorresponds to a ∗ -homomorphism π : A → L C ( X ) ( E ), which in turn corresponds to a field of representations π x : A → L ( E x ), where E x := E ⊗ ev x C is the Hilbert space fibre over x associated with the Hilbert module E . Recall that the field ( E x ) x ∈ X is then a continuous field of Hilbert spaces in the sense of [Dix:77]. Onlythe C ( X )-algebra C ( X, A ) will be needed in the present paper, meaning a constant field, and we shallalways use in the sequel this notation of adding a hat for the C ( X )-representation of C ( X, A ) associatedwith a given ∗ -homomorphism from A to L C ( X ) ( E ). We have chosen to state our results in this language of C ( X )-representations for the sake of possible generalizations, see [BR1:20, BR2:20].Once such representation π is fixed and Z is metric-proper as above, an operator T ∈ L C ( X ) ( E ) will besaid to have finite propagation ≤ R (with respect to π ) if π ( a ) T π ( a ) = 0 for any a , a ∈ A such that d (Supp( a ) , Supp( a )) > R. Recall that the support Supp( a ) of an element a ∈ A is the complement of the largest open subspace U of Z such that f a = 0 for any f ∈ C ( U ). When Z is not necessarily a proper-metric space, the supportSupp( T ) of the operator T ∈ L C ( X ) ( E ) itself with respect to the representation π can still be defined as thecomplement in Z × Z of the union of all open sets of the form U × V , where U and V are open in Z , suchthat π ( a ) T π ( a ) = 0 for any a ∈ C ( U ) A and a ∈ C ( V ) A . We denote from now on for k ≥ χ ∈ C ( Z ) for the proper Γ-action on Z with W χ := { χ = 0 } , by Γ ( k ) χ thesubset of Γ given byΓ ( k ) χ := { ( g, g ′ ) ∈ Γ |∃ ( g i ) ≤ i ≤ k − such that g i W χ ∩ g i +1 W χ = ∅ and g = g, g k = g ′ } . For k = 0, we simply take for Γ (0) χ the diagonal in Γ which is isomorphic to Γ. We point out that when Z/ Γis compact, the first and second projections Γ ( k ) χ → Γ are proper. This is obviously equivalent to the samerequirement for k = 1. Definition 1.1.
We shall say that the proper Γ -action on Z is uniformly proper if we can find such a cutofffunction χ so that the first (and/or second) projection Γ (1) χ → Γ is proper. Notice that if Γ is finite then any action of Γ on Z is uniformly proper. In general if Γ i acts uniformlyproperly on Z i for i = 1 , × Γ on Z × Z will automatically be uniformlyproper. Examples of proper non uniformly proper actions can be found in the literature, they are given byfinitely generated infinite torsion groups, see for instance [Osin:16].In the present paper, when the action is uniformly proper with a chosen adapted continuous cutoff function χ as above, a given operator T will have localized support if there exists an integer k ≥ T is contained in the closure of [ ( g,g ′ ) ∈ Γ ( k ) χ gW χ × g ′ W χ . As explained in Appendix A, this is equivalent to the existence of k ′ ≥ T iscontained in S ( g,g ′ ) ∈ Γ ( k ′ ) χ gW χ × g ′ W χ . The propagation index of T is then the least such k ′ .When the action of Γ is cocompact and Z is endowed with a Γ-invariant metric which endows it with thetopology of a proper-metric space, it is easy to see that localized operators coincide with finite propagationoperators, see again Appendix A.Let us recall now the notion of fibrewise ample representation, see [PPV:79, BR2:20]. Definition 1.2. A C ( X ) -representation b π : C ( X, A ) → L C ( X ) ( E ) will be called a fibrewise ample representa-tion if for any x ∈ X , the representation π x : A → L ( E x ) is ample, i.e. for any x ∈ X , π x is non-degenerateand one has for a ∈ A : π x ( a ) ∈ K ( E x ) = ⇒ a = 0 . Here and as usual K ( E x ) denotes the elementary C ∗ -algebra of compact operators on the Hilbert space E x . QUIVARIANT PPV AND PASCHKE DUALITY 5
Given a Hilbert space unitary representation U : Γ → U ( H ), we denote as usual by H ∞ the Hilbert space H ⊗ ℓ N endowed with the unitary representation U ⊗ id ℓ N . Unless otherwise specified, the Hilbert space ℓ Γ will be endowed with the right regular representation of Γ, so ℓ Γ ∞ is endowed with the correspondingrepresentation. Our extended PPV theorem can be stated as follows: Theorem 1.3.
Assume that the action of Γ on Z is uniformly proper and choose an adapted cutoff func-tion χ . Let H and H be two infinite-dimensional separable complex Hilbert spaces, endowed with unitaryrepresentations of Γ . Let b π and b π be as above two fiberwise ample Γ -equivariant C ( X ) -representations of C ( X, A ) in the Hilbert Γ -modules C ( X ) ⊗ H and C ( X ) ⊗ H respectively. Then, identifying each b π i withthe trivially extended representation (cid:18) b π i
00 0 (cid:19) that is further tensored by the identity of ℓ Γ ∞ there existsa sequence { W n } n ∈ N of Γ -invariant unitary operators W n ∈ L C ( X ) (cid:0) [( H ⊕ H ) ⊗ ℓ Γ ∞ ] ⊗ C ( X ) , [( H ⊕ H ) ⊗ ℓ Γ ∞ ] ⊗ C ( X ) (cid:1) , such that W ∗ n b π ( ϕ ) W n − b π ( ϕ ) is compact, and lim n →∞ || W ∗ n b π ( ϕ ) W n − b π ( ϕ ) || = 0 . Moreover, we can ensure that the operators W n are localized with uniform propagation index, actually ≤ .In particular, if Z is proper-metric such that Z/ Γ is compact then we can always ensure that the unitaries W n have (uniform) finite propagation. In the next section, Theorem 1.3 is first partially proved, more precisely we prove the weaker versionstated as Theorem 2.1, which only constructs one unitary W with the allowed properties. It is only later onin Subsection 2.2 that the construction of the sequence ( W n ) n is carried out with the norm-control. In thesequel, an isometry (resp. unitary) S satisfying the (up to compact operators) intertwining property (2.2)will be referred to as a PPV-isometry (resp. PPV-unitary).1.2. Equivariant Paschke-Higson duality.
As an important application, we deduce the Paschke-Higsonduality isomorphism for Γ-families. More precisely, we assume now and for simplicity that the action of Γon Z is cocompact and that Z is a proper-metric space with a chosen Γ-invariant distance. Notice thoughthat the general case can be treated similarly using the generalized Roe algebras replacing finite propagationby localized operators under the assumption of uniform properness of the Γ-action, see Remark A.4. In[BR2:20], we defined in the cocompact case a generalization of the classical equivariant Roe algebras ofpseudolocal and locally compact operators associated with a fiberwise ample representation of C ( X, A ) onthe Hilbert C ( X )-module ( ℓ Γ ∞ ⊗ H ) ⊗ C ( X ) induced by a given ample representation of A in a fixed H . The Roe algebra of pseudolocal operators is denoted by D ∗ Γ ( X, A ; ( ℓ Γ ∞ ⊗ H )), and the Roe algebra oflocally compact operators is denoted C ∗ Γ ( X, A ; ( ℓ Γ ∞ ⊗ H )). More precisely, D ∗ Γ ( X, A ; ( ℓ Γ ∞ ⊗ H )) is bydefinition the norm closure in L C ( X ) (cid:0) C ( X ) ⊗ ( ℓ Γ ∞ ⊗ H ) (cid:1) of the space of Γ-invariant operators with finitepropagation and whose commutators with the elements of C ( X, A ) are compact operators. The C ∗ -algebra C ∗ Γ ( X, A ; ( ℓ Γ ∞ ⊗ H )) is on the other hand its subspace which is composed of those operators which satisfythe additional condition that their composition with the elements of C ( X, A ) are already compact operators.An obvious observation is that C ∗ Γ ( X, A ; ( ℓ Γ ∞ ⊗ H )) is a 2-sided closed ideal in the unital C ∗ -algebra D ∗ Γ ( X, A ; ( ℓ Γ ∞ ⊗ H )). Our Paschke-Higson duality theorem identifies the K -theory of the quotient Roealgebra Q ∗ Γ ( X, A ; (
Z, ℓ Γ ∞ ⊗ L Z )) with the Γ-equivariant KK -theory of the pair of Γ-algebras ( A, C ( X )).More precisely: Theorem 1.4.
Suppose again that the isometric action of Γ on Z is proper and cocompact. Then we havea group isomorphism P ∗ : K ∗ ( Q ∗ Γ ( X, A ; (
Z, ℓ Γ ∞ ⊗ H ))) ∼ = −→ KK Γ ∗ +1 ( A, C ( X )) , ∗ = 0 , . M-T. BENAMEUR AND I. ROY
Notice that Z does not appear in the RHS, only its existence is supposed so that the LHS does not dependon the choice of such Z . The fact that the ample representation does not appear in the RHS is standard dueto our PPV theorem. In the case of trivial Γ, this theorem is well known, see for instance [Hig:95, V:83].An interesting case corresponds to the case A = C ( Z ). Then we get using the notations from [BR2:20] thefollowing theorem which was fully used there to deduce the Higson-Roe sequence for the groupoid G = X ⋊ Γ: Theorem 1.5.
Suppose again that the isometric action of Γ on Z is proper and cocompact. Then we havea group isomorphism P ∗ : K ∗ ( Q ∗ Γ ( X ; ( Z, ℓ Γ ∞ ⊗ H ))) ∼ = −→ KK ∗ +1Γ ( Z, X ) , ∗ = 0 , . The proof of our PPV theorem as well as the deduction of the Paschke-Higson isomorphism, are carriedout in the next sections. 2.
Proof of the extended PPV theorem
We devote this section to the proof of our G -equivariant, norm-controlled and support-localized, versionof the PPV theorem, say Theorem 1.3. Inorder to simplify the reading of the this technical proof, we havefirst given the proof of a weaker version which does not adress the norm-control question.2.1. The support-localized PPV theorem.
We first forget the norm-control and prove the followingweaker version of Theorem 1.3.
Theorem 2.1 (Extended PPV theorem) . Assume that the action of Γ on Z is uniformly proper with achosen adapted cutoff function χ . Let H and H be two infinite-dimensional separable complex Hilbert spaces,endowed with unitary representations of Γ . Let b π and b π be as above two fiberwise ample Γ -equivariant C ( X ) -representations of C ( X, A ) in the Hilbert Γ -modules C ( X ) ⊗ H and C ( X ) ⊗ H respectively. Then, identifyingeach b π i with the trivially extended representation (cid:18) b π i
00 0 (cid:19) that is further tensored by the identity of ℓ Γ ∞ ,there exists a Γ -invariant unitary operator W ∈ L C ( X ) (cid:0) [( H ⊕ H ) ⊗ ℓ Γ ∞ ] ⊗ C ( X ) , [( H ⊕ H ) ⊗ ℓ Γ ∞ ] ⊗ C ( X ) (cid:1) , which essentially intertwines the extended representations, i.e. such that W ∗ b π ( ϕ ) W − b π ( ϕ ) ∈ K C ( X ) (cid:0) [( H ⊕ H ) ⊗ ℓ Γ ∞ ] ⊗ C ( X ) (cid:1) , for all ϕ ∈ C ( X, A ) . Moreover, we can ensure that the operator W is localized. In particular, if the proper Γ -space Z is cocompactthen we can ensure that the unitary W has finite propagation. Under the assumption that A = C ( Z ) for a proper and cocompact Γ-space Z , a striking application ofTheorem 2.1 is to the equivariant family Paschke-Higson duality Theorem 1.4, as stated in Section 3 andwhich allows to incorporate the Baum-Connes map for the groupoid G = X ⋊ Γ in a long six-term exactsequence, see [BR2:20]. Notice that if Γ is a finite group then any separable Γ-algebra is a proper Γ-algebraover the trivial space Z = { ⋆ } , and the theorem is valid for any such Γ-algebra. This is well known, see[PPV:79] for trivial Γ and unital A , and [Ka:81] for the general case of compact group actions. Forgettingfirst the Γ-invariance of the intertwining unitary, we shall first prove the following independent result: Theorem 2.2.
Under the assumption and notations of Theorem 2.1 but for any proper (not necessarilyuniformly proper) Γ -action on Z , there exists a unitary U ∈ L C ( X ) (cid:0) [ ℓ Γ ∞ ⊗ ( H ⊕ H )] ⊗ C ( X ) , [ ℓ Γ ∞ ⊗ ( H ⊕ H )] ⊗ C ( X ) (cid:1) such that (2.1) U ∗ b π ∞ ( ϕ ) U − b π ∞ ( ϕ ) ∈ K C ( X ) (cid:0) [ ℓ Γ ∞ ⊗ ( H ⊕ H )] ⊗ C ( X ) (cid:1) for all ϕ ∈ C ( X, A ) . Moreover, we can ensure that the operator the operator U is localized. In particular, if the proper Γ -space Z is cocompact, then we can ensure that U has finite propagation. QUIVARIANT PPV AND PASCHKE DUALITY 7
We thus fix two fiberwise ample G -equivariant representations b π and b π of C ( X, A ) in the Hilbert G -modules H ⊗ C ( X ) and H ⊗ C ( X ) respectively. In the sequel, we shall denote for x ∈ X , by q x thecomposite map q x : L C ( X ) ( H i ⊗ C ( X )) d x −→ L ( H i ) pr −→ Q ( H i ) , where d x is evaluation at x while the map L ( H i ) pr −→ Q ( H i ) is the quotient projection onto the Calkin algebra Q ( H i ) = L ( H i ) / K ( H i ). We begin with the construction of a finite-propagation PPV- isometry . Lemma 2.3.
Under the assumptions of Theorem 2.2, there exists an isometry ˆ S ∈ L C ( X ) (( H ⊕ H ) ⊗ C ( X ) , ( H ⊕ H ) ⊗ ℓ Γ ⊗ C ( X )) such that ˆ S ∗ (( b π ( f ) ⊗ id ℓ Γ ) ⊕
0) ˆ S − ( b π ( f ) ⊕ ∈ K C ( X ) (( H ⊕ H ) ⊗ C ( X )) , ∀ f ∈ C ( X × Z ) Moreover, we can ensure that the operator ˆ S is localized. In particular, when Z/ Γ is compact, we can ensurethat ˆ S has finite propagation.Proof. When Γ is a finite group, this result is well known, see for instance [Ka:81], and we give the proofunder the assumption that Γ is infinite. We shall sometimes identify C ( Z ) with its range in the center of themultiplier algebra A when no confusion can occur. Fix a cutoff function χ ∈ C ( Z ) for the proper Γ-action on Z . The quotient projection Z → Z/ Γ then restricts into a proper map Supp( χ ) → Z/ Γ. Denote by V χ theinterior of Supp( χ ) and set A χ := C ( V χ ) A + which is a two-sided self-adjoint ideal in A by our assumptionthat C ( Z ) maps to the center ZA of A . Recall that the ∗ -homomorphism π i : A → L C ( X ) ( H i ⊗ C ( X )) isnon-degenerate and extends to a unital ∗ -homomorphism, still denoted π i , from M ( A ) to L C ( X ) ( H i ⊗ C ( X )).The proof of Lemma 2.3 will be split into 3 steps. Step 1 (Apply PPV) : Consider the Hilbert submodules E iχ := π i ( A χ )( H i ⊗ C ( X )) for i = 1 , Claim : The ∗ -homomorphism π i preserves E iχ and its restriction to A χ (acting on E iχ ) is denoted π χi .This is again associated with a fibrewise ample representation b π χi .To check this, note that the field of Hilbert spaces associated to the Hilbert module E iχ is given by ( R i,x :=[ π i,x ( A χ ) H i ]) x ∈ X , where ( π i,x ) x ∈ X is the field of representations associated with the ∗ -homomorphism π i : A → L C ( X ) ( H i ⊗ C ( X )). Then, the field of Hilbert space representations associated with π χi is given at x ∈ X by the restriction of the representations π i,x and denoted π χi,x : A χ → L ( R i,x ). For each x ∈ X , therestricted representations π χi,x are also clearly ample.The C ( X )-module E iχ is countably generated and we may assume that it is an orthocomplemented sub-module of H i ⊗ C ( X ), respectively for i = 1 ,
2. The ∗ -homomorphism π χi will be extended by zero on theorthocomplement, this corresponds to extending each π χi,x by zero on the orthogonal Hilbert subspace of R i,x . The corresponding extended representation b π χi ⊕ C ( X, A χ ) then satisfies the following properties[PPV:79]:(1) it is lower semi-continuous, i.e. for any convergent sequence x n → x in X , we have T n I x n ⊆ I x where I x := ker( q x ◦ ( b π χi ⊕ ⊆ A χ ;(2) it is exact, i.e. ∩ x ∈ X I x = { } , and(3) it is trivial, i.e. ker( q x ◦ ( b π χi ⊕ d x ◦ ( b π χi ⊕ b π iχ is fibrewise ample. To check (1), just notice that d x ◦ ( π χi ( a ) ⊕
0) = ( π χi,x ⊕ a ) , and that each π i,x is ample here. A similar but easier argument can be used to prove (2).The C ( X )-representation b π χi ⊕ ∗ -homomorphism π χi ⊕ A χ → L C ( X ) ( H i ⊗ C ( X )). Inorder to apply the main PPV theorem about trivial X -extensions , we check now the same properties for theunique extension of π χi ⊕ C ∗ -algebra unitalization A χ ⊕ C of A χ . In terms of C ( X )-representations,we thus obtain the extended representation to C ( X, A χ ) ⊕ C ( X ) given by( b π χi ⊕ + ( f ⊕ λ ) := ( b π χi ⊕ f ) + ρ ( λ ) for f ∈ C ( X, A χ ) and λ ∈ C ( X ) . M-T. BENAMEUR AND I. ROY
Here λ acts on the Hilbert C ( X )-module H i ⊗ C ( X ) by the adjointable operator ρ ( λ ) corresponding to theright module multiplication.Since Γ is infinite all the Hilbert spaces R ⊥ i,x are infinite dimensional separable Hilbert spaces, we mayuse the Kasparov stabilisation theorem to replace ( E iχ ) ⊥ by the standard infinite dimensional countablygenerated Hilbert C ( X )-module H i ⊗ C ( X ), so as to be able to apply the PPV theorem, see [Ka:81].Hence the verification of the three properties for ( b π χi ⊕ + is obvious. If for instance ( f, λ ) ∈ ker( q x ◦ ( b π χi ⊕ + ), then λ = 0 since ( E iχ ) ⊥ x is infinite-dimensional. Thus, we again get π χi,x ( f ( x )) ∈ K ( R i,x ) = ⇒ f ( x ) = 0,so ( f, λ ) ∈ ker( d x ◦ ( b π χi ⊕ + ). Therefore, we get the triviality property. Lower-semicontinuity and exactnessare proved similarly and are left as an exercise.Therefore, the representations ( b π χi ⊕ + are essentially unitarily equivalent by the PPV theorem [PPV:79],i.e. there exists a unitary S χ ∈ L C ( X ) ( H ⊗ C ( X ) , H ⊗ C ( X )) such that we have in particular for any f ∈ C ( X, A χ ): S ∗ χ (cid:20)b π χ ( f ) 00 0 (cid:21) S χ − (cid:20)b π χ ( f ) 00 0 (cid:21) ∈ K C ( X ) ( H ⊗ C ( X )) . Notice that we also have the same relation for f ∈ C ( X × V χ ) viewed in C ( X, A χ ), due to our assumptionthat C ( Z ) ⊂ ZA . Step 2 (
First modification ): Consider the operator s χ ∈ L ( E χ , E χ ) which is the (1 , S χ : E χ ⊕ ( E χ ) ⊥ → E χ ⊕ ( E χ ) ⊥ . It satisfies the following properties for any f ∈ C ( X, A χ ):(1) s ∗ χ b π χ ( f ) s χ − b π χ ( f ) ∼ b π χ ( f )( s χ s ∗ χ − id) ∼ s ∗ χ s χ − id) b π χ ( f ) ∼ s χ s ∗ χ , b π χ ( f )] ∼ s ∗ χ s χ , b π χ ( f )] ∼ − s ∗ χ s χ ) and (1 − s χ s ∗ χ ) are positive operators.Therefore we can form the unitary ˆ s χ : E χ ⊕ E χ → E χ ⊕ E χ given by the matrixˆ s χ := (cid:20) s χ (1 − s χ s ∗ χ ) / − (1 − s ∗ χ s χ ) / s ∗ χ (cid:21) Properties (1) , (2) and (3) above, imply that ˆ s χ intertwines the representations b π χ ⊕ b π χ and b π χ ⊕ b π χ up tocompacts on E χ ⊕ E χ and E χ ⊕ E χ , respectively. Extending the unitary ˆ s χ by zero, we get a partial isometryin L C ( X ) (( H ⊕ H ) ⊗ C ( X ) , ( H ⊕ H ) ⊗ C ( X )), that we still denote by ˆ s χ , given by: s χ p − s χ s ∗ χ
00 0 0 0 − p − s ∗ χ s χ s ∗ χ
00 0 0 0 where we have written( H ⊕ H ) ⊗ C ( X ) = [ E χ ⊕ ( E χ ) ⊥ ] ⊕ [ E χ ⊕ ( E χ ) ⊥ ] , and similarly for ( H ⊕ H ) ⊗ C ( X ) . Step 3 (
Second modification) : For ( x, g ) ∈ G , we denote by V i ( x,g ) the unitary implementing the G -action on H i ⊗ C ( X ). Then we define an operator ˆ S ∈ L C ( X ) (( H ⊕ H ) ⊗ C ( X ) , ℓ Γ ⊗ ( H ⊕ H ) ⊗ C ( X ))by setting the following pointwise formula: h ˆ S x i g := (cid:16) V x,g ) ⊕ V x,g ) (cid:17) ˆ s χ (cid:16)b π ( χ / ) ⊕ b π ( χ / ) (cid:17) (cid:16) V x,g ) ⊕ V x,g ) (cid:17) − . The operator ˆ S then satisfies the allowed properties in the statement Lemma 2.3, as we prove it in Lemma2.4 below. Therefore, the proof of Lemma 2.3 is now complete. (cid:3) Lemma 2.4.
The operator ˆ S satisfies the following properties:(1) ˆ S is an isometry. QUIVARIANT PPV AND PASCHKE DUALITY 9 (2) ˆ S intertwines b π ⊕ and (id ℓ Γ ⊗ b π ) ⊕ up to compacts.(3) The support of ˆ S is contained in the closure of A = S g ∈ Γ gW χ × gW χ and hence has propaga-tion index ≤ . In particular, if Z/ Γ is compact then ˆ S has finite propagation (bounded above by diam Z ( V χ ) ).Proof. (1) Since ˆ s χ is an isometry in restriction to the range of b π ( χ / ) ⊕ b π ( χ / )), a straightforwardverification using the relation P g ∈ Γ g ∗ χ = 1 Z shows that ˆ S is an isometry.(2) It suffices to check this condition for elements f in C ( X, A c ) where A c := C c ( Z ) A ⊂ A . Then, usingthe previously listed properties of ˆ s χ , we have:[ ˆ S ∗ (id ℓ Γ ⊗ b π ( f ) ⊕
0) ˆ S ] x = X g ∈ Γ ( V x,g ) ⊕ V x,g ) )( π ( χ / ) ⊕ π ( χ / ))ˆ s ∗ χ (( V ) − x,g ) ⊕ ( V ) − x,g ) )( b π ( f ) ⊕ V x,g ) ⊕ V x,g ) )ˆ s χ ( π ( χ / ) ⊕ π ( χ / ))(( V ) − x,g ) ⊕ ( V ) − x,g ) )= X g ∈ Γ ( V x,g ) ⊕ V x,g ) )( π ( χ / ) ⊕ π ( χ / ))(ˆ s ∗ χ ( b π ( g ∗ f ) ⊕ s χ )( π ( χ / ) ⊕ π ( χ / ))(( V ) − x,g ) ⊕ ( V ) − x,g ) ) ∼ X g ∈ Γ ( V x,g ) ⊕ V x,g ) )( π ( χ / ) ⊕ π ( χ / ))(ˆ s ∗ χ ( b π ( χ / g ∗ f ) ⊕ s χ (( V ) − x,g ) ⊕ ( V ) − x,g ) )The last equivalence is a consequence of the fact that ˆ s χ commutes up to compacts with π ( χ / ) ⊕ π ( χ / ).Hence we deduce[ ˆ S ∗ (id ℓ Γ ⊗ b π ( f ) ⊕
0) ˆ S ] x ∼ X g ∈ Γ ( V x,g ) ⊕ V x,g ) )( π ( χ / ) ⊕ π ( χ / ))( b π ( χ / g ∗ f ) ⊕ V ) − x,g ) ⊕ ( V ) − x,g ) ) ∼ X g ∈ Γ ( V x,g ) ⊕ V x,g ) )( π ( χ / ) ⊕ π ( χ / ))( b π ( g ∗ f ) ⊕ π ( χ / ) ⊕ π ( χ / ))(( V ) − x,g ) ⊕ ( V ) − x,g ) )= ( b π ( f ) ⊕ X g ∈ Γ ( V x,g ) ⊕ V x,g ) )( π ( χ ) ⊕ π ( χ ))(( V ) − x,g ) ⊕ ) V ) − x,g ) )= ( b π ( f ) ⊕ g ∈ Γ such thatSupp( g ∗ f ) ∩ ( X × Supp( χ )) = ∅ is finite, due to the properness of the Γ-action and the fact that χ is a cut-off function. Indeed, we knowfrom the very definition of χ that for any compact subspace K of Z , the subset { g ∈ Γ | Supp( χ ) ∩ gK = ∅} is finite, see for instance [Tu:99].(3) Assume now that W and W are two open subspaces of Z such that W × W does not intersect anysubspace of Z of the form gW χ × gW χ , where g runs over Γ, then for a i ∈ C ( W i ) A , we can compute h (id ℓ Γ ⊗ π ( a ) ⊕
0) ˆ S ( π ( a ) ⊕ i g = ( π ( a ) ⊕ (cid:16) V x,g ) ⊕ V x,g ) (cid:17) ˆ s χ (cid:16) π ( χ / ) ⊕ π ( χ / ) (cid:17) (cid:16) V x,g ) ⊕ V x,g ) (cid:17) − ( π ( a ) ⊕ (cid:16) V x,g ) ⊕ V x,g ) (cid:17) (cid:0) π ( g − a ) ⊕ (cid:1) ˆ s χ (cid:16) π ( χ / ) ⊕ π ( χ / ) (cid:17) (cid:0) π ( g − a ) ⊕ (cid:1) (cid:16) V x,g ) ⊕ V x,g ) (cid:17) − = (cid:16) V x,g ) ⊕ V x,g ) (cid:17) (cid:0) π ( g − a ) ⊕ (cid:1) ˆ s χ (cid:16) π ( χ / g − a ) ⊕ (cid:17) (cid:16) V x,g ) ⊕ V x,g ) (cid:17) −
10 M-T. BENAMEUR AND I. ROY
Therefore, we see that if for a given g ∈ Γ, we have χ / g − a is non-zero then Supp( a ) ∩ g Supp( χ ) = ∅ .But then by hypothesis we know that since W and W are open we also have W × W \ g Supp( χ ) × g Supp( χ ) = ∅ , and hence necessarily Supp( a ) ∩ g Supp( χ ) = ∅ , say that Supp( g − a ) ∩ Supp( χ ) = ∅ . This in turn impliesthat (cid:0) π ( g − a ) ⊕ (cid:1) ˆ s χ = 0 since the range of ˆ s χ is contained in E χ ⊕ E χ . Therefore we conclude that the operator (id ℓ Γ ⊗ π ( a ) ⊕
0) ˆ S ( π ( a ) ⊕
0) is trivial.If we assume that Z/ Γ is compact and that Z is a metric-proper space with the above properties, thensetting κ := diam Z (Supp( χ )) which is now a finite positive number, we can deduce by the same calculationthat whenever a , a ∈ C c ( Z ) A are such that d (Supp( a ) , Supp( a )) > κ , one has by the Γ-invariance of thedistance d the same relation (id ℓ Γ ⊗ π ( a ) ⊕
0) ˆ S ( π ( a ) ⊕
0) = 0 . (cid:3) Corollary 2.5.
There exists an isometry S ∈ L C ( X ) ( ℓ Γ ∞ ⊗ ( H ⊕ H ) ⊗ C ( X ) , ℓ Γ ∞ ⊗ ( H ⊕ H ) ⊗ C ( X )) such that S ∗ ( b π ∞ ( f ) ⊕ S − ( b π ∞ ( f ) ⊕ ∈ K C ( X ) (cid:0) [ ℓ Γ ∞ ⊗ ( H ⊕ H )] ⊗ C ( X ) (cid:1) . Moreover, we can ensure that the operator S is localized with support contained in the closure of A = S g ∈ Γ gW χ × gW χ and hence with propagation index ≤ . In particular, if Z/ Γ is compact then ˆ S has finitepropagation (bounded above by diam( V χ ) ).Proof. From Lemma 2.3,we deduce an isometry ˆ S ∈ L C ( X ) ( ℓ Γ ∞ ⊗ ( H ⊕ H ) ⊗ C ( X ) , ℓ Γ ⊗ ℓ Γ ∞ ⊗ ( H ⊕ H ) ⊗ C ( X )), such that ˆ S intertwines the representations b π ∞ ⊕ ℓ Γ ⊗ b π ∞ ) ⊕ u ∞ : ℓ Γ ⊗ ℓ Γ ∞ → ℓ Γ ∞ . The isometry S := ( u ∞ ⊗ id ( H ⊕ H ) ⊗ C ( X ) ) ◦ ˆ S thenintertwines b π ∞ ⊕ b π ∞ ⊕
0, up to compacts and still has the same support as ˆ S . In particular, it hasuniform finite propagation when Z/ Γ is compact. (cid:3)
We are now ready to prove Theorem 2.2.
Proof. (of Theorem 2.2) Replacing, in the statement of Corollary 2.5, H i by ℓ N ⊗ H i and b π i by b π ∞ i =id ℓ N ⊗ b π i for i = 1 ,
2, we obtain an isometry with the prescribed support condition (finite-propagation when Z/ Γ is compact and Z is metric-proper) S ∈ L C ( X ) (cid:0) ( ℓ Γ ∞ ) ∞ ⊗ ( H ⊕ H ) ⊗ C ( X ) , ( ℓ Γ ∞ ) ∞ ⊗ ( H ⊕ H ) ⊗ C ( X ) (cid:1) such that for any f ∈ C ( X, A ): S ∗ (( b π ∞ ) ∞ ( f ) ⊕ S − ( b π ∞ ) ∞ ( f ) ⊕ ∈ K C ( X ) (cid:0) ( ℓ Γ ∞ ) ∞ ⊗ ( H ⊕ H ) ⊗ C ( X ) (cid:1) . The support of S is more precisely contained in S g ∈ Γ Supp( gχ ) × Supp( gχ ) and hence S has finite prop-agation in the metric and cocompact case. Indeed, in this case and since the distance is Γ-invariant, thepropagation is ≤ the diameter of Supp( χ ). Let r ∞ : ℓ N ⊗ ℓ N → ℓ N be a unitary. Composing S with r ∞ ⊗ id ℓ Γ ⊗ ( H ⊕ H ) ⊗ C ( X ) , we get an isometry S := (cid:0) r ∞ ⊗ id ℓ Γ ⊗ ( H ⊕ H ) ⊗ C ( X ) (cid:1) ◦ S ∈ L C ( X ) (cid:0) ( ℓ Γ ∞ ) ∞ ⊗ ( H ⊕ H ) ⊗ C ( X ) , ℓ Γ ∞ ⊗ ( H ⊕ H ) ⊗ C ( X ) (cid:1) which satisfies S ∗ ( b π ∞ ( f ) ⊕ S − (( b π ∞ ) ∞ ( f ) ⊕ ∈ K C ( X ) (cid:0) ( ℓ Γ ∞ ) ∞ ⊗ ( H ⊕ H ) ⊗ C ( X ) (cid:1) and has the same support.Consider the operator R : ℓ N ⊗ ℓ Γ ∞ ⊗ ( H ⊕ H ) → ℓ N ⊗ ℓ Γ ∞ ⊗ ( H ⊕ H ) defined by the followingformula: R ( h ⊕ h ⊕ · · · ) = 0 ⊕ h ⊕ h ⊕ · · · QUIVARIANT PPV AND PASCHKE DUALITY 11
Then R induces a C ( X )-linear isometry on ℓ N ⊗ ℓ Γ ∞ ⊗ ( H ⊕ H ) ⊗ C ( X ). Consider also the operator R : ℓ N ⊗ ℓ Γ ∞ ⊗ ( H ⊕ H ) → ℓ Γ ∞ ⊗ ( H ⊕ H ) defined by the formula: R ( h ⊕ h ⊕ · · · ) = h Then R induces a C ( X )-adjointable co-isometry R ∈ L C ( X ) (cid:0) ( ℓ Γ ∞ ) ∞ ⊗ ( H ⊕ H ) ⊗ C ( X ) , ℓ Γ ∞ ⊗ ( H ⊕ H ) ⊗ C ( X ) (cid:1) . Notice that we have the convenient relations R R = 0 and R R ∗ + R ∗ R = id ℓ N ⊗ ℓ Γ ∞ ⊗ ( H ⊕ H ) ⊗ C ( X ) . We are now in position to define the unitary S ∈ L C ( X ) (cid:0) ℓ Γ ∞ ⊗ ( H ⊕ H ) ⊗ C ( X ) , ℓ Γ ∞ ⊗ (( H ⊕ H ) ⊕ ( H ⊕ H )) ⊗ C ( X ) (cid:1) by using the following formula: S := (cid:18) I − S S ∗ + S R ∗ S ∗ R S ∗ (cid:19) . It is a straightforward computation to show that S is a unitary and that it intertwines b π ∞ ⊕ b π ∞ ⊕ ⊕ b π ∞ ⊕ R commutes with the representation id ℓ N ⊗ b π ∞ ⊕
0, and R intertwines (exactly) the representations b π ∞ ⊕ ℓ N ⊗ b π ∞ ) ⊕
0, and therefore have support containedin the diagonal of Z . Whence, the operator S is localized by composition with the propagation index whichis ≤
7. Again in the cocompact case and with the Γ-invariant distance on Z , we see that the operator S hasfinite propagation which is ≤ to the diameter of the compact space ∪ g | g Supp( χ ) ∩ Supp( χ ) = ∅ Supp( gχ ). Indeed,this is a finite union by definition of the cutoff function.A similar unitary T exists between b π ∞ ⊕ b π ∞ ⊕ ⊕ b π ∞ ⊕ α in the target space of these unitaries, which exchanges the first two andthe last two factors (i.e. α is given by ( u , u , u , u ) ( u , u , u , u )), and then taking the composition U = T ∗ α ∗ S , one gets the desired unitary which intertwines b π ∞ ⊕ b π ∞ ⊕ α has support inside the diagonal of Z . In conclusion, the unitary U is also localized with the same propagation index, which is hence ≤
7. As a consequence in the cocompactcase with the Γ-invartiant distance, we conclude again by an easy verification that the unitary U has finitepropagation as desired. (cid:3) Let us now take into account the action of our discrete countable group Γ by homeomorphisms on X .Recall that A is a proper Γ-algebra over Z and that C ( Z ) maps inside the center of A itself. We denoteas before by G the action groupoid X ⋊ Γ or its space of arrows, since no confusion can occur. A specificunitary representation of Γ is the (right) regular representation ρ in the Hilbert space ℓ Γ, which can betensored by the identity of ℓ N to get the unitary representation ρ ∞ of Γ in ℓ Γ ∞ = ℓ Γ ⊗ ℓ N . Recall thatthe Γ-action on Z is uniformly proper if there exists a cutoff function χ ∈ C ( Z ) such that the first (and/orthe second) projection Γ (1) χ → Γ is proper. This is automatically satisfied for any proper cocompact actionbut is an assumption in general. We are now in position to state Theorem 2.1, that we restate using thegroupoid language, so as to fit with possible generalizations, as follows:
Theorem 2.6.
Assume that the action groupoid G = X ⋊ Γ acts uniformly proper on the G -space Y = X × Z ,meaning here that Γ acts uniformly properly on Z with a chosen adapted cutoff function χ ∈ C ( Z ) . Let b π and b π be two fiberwise ample G -equivariant representations of C ( X, A ) in the Hilbert G -modules H ⊗ C ( X ) and H ⊗ C ( X ) respectively. Then, identifying each b π i with the trivially extended representation (cid:18) b π i
00 0 (cid:19) that is further tensored by the identity of ℓ Γ ∞ , there exists a G -invariant unitary operator W ∈ L C ( X ) (cid:0)(cid:2) ℓ Γ ∞ ⊗ ( H ⊕ H ) (cid:3) ⊗ C ( X ) , (cid:2) ℓ Γ ∞ ⊗ ( H ⊕ H ) (cid:3) ⊗ C ( X ) (cid:1) , such that for any ϕ ∈ C ( X, A ) W ∗ b π ( ϕ ) W − b π ( ϕ ) ∈ K C ( X ) (cid:0) [ ℓ Γ ∞ ⊗ ( H ⊕ H )] ⊗ C ( X ) (cid:1) . Moreover, we can ensure that the operator W is localized with the propagation index ≤ . In particular, if Z/ Γ is compact with the previous metric assumption on Z , then we can ensure that W has finite propagation.Proof. Since the extended representations (of the unitalization C ( X, A + )) are fiberwise ample (say homo-geneous in the terminology used in [PPV:79]), by “forgetting” the right regular Γ-action on ℓ Γ ∞ , fromTheorem 2.2 we deduce again the existence of a unitary that we rather denote in this proof by S ( U willdenote below another family of isometries) with support within A (so with finite propagation when Z/ Γ iscompact and Z is metric-proper): S ∈ L C ( X ) (cid:0)(cid:2) ℓ Γ ∞ ⊗ ( H ⊕ H ) (cid:3) ⊗ C ( X ) , (cid:2) ℓ Γ ∞ ⊗ ( H ⊕ H ) (cid:3) ⊗ C ( X ) (cid:1) such that for any f ∈ C ( X, A + ), S ∗ b π ( f ) S − b π ( f ) ∈ K C ( X ) (cid:0) [ ℓ Γ ∞ ⊗ ( H ⊕ H )] ⊗ C ( X ) (cid:1) . In particular this property holds for the restrictions of b π and b π to C ( X, A ).The unitary U obtained in this way is of course a priori not Γ-invariant. To remedy this, we shall use aclassical trick which allows to “average”. Using Fell’s trick, one can construct a family of operators ( U g ) g ∈ Γ acting on ℓ (Γ) ∞ , such that (see for instance [GWY:16] or [BR2:20]): • for g, g ′ ∈ Γ, U ∗ g U g ′ = δ g,g ′ id ℓ (Γ) ∞ , in particular each U g is an isometry; • P g ∈ Γ U g U ∗ g = id ℓ (Γ) ∞ ; and • (Γ-equivariance) U g ′ g = ρ ∞ g U g ′ ρ ∞ g − for any ( g, g ′ ) ∈ Γ .Here of course ρ is the right regular representation of Γ. Recall the cutoff funtion χ ∈ C ( Z ) defined usingthe properness of the Γ-action on Z and which is compactly supported when Z/ Γ is assumed compact. Weproceed now to define the allowed field W x : ( H ⊕ H ) ⊗ ℓ (Γ) ∞ −→ ( H ⊕ H ) ⊗ ℓ (Γ) ∞ or equivalentlythe corresponding operator W obtained by the averaging trick.Consider the dense submodule E ′ of (cid:2) ℓ Γ ∞ ⊗ ( H ⊕ H ) (cid:3) ⊗ C ( X ) which is given by E ′ := ( b π ⊕ b π )( C ( X, A c )) (cid:0)(cid:2) ℓ Γ ∞ ⊗ ( H ⊕ H ) (cid:3) ⊗ C ( X ) (cid:1) In this notation, A c = C c ( Z ) A as before, and b π i is the original representation of C ( X, A ) on C ( X ) ⊗ H i thatwe have tensored with the identity in ℓ Γ ∞ . We similarly define E ′ .Notice that, b π i also denotes the extended representation of C ( X, A ) in (cid:2) ℓ Γ ∞ ⊗ ( H ⊕ H ) (cid:3) ⊗ C ( X ) (resp. (cid:2) ℓ Γ ∞ ⊗ ( H ⊕ H ) (cid:3) ⊗ C ( X )) obtained as b π i ⊕ i = 1 ,
2, respectively. On the other hand, an operator T ∈ L C ( X ) (cid:0)(cid:2) ℓ Γ ∞ ⊗ ( H ⊕ H ) (cid:3) ⊗ C ( X ) , (cid:2) ℓ Γ ∞ ⊗ ( H ⊕ H ) (cid:3) ⊗ C ( X ) (cid:1) is G -invariant if for any g ∈ Γ, we have for ( x, g ) ∈ X × Γ: T x = ( gT ) x := ( V x,g ) ⊕ V x,g ) ) T xg ( V x,g ) ⊕ V x,g ) ) − . Recall that V i denotes the extensions of the X ⋊ Γ-actions on H i ⊗ C ( X ) by tensoring with the right regularrepresentation of Γ, ρ ∞ on ℓ Γ ∞ . So, choosing a cutoff function χ as before with the extra property thatthe first projection Γ (1) χ → Γ is proper, we now replace S by the (well defined) average operator W = X g ∈ Γ (id H ⊕ H ⊗ U g ) ◦ g ( Sπ ( √ χ )) . Here ( U g ) g ∈ Γ is the family of isometric operators on the Hilbert space ℓ (Γ) ∞ defined above. For e ∈ E ′ , wethus have defined W x ( e x ) := X g ∈ Γ (id H ⊕ H ⊗ U g ) (cid:0) V ⊕ V (cid:1) ( x,g ) S xg (cid:0) V ⊕ V (cid:1) ( xg,g − ) π ,x ( g ∗ √ χ )( e x )The sum defining W x is then finite since for any ϕ ∈ C c ( Z ) that is viewed in A , we have( π ,x ⊕ g ∗ √ χ )( π ,x ⊕ π ,x )( ϕ ) = ( π ,x ( ϕ √ g ∗ χ ) ⊕ , QUIVARIANT PPV AND PASCHKE DUALITY 13 and the number of g ∈ Γ such that ϕ √ g ∗ χ = 0 is finite by the properness of the Γ-action on Z . Hence, W ( e ) is well defined on the elements e ∈ E ′ . Moreover, an easy inspection, using the properties of the family( U g ) g ∈ Γ , shows that the relation W ∗ x W x = id holds on E ′ .This shows that W x automatically extends to an isometry between the corresponding Hilbert spaces thatwe still denote W x . Moreover, when e = [ b π ( f ) ⊕ b π ( f )] e with f ∈ C ( X, A c ), there is a finite subset I e ofΓ, which does not depend on the variable x ∈ X , such that, W ( e ) x = X g ∈ I e T g,x ( e x ) ∀ x ∈ X. Here each of the maps x T g,x ( e x ) and x T ∗ g,x ( e x ) is of course norm-continuous. We thus end up withthe adjointable isometry, still denoted W , between the Hilbert modules (cid:2) ( H ⊕ H ) ⊗ ℓ Γ ∞ (cid:3) ⊗ C ( X ) and (cid:2) ( H ⊕ H ) ⊗ ℓ Γ ∞ (cid:3) ⊗ C ( X ) as announced. Notice that in the cocompact case, W has finite propagation byconstruction.Now, W satisfies the following properties:(1) W ∗ b π ( f ) W − b π ( f ) ∈ K C ( X ) (cid:0) [( H ⊕ H ) ⊗ ℓ (Γ) ∞ ] ⊗ C ( X ) (cid:1) , for any f ∈ C ( X, A c );(2) W is G -invariant.Once these properties have been verified, a standard trick as in the proof of Theorem 2.2 using the directsum of the representations allows to find in place of the isometry W , a unitary which will also satisfy thesame two properties. Note (see the notation in the proof of Theorem 2.2) that if the initial isometry S is G -invariant, then the operators S , as well as R and R appearing in the proof of Theorem 2.2 are all G -invariant by construction.Regarding the first item, notice that we have for any f ∈ C ( X, A c ): b π ( f ) W = X g ∈ Γ (id H ⊕ H ⊗ U g ) ◦ g (cid:0)b π ( g − f ) Sπ ( √ χ ) (cid:1) , But an easy inspection of the support of the operator S (which is contained in A ), using that the action isuniformly proper, we deduce that b π ( g − f ) Sπ ( √ χ ) is only non-zero for a finite number of elements g ∈ Γ.The similar statement holds for W b π ( f ). An ad hoc consequence is that the support of W is also containedin A . When Z/ Γ is compact, this is more obvious since the operator S has finite propagation. Hence thesum defining the operator b π ( f ) W is finite independently of the test vector e and therefore makes sense inthe uniform operator topology.Therefore, we may compute using the G -equivariance of the representations b π i : b π ( f ) W = X g ∈ Γ (id H ⊕ H ⊗ U g ) ◦ g (cid:0)b π ( g − f ) Sπ ( √ χ ) (cid:1) ∼ X g ∈ Γ (id H ⊕ H ⊗ U g ) ◦ g (cid:0) S b π ( g − f ) π ( √ χ ) (cid:1) = X g ∈ Γ (id H ⊕ H ⊗ U g ) ◦ g ( Sπ ( √ χ )) ◦ g b π ( g − f )= W b π ( f ) . The sign ∼ again refers to equality modulo the compact operators of the corresponding Hilbert modules andsince the sum is finite, the operator b π ( f ) W − W b π ( f ) is clearly compact. Now, since W ∗ is an adjointableoperator, composing with W ∗ on the left yields to the conclusion.Finally, W was indeed born to be G -invariant. Since the submodule E ′ is a G -submodule, we may prove G -invariance strongly on the vectors of E ′ . Let us denote the G -actions on H ⊕ H by b V := V ⊕ V and similarly by b V := V ⊕ V the G -action on H ⊕ H . We then compute for any ( x, h ) ∈ G : W x b V x,h ) = X g ∈ Γ (id ⊗ U g ) b V x,g ) S xg π ,x ( √ χ ) b V xg,g − ) b V x,h ) = X g ∈ Γ (id ⊗ U g ) b V x,g ) S xg π ,x ( √ χ ) b V xg,g − h ) = X l ∈ Γ (id ⊗ U hl ) b V x,hl ) S xhl π ,x ( √ χ ) b V xhl,l − ) Hence W x b V x,h ) = X l ∈ Γ (id ⊗ U hl ) b V x,h ) b V xh,l ) S xhl π ,x ( √ χ ) ( b V xh,l ) ) − = X l ∈ Γ b V x,h ) (id ⊗ U l ) b V xh,l ) S xhl π ,x ( √ χ ) ( b V xh,l ) ) − = b V x,h ) W xh which is the required right hand side. (cid:3) It is worthpointing out that all the previous theorems apply to the case of A = C ( Z, B ) where B is anyseparable unital Γ-algebra. An already interesting application is when A = C ( Z ) as we shall see in the nextsection.2.2. The general case.
By using an easy generalization of the PPV work, expanded in Appendix B, wenow state the norm-controlled version of our main Theorem 2.1, say Theorem 1.3 which gives the precisegeneralization of results in [V:76], compare also with [Ka:80]. So the goal of this section is to explain how toadapt the proof of the previous section so as to construct the sequence of unitaries of Theorem 1.3. For anoperator T ∈ C ( X, L ( H ) ∗ s ), we shall use the notation T ǫ ∼ T is compact and hasnorm at most a constant multiple of ǫ ; the constant may depend on T . Recall that A is a separable properΓ-algebra over Z .Let Σ be a countable dense subset of the separable C ∗ -algebra C ( X, A ), which is closed under the invo-lution a a ∗ and globally Γ-invariant. Such Σ always exists since we can for instance take the union of theΓ-orbits of a countable dense self-adjoint subset of C ( X, A ). Theorem 2.7. [Controlled version of Theorem 2.1]Under the assumptions of Theorem 2.1, if we fix ǫ > then we can ensure that the Γ -invariant unitaryoperator W = W ǫ ∈ L C ( X ) (cid:0) [( H ⊕ H ) ⊗ ℓ Γ ∞ ] ⊗ C ( X ) , [( H ⊕ H ) ⊗ ℓ Γ ∞ ] ⊗ C ( X ) (cid:1) , obtained in that theorem, satisfies in addition the following control condition: ∀ ϕ ∈ Σ , ∃ C ϕ independent of ǫ such that || W ∗ ǫ b π ( ϕ ) W ǫ − b π ( ϕ ) || ≤ C ϕ ǫ. Said differently, W ǫ satisfies the support condition plus the relation W ∗ ǫ b π ( ϕ ) W ǫ − b π ( ϕ ) ǫ ∼ , for all ϕ ∈ Σ . We only need to explain how to complete the proof given for Theorem 2.1 so that the control is ensured.We thus start by stating the following Lemma which generalizes Lemma 2.3.
Lemma 2.8. [Controlled version of Lemma 2.3]Under the assumptions of Theorem 2.7 and given ǫ > , there exists an isometry ˆ S ǫ ∈ L C ( X ) (( H ⊕ H ) ⊗ C ( X ) , ( H ⊕ H ) ⊗ ℓ Γ ⊗ C ( X )) QUIVARIANT PPV AND PASCHKE DUALITY 15 with the same support as in Lemma 2.3 such that ˆ S ∗ ǫ (( b π ( ϕ ) ⊗ id ℓ Γ ) ⊕
0) ˆ S ǫ − ( b π ( ϕ ) ⊕ ǫ ∼ , ∀ ϕ ∈ Σ . Proof.
We explain the needed complements to the steps in the proof of Lemma 2.3, which exploit the normcontrol on the residual compact operators as in Theorem B.4 of the appendix. We also forget the supportcondition for ˆ S which is again satisfied as one can check easily. We again fix a cut-off function χ ∈ C ( Z ).Let V χ is the non-empty interior of the support of χ . Step 1:
We need to apply Corollary B.7. Let Λ be a countable dense subset of C ( X ), containing 0 and1 X , which is closed under adjoints. Consider the subset Σ χ := { r n } n ∈ N of C ( X, A χ ) ⊕ C ( X ) composed ofelements r n which either belong to (Σ ∩ C ( X, A χ ) , Λ) or are of the form ( f χ / , f ∈ Σ. Since thesequence { r ′ n = r n / ( n + || r n || ) for r n ∈ Σ χ } ∞ n =1 is convergent to 0 in C ( X, A χ ) ⊕ C ( X ), the collection Σ χ = { ( χ / , , (0 , X ) } ∪ { r ′ n } n ∈ N is a compactself-adjoint total subset of C ( X, A χ ) ⊕ C ( X ).Then the image ( b π χ ⊕ + (Σ χ ) is a self-adjoint compact subset of C χ := ( b π χ ⊕ + ( C ( X, A χ ) ⊕ C ( X )),which is total in C χ and contains the identity. Let B χ be the algebra generated by C χ and C ( X, K ( H )),which defines an X -extension algebra for the unital algebra C ( X, A χ ) ⊕ C ( X ). Since ( b π χ ⊕ + is fibrewiseample, using the same arguments as in the proof of Lemma 2.3, we conclude that B χ is a trivial X -extension.Using the separability of C ( X, K ( H )), we fix a compact self-adjoint total subset F χ of B χ which containsΣ χ .Consider also the trivial X -extension obtained analogously by ( b π χ ⊕ + . Then we get, using the notationsin the proof of Lemma 2.3 and Corollary B.7 (for the compact subset F χ ), a unitary S χ ∈ L C ( X ) ( H ⊗ C ( X ) , H ⊗ C ( X )) depending on ǫ , such that we have in particular for any ˜ f ∈ Σ χ of the form ˜ f = ( f, S ∗ χ (cid:20)b π χ ( f ) 00 0 (cid:21) S χ − (cid:20)b π χ ( f ) 00 0 (cid:21) ǫ ∼ C ( X )-modules H i ⊗ C ( X ) is decomposed as E iχ ⊕ ( H i ⊗ C ( X )) for i = 1 , Step 2:
Let s χ : E χ → E χ be the (1 , S χ . Then we have for˜ f = ( f, ∈ Σ χ :(1) s ∗ χ b π χ ( f ) s χ − b π χ ( f ) ǫ ∼ b π χ ( f )( s χ s ∗ χ − id) ǫ ∼ s ∗ χ s χ − id) b π χ ( f ) ǫ ∼ s χ s ∗ χ , b π χ ( f )] ǫ ∼ − s ∗ χ s χ ) and (1 − s χ s ∗ χ ) are positive operators.Indeed, for the first item, it suffices to observe that s ∗ χ b π χ ( f ) s χ − b π χ ( f ) is the (1 , S ∗ χ (cid:20)b π χ ( f ) 00 0 (cid:21) S χ − (cid:20)b π χ ( f ) 00 0 (cid:21) Since the norms of the elements constituting a 2 × s χ : E χ ⊕ E χ → E χ ⊕ E χ as follows: ˆ s χ := (cid:20) s χ (1 − s χ s ∗ χ ) / (1 − s ∗ χ s χ ) / s ∗ χ (cid:21) We have the following formula for any ( f, ∈ Σ χ :ˆ s ∗ χ ( b π χ ( f ) ⊕ s χ − ( b π χ ( f ) ⊕
0) = (cid:20) s ∗ χ b π χ ( f ) s χ − b π χ ( f ) s ∗ χ b π χ ( f )(1 − s χ s ∗ χ ) / (1 − s χ s ∗ χ ) / b π χ ( f ) s χ (1 − s χ s ∗ χ ) / b π χ ( f )(1 − s χ s ∗ χ ) / (cid:21) Note that || s χ || ≤
1, and || (1 − s χ s ∗ χ ) / || ≤
1. We also have || (1 − s χ s ∗ χ ) / b π χ ( f ) || = || b π χ ( f ∗ )(1 − s χ s ∗ χ ) b π χ ( f ) || ≤ || f || . || (1 − s χ s ∗ χ ) b π χ ( f ) || Thus we get from properties (2) and (3) above that (1 − s χ s ∗ χ ) / b π χ ( f ) ǫ ∼
0. Therefore all the matrixentries A ij in the above matrix satisfy A ij ǫ ∼
0. Thus we get:ˆ s ∗ χ ( b π χ ( f ) ⊕ s χ − ( b π χ ( f ) ⊕ ǫ ∼ s χ by zero, we get a partial isometry in L C ( X ) (( H ⊕ H ) ⊗ C ( X ) , ( H ⊕ H ) ⊗ C ( X )),that we still denote by ˆ s χ . Step 3 :
For ( x, g ) ∈ G , we denote by V i ( x,g ) the unitary implementing the G -action on H i ⊗ C ( X ). Thenwe define an operator ˆ S ∈ L C ( X ) (( H ⊕ H ) ⊗ C ( X ) , ℓ Γ ⊗ ( H ⊕ H ) ⊗ C ( X )) by setting the followingpointwise formula: h ˆ S x i g := (cid:16) V x,g ) ⊕ V x,g ) (cid:17) ˆ s χ (cid:16)b π ( χ / ) ⊕ b π ( χ / ) (cid:17) (cid:16) V x,g ) ⊕ V x,g ) (cid:17) − . The operator ˆ S then satisfies the allowed properties in the statement of Lemma 2.8, as we prove below. Letus show that we also have: ˆ S ∗ (( b π ( f ) ⊗ id ℓ Γ ) ⊕
0) ˆ S − ( b π ( f ) ⊕ ǫ ∼ , ∀ f ∈ Σ . Replacing f by a compactly supported element which is as uniformly close as we please to f , we may assumethat f is itself compactly supported. Denote then by Γ( χ, f ) the set of g ∈ Γ such that Supp( g ∗ f ) ∩ Supp( χ ) = ∅ . Due to the properness of the Γ-action, this is a finite set. Consider the compact operators for g ∈ Γ , f ∈ Σ: K χ := ˆ s χ (cid:16)b π ( χ / ) ⊕ b π ( χ / ) (cid:17) − (cid:16)b π ( χ / ) ⊕ b π ( χ / ) (cid:17) ˆ s χ , and K g ( χ, f ) = ˆ s ∗ χ ( b π χ (( g ∗ f ) χ / ) ⊕ s χ − ( b π χ (( g ∗ f ) χ / ) ⊕ || K g ( χ, f ) || ≤ C g ǫ for some constant C g > ǫ and similarly for K χ with constant C χ .Then from the computation in item (2) in the proof of Lemma 2.4, we get:[ ˆ S ∗ (( b π ( f ) ⊗ id ℓ Γ ) ⊕
0) ˆ S − ( b π ( f ) ⊕ x = X g ∈ Γ( χ,f ) (cid:16) V x,g ) ⊕ V x,g ) (cid:17) (cid:16)b π ( χ / ) ⊕ b π ( χ / ) (cid:17) ( K χ + K g ( χ, f )) x (cid:16)b π ( χ / ) ⊕ b π ( χ / ) (cid:17) (cid:16) V x,g ) ⊕ V x,g ) (cid:17) − Thus one gets for any f ∈ Σ, || ˆ S ∗ (( b π ( f ) ⊗ id ℓ Γ ) ⊕
0) ˆ S − ( b π ( f ) ⊕ || ≤ (cid:18) C χ + max g ∈ Γ( χ,f ) C g (cid:19) . | Γ( χ, f ) | .ǫ This proves the claim. (cid:3)
Corollary 2.9 (Norm-controlled version of Theorem 2.2) . There exists a unitary ˆ S ǫ ∈ L C ( X ) ( ℓ Γ ∞ ⊗ ( H ⊕ H ) ⊗ C ( X ) , ℓ Γ ∞ ⊗ ( H ⊕ H ) ⊗ C ( X )) as in Theorem 2.2 such that ˆ S ∗ ǫ (( b π ( f ) ⊗ id ℓ Γ ) ⊕
0) ˆ S ǫ − ( b π ( f ) ⊕ ǫ ∼ for any f ∈ Σ . Proof.
By directly verifying the constructions in Corollary 2.5 and the proof of Theorem 2.2, we see thatif the initial isometry is chosen to satisfy the conditions in Lemma 2.8, then all the intertwining isometriesand unitaries that appear in the proofs of Corollary 2.5 and Theorem 2.2 must also satisfy the analogouscondition on the norms of the residual compact operators. (cid:3)
We are now ready to prove Theorem 2.7.
QUIVARIANT PPV AND PASCHKE DUALITY 17
Proof of Theorem 2.7:
Let S ǫ ∈ L C ( X ) ( ℓ Γ ∞ ⊗ ( H ⊕ H ) ⊗ C ( X ) , ℓ Γ ∞ ⊗ ( H ⊕ H ) ⊗ C ( X ) be a unitary,obtained from Corollary 2.9, such that(2.2) S ∗ ǫ b π ( f ) S ǫ − b π ( f ) ǫ ∼ ∀ f ∈ Σ . Observe that if f ′ ∈ C ( X, A c ), with A c := C c ( Z ) A , satisfies || f − f ′ || ∞ ≤ ǫ , then the analogous relation to2.2 also holds for f ′ , and vice versa, if the relation holds for f ′ it also holds for f . Also note that since Σis globally Γ-invariant, the construction of the Γ-invariant unitary W ǫ which intertwines the representations b π and b π then follows from Theorem 2.1, using the norm-controlled operator S ǫ . The only thing to checkis that for all f ∈ Σ, b π ( f ) W ǫ − W ǫ b π ( f ) ǫ ∼ . Let f ∈ Σ, we first show that the required relation holds for any f ′ ∈ C ( X, A c ) such that || f − f ′ || ∞ ≤ ǫ .First note that by the localization of the support of S ǫ , for f ′ ∈ C ( X, A c ), the sum defining b π ( f ′ ) W ǫ isagain finite. Moreover, the number of terms in the finite sum is independent of the operator S ǫ itself, andtherefore independent of ǫ .We have from the computations in the proof of Theorem 2.1, for f ′ as above, b π ( f ′ ) W − W b π ( f ′ ) = X g ∈ Γ (id H ⊕ H ⊗ U g ) ◦ g (cid:2)(cid:0)b π ( g − f ′ ) S − S b π ( g − f ′ ) (cid:1) π ( √ χ ) (cid:3) This sum is again over a finite subset Γ( χ, f ′ ) of Γ, due to the assumption of uniform proper action and giventhe support condition for S . As before, if we let K g ( χ, f ′ ) denote the compact operator [ b π (( g − ) ∗ f ′ ) S ǫ − S ǫ b π (( g − ) ∗ f ′ )], we have K g ( χ, f ′ ) ǫ ∼ g ∈ Γ, say with the constant of inequality C g >
0, and hencewe have || b π ( f ′ ) W − W b π ( f ′ ) || ≤ (cid:18) max g ∈ Γ( χ,f ′ ) C g (cid:19) . | Γ( χ, f ′ ) | .ǫ. Now as || f − f ′ || ≤ ǫ , we also have: b π ( f ) W − W b π ( f ) ǫ ∼ χ, f ′ ) is contained in the set of g ∈ Γ such thatSupp(( g − ) ∗ f ′ ) ∩ B κ (Supp( χ )) = ∅ , where κ is the diameter of the cutoff function χ . This ends the proof. (cid:3) We have now completed the proof of Theorem 1.3.3.
Application to equivariant Paschke duality
As an application of our equivariant version of the PPV theorem, stated in Theorem 2.1, we now provethe Paschke-Higson duality theorem in this context. We assume in this section that the proper Γ-space Z is cocompact and endowed as before with the Γ-equivariant metric d so that closed balls are compactsubspaces of Z , said differently the metric space ( Z, d ) is proper. Recall that A is a proper Γ-algebra over Z and that we have assumed that C ( Z ) maps inside the multipliers of A itself. Recall from [BR2:20], thatassociated with the proper metric space Z and a proper action of the groupoid G = X ⋊ Γ on the C ∗ -algebra C ( X, A ), we can define the G -equivariant Roe algebras, which will be denoted as D ∗ Γ ( X, A ; ℓ Γ ∞ ⊗ H ) and C ∗ Γ ( X, A ; ℓ Γ ∞ ⊗ H ) associated with a given ample Γ-equivariant representation of A in H . The first one isthe closure of the space of pseudo-local Γ-invariant operators, while the second one is the ideal in the firstone composed of those operators that are moreover locally compact. The quotient algebra is denoted as Q ∗ Γ ( X ; ( Z, ℓ Γ ∞ ⊗ L Z )).Let us recall the precise definitions which are the immediate generalizations of the ones given in [BR1:20]and [BR2:20] when A = C ( Z ). Let ( H, U ) be a unitary Hilbert space representation of Γ together withan ample Γ-equivariant representation π of A . This is equivalent to the datum of a G -equivariant C ( X )-representation ˆ π of C ( X, A ). Recall that any adjointable operator T of L C ( X ) ( C ( X ) ⊗ H ) is given by a ∗ -strongly continuous field ( T x ) x ∈ X of bounded operators on H . An adjointable operator is G -equivariant,if the field ( T x ) x ∈ X satisfies the relations T xg = U − g T x U g , ( x, g ) ∈ X × Γ . The space of G -equivariant adjointable operators is denoted as usual L C ( X ) ( C ( X ) ⊗ H ) Γ . Recall also fromthe previous section the notion of propagation of a given operator with respect to the C ( X )-Γ-equivariantrepresentation ˆ π of C ( X, A ). We denote by D ∗ Γ ( X, A ; H ) and C ∗ Γ ( X, A ; H ) the corresponding Roe algebrasas defined in [BR1:20], but for our groupoid G and our specific Hilbert G -module C ( X ) ⊗ H . More precisely, D ∗ Γ ( X, A ; H ) is defined as the norm closure in L C ( X ) ( C ( X ) ⊗ H ) of the following space { T ∈ L C ( X ) ( C ( X ) ⊗ H ) G , T has finite propagation and [ T, π ( f )] ∈ C ( X, K ( H )) for any f ∈ C ( X, A ) } . The ideal C ∗ Γ ( X, A ; H ) is composed of all the elements T of D ∗ Γ ( X, A ; H ) which satisfy in addition that T π ( f ) ∈ C ( X, K ( H )) for any f ∈ C ( X, A ) . The finite propagation property here is supposed to hold uniformly on X , so ( T x ) x ∈ X has finite propagationif there exists a constant M ≥ ϕ, ψ ∈ C ( X, A ) with d (Supp( ϕ ) , Supp( ψ )) > M , we have π x ( ϕ ) T x π x ( ψ ) = 0 , ∀ x ∈ X. We thus have the short exact sequence of C ∗ -algebras0 → C ∗ Γ ( X, A ; H ) ֒ → D ∗ Γ ( X, A ; H ) −→ Q ∗ Γ ( X, A ; H ) → , where we have denoted by Q ∗ Γ ( X, A ; H ) the quotient C ∗ -algebra of D ∗ Γ ( X, A ; H ) by its two-sided closedinvolutive ideal C ∗ Γ ( X, A ; H ). The notation here is ambiguous as we don’t mention the space Z while thenotion of propagation with respect to the representation of A depends a priori on the choice of ( Z, d ). Thereason for this simplified notation is that the K -groups will not depend on this choice as we shall see below,although the identifications are not natural.The Paschke-Higson duality theorem identifies the K -theory of the quotient algebra Q ∗ Γ ( X, A ; ℓ Γ ∞ ⊗ H ) with the G -equivariant KK -theory of the pair C ( X, A ) , C ( X ). For details about the definition of G -equivariant KK -theory the reader is referred to the fundamental paper of Le Gall [LeGall:99]. Since X iscompact here, notice though that the latter group is naturally isomorphic to the Γ-equivariant KK -theoryof the pair ( A, C ( X )), see [BR2:20], section 4, for more details.When A = C ( Z ), we can for instance make use of the representation π X × Z which is induced by multi-plication on ℓ Γ ∞ ⊗ L ( Z ) ⊗ C ( X ), where L Z = L ( Z, µ Z ) is defined for a choice of a Borel Γ-invariantmeasure µ Z on Z , which we shall always assume to be fully supported. This representation is fibrewiseample in the sense of Definition 1.2.We are now in position to prove Theorem 1.4. We need to construct a group isomorphism P ∗ : K ∗ ( Q ∗ Γ ( X, A ; ℓ Γ ∞ ⊗ H )) ∼ = −→ KK Γ ∗ +1 ( A, C ( X )) , ∗ = 0 , . We only treat the case ∗ = 0. The proof is again a repetition of the proof given in [BR2:20], Theorem4.1, and adapted to the more general proper Γ-algebra A ; we sketch it here only for completeness. Weconstruct a group homomorphism P ′ : KK ( A, C ( X )) → K ( Q ∗ Γ ( X, A ; ℓ Γ ∞ ⊗ H )), using the equivariantPPV Theorem 2.1. The homomorphism P ′ will then be an inverse to the natural Paschke-Higson map P : K ( Q ∗ Γ ( X, A ; ℓ Γ ∞ ⊗ H )) → KK Γ1 ( A, C ( X )) Step 1:
Let [( σ, E, F )] ∈ KK Γ1 ( A, C ( X )). We may assume as usual that σ is non-degenerate and that F isself-adjoint. Using Kasparov’s stabilization theorem, we obtain a cycle of the form [ σ , H ⊗ C ( X ) , F ], whichis endowed with the transported G -action via the Kasparov isomorphism E ⊕ ( H ⊗ C ( X )) ∼ = H ⊗ C ( X ). Notethat the summand H ⊗ C ( X ) appearing on the left side of the isomorphism is endowed with its canonical G -action induced by the action of G on C ( X, A ). It is easy to check that the latter cycle lies in the same KK Γ1 -class as [ σ, E, F ]. Step 2:
Embed H ⊗ C ( X ) equivariantly in ℓ Γ ⊗ H ⊗ C ( X ) via an equivariant isometry S : H ⊗ C ( X ) → ℓ (Γ) ⊗ H ⊗ C ( X ), defined by the following formula which uses the cut-off function χ ∈ C c ( Z ) as in theprevious section: S ( e ) = X g ∈ Γ δ g − ⊗ σ ( g √ χ )( e ) for e ∈ E c , QUIVARIANT PPV AND PASCHKE DUALITY 19 where E c = π ( A c )( H ⊗ C ( X )), the G -action on H ⊗ C ( X ) is given by the action V from Step 1, while the G -action on ℓ (Γ) ⊗ H ⊗ C ( X ) is given by the right regular representation of Γ on ℓ Γ tensored by the sameaction V .Now, (cid:0) Sσ ( • ) S ∗ , SS ∗ ( ℓ Γ ⊗ H ⊗ C ( X )) , SF S ∗ (cid:1) is equivalent to ( σ , H ⊗ C ( X ) , F ), and after adding asuitable degenerate cycle, we get the cycle (cid:0) σ := id ℓ Γ ⊗ σ , ℓ Γ ⊗ H ⊗ C ( X ) , F := ( F ⊕ id) (cid:1) which is stillin the same KK Γ1 -class as ( σ , H ⊗ C ( X ) , F ). For details of this construction we refer the reader to [BR2:20],see Step 2 of the proof of Theorem 4.1 there. Step 3 : Add further degenerate cycles to [ σ , ℓ Γ ⊗ H ⊗ C ( X ) , F ] we may pass to a new Γ-equivariantKasparov cycle (cid:0) σ ∞ := id ℓ N ⊗ σ , ℓ Γ ∞ ⊗ H ⊗ C ( X ) , F ∞ := diag( F , id , id ... ) (cid:1) which represents the same KK Γ1 -class. We further add the degenerate cycle (cid:0) , ℓ Γ ∞ ⊗ H ⊗ C ( X ) , (cid:1) to [ σ ∞ , ℓ Γ ∞ ⊗ H ⊗ C ( X ) , F ∞ ]with the Γ-action now taken as the one coming canonically from the Γ-action on H ⊗ C ( X ) tensored withthe right regular representation on the factor ℓ Γ and extended trivially on ℓ N . We obtain in this way anew Γ-equivariant Kasparov cycle (cid:0) σ := σ ∞ ⊕ , ℓ Γ ∞ ⊗ ( H ⊕ H ) ⊗ C ( X ) , F := F ∞ ⊕ (cid:1) still remaining in the same KK Γ1 -class. Step 4:
We can now apply Theorem 2.1, to get a Γ-invariant C ( X )-adjointable unitary W such that W σ ( f ) W ∗ − ( b π ∞ ( f ) ⊕ ∈ K C ( X ) (cid:0) ℓ Γ ∞ ⊗ ( H ⊕ H ) ⊗ C ( X ) (cid:1) , for all f ∈ C ( X, A ) . where π ∞ : C ( X, A ) → L C ( X ) ( ℓ Γ ∞ ⊗ H ⊗ C ( X )) is induced by the ample representation π of A in theHilbert module H ⊗ C ( X ) and extended by the identity on ℓ Γ ∞ . By Kasparov’s homological equivalenceLemma(see [BR2:20], Appendix B), the cycles (cid:0) σ , ℓ Γ ∞ ⊗ ( H ⊕ H ) ⊗ C ( X ) , F (cid:1) and (cid:0) π ∞ ⊕ , ℓ Γ ∞ ⊗ ( H ⊕ H ) ⊗ C ( X ) , F (cid:1) , live in the same KK Γ1 -class, where F := W F W ∗ . Step 5:
Let ˜ F and W be the (1 , × F , corresponding tothe direct sum ℓ Γ ∞ ⊗ ( H ⊕ H ) ⊗ C ( X ) . Then the cycle[ b π ∞ ⊕ , ℓ Γ ∞ ⊗ ( H ⊕ H ) ⊗ C ( X ) , F ]is in the same KK Γ1 -class as the cycle [ b π ∞ , ℓ Γ ∞ ⊗ H ⊗ C ( X ) , F := W ˜ F W ∗ ]. Step 6:
Replace the operator F by a Γ-invariant finite propagation operator F as usual by averaging √ χF √ χ . We define the inverse map P ′ : KK Γ1 ( C ( X, A ) , C ( X )) → K ( Q ∗ Γ ( X, A ; ℓ Γ ∞ ⊗ H ) by setting P ′ ([ σ, E, F ]) := (cid:20) q (cid:18)
12 ( W W ∗ + F ) (cid:19)(cid:21) where q : D ∗ Γ ( X, A ; ℓ Γ ∞ ⊗ H ) → Q ∗ Γ ( X, A ; ℓ Γ ∞ ⊗ H ) is the quotient projection.The map P ′ is well-defined and a bijective group homomorphism, following the same arguments as in thecompact case in [BR2:20], Theorem 4.1. Hence the proof of our Paschke-Higson theorem is now complete. Appendix A. Localized operators on uniformly proper Γ -spaces We prove in this appendix some standard results about supports of our localized operators that are usedin some proofs. Let us fix a non-degenerate ∗ -representation π : C ( Z ) → L ( H ) of the C ∗ -algebra C ( Z ) inthe separable Hilbert space H , that we extend to C b ( Z ) as usual. Recall that Γ acts uniformly properly on Z and that χ is a chosen adapted continuous cutoff function.We shall use the following notations for an operator T ∈ L ( H ):Supp( T ) z := { z ′ ∈ Z | ( z ′ , z ) ∈ Supp( T ) } , Supp( T ) z ′ := { z ∈ Z | ( z ′ , z ) ∈ Supp( T ) } Notice that if W χ = { χ = 0 } then Z = S g ∈ Γ gW χ . We denote as in Section 1 for any k ≥ ( k ) χ := { ( g, g ′ ) ∈ Γ |∃ ( g i ) ≤ i ≤ k − such that g i W χ ∩ g i +1 W χ = ∅ and g = g, g k = g ′ } . For k = 0, we set Γ (0) χ = Γ viewed as the diagonal of Γ . Notice that Γ ( k ) χ ⊂ Γ ( k +1) χ for any k , and that S k ≥ Γ ( k ) χ = Γ . Recall that the uniform properness of the action means that the first (or the second)projection Γ → Γ becomes proper when restricted to Γ (1) χ . It is an obvious observation that if the properΓ-space Z is cocompact, then the action of Γ on Z is automatically uniformly proper since the support of χ can then be taken compact, so that { g ∈ Γ | g Supp( χ ) ∩ Supp( χ ) = ∅} is finite.Set A k := S ( g,g ′ ) ∈ Γ ( k ) χ gW χ × g ′ W χ , then it is easy to check using the properties of W χ that for any k ≥ A k is contained A k +2 . Definition A.1 (Localized operators) . An operator T ∈ L ( H ) is said to have localized support if thereexists k ≥ so that Supp( T ) is contained in (the closure of ) some A k with k ≥ .The least k such that the support of T is contained in A k will be called the propagation index of T (withrespect to χ ). For brevity, we shall call an operator with finite propagation index a localized operator. For a localized operator T with propagation index k and if we denote by Γ z the finite subset of Γ composedof those g for which z ∈ gW χ , then for any z ∈ Z we have:Supp( T ) z ⊂ [ g ∈ Γ z [ g ′ | ( g,g ′ ) ∈ Γ ( k ) χ g ′ W χ . Proposition A.2.
Assume that Z is a proper cocompact Γ -space with a Γ -invariant distance d s that Z isa metric-proper space. Then localized operators coincide with finite propagation operators.Proof. We can find a cutoff function χ which is compacty supported in Z and hence whose support has finitediamater. An operator T is localized with propagation index ≤ k if and only if its support is contained in A k . Hence denoting by d χ the diamater of W χ in Z which is equal to the diameter of any translate gW χ for g ∈ Γ, we see that for any ( z, z ′ ) ∈ Supp( T ), we have by d ( z, z ′ ) ≤ kd χ . Hence T has finite propagation ≤ kd χ . If conversely T has finite propagation κ . For any ( z, z ′ ) ∈ Supp( T ),we have d ( z, z ′ ) ≤ κ and we also know that there exists g ∈ Γ such that z ∈ g W χ . Since Z is metric-proper,there exists a finite subset Γ κ of Γ such that the closed ball neighborhood B κ := { z ∈ Z | d ( z, Supp( χ )) ≤ κ } of the compact space Supp( χ ) is contained in ∪ g ∈ Γ κ gW χ . Moreover, let us denote by k the least integer suchthat for any g ∈ Γ κ , we have ( e, g ) ∈ Γ ( k ) χ , with e being the neutral element of Γ. To sum up we know that z ∈ g W χ while d ( z, z ′ ) ≤ κ so that z ′ ∈ S g ∈ Γ κ g gW χ , and henceforth( z, z ′ ) ∈ [ g ∈ Γ κ g W χ × g gW χ ⊂ [ ( g,g ′ ) ∈ Γ ( k ) χ gW χ × g ′ W χ = A k , and k is of course independent of the chosen ( z, z ′ ) ∈ Supp( T ). (cid:3) Proposition A.3.
The space of localized operators is unital ∗ -subalgebra of L ( H ) . Moreover,(1) the propagation index of the adjoint is equal to the propagation index of the given localized operator.(2) the propagation index of the sum of two localized operators is ≤ to the maximum of the propagationindices.(3) the propagation index of the composition of two localized operators is ≤ the sum of the propagationindices.Proof. The first item is clear since the one has the relation Supp( T ∗ ) = σ (Supp( T )), where σ : Z × Z → Z × Z is the involution ( z, z ′ ) ( z ′ , z ). The support of the identity operator is the diagonal in Z which is containedin Γ (0) χ . Take two localized operators T and S with propagation indices k and k ′ respectively. The supportof the sum T + S is obviously contained in Supp( T ) ∪ Supp( S ). ThereforeSupp( T + S ) ⊂ A k ∪ A k ′ = A max( k,k ′ ) . QUIVARIANT PPV AND PASCHKE DUALITY 21
On the other hand, for any ( z, z ′′ ) such that Supp( T ) z ∩ Supp( S ) z ′′ = ∅ , and denoting the propagation indexof T by k and the propagation index of S by k ′ , there exists ( g , · · · , g k ) ∈ Γ k and ( g ′ , · · · , g ′ k ′ ) ∈ Γ k ′ suchthat z ∈ g W χ , z ′′ ∈ g ′ k ′ W χ , g i W χ ∩ g i +1 W χ = ∅ for 0 ≤ i ≤ k − ,g k W χ ∩ g ′ W χ = ∅ and g ′ j W χ ∩ g ′ j +1 W χ = ∅ for 0 ≤ j ≤ k ′ − . Hence ( z, z ′′ ) ∈ A k + k ′ +1 . Hence, using that the support of T S is contained the closure of { ( z, z ′′ ) ∈ Z | Supp( T ) z ∩ Supp( S ) z ′′ = ∅} and the inclusion A k + k ′ +1 ⊂ A k + k ′ +3 we deduce that the support of T S is contained in A k + k ′ +3 . (cid:3) Remark A.4.
The analogously defined Roe C ∗ -algebras of locally compact and pseudolocal operators with localized support , can hence be defined in our more general setting of non-cocompact uniformly proper actions. Appendix B. The norm-controlled PPV theorem
In this section we give a norm-controlled version of the PPV theorem [PPV:79][Theorem 2.10]. This is afolklore-type result which nevertheless is not found in the literature to the best of our knowledge.Let X be a finite dimensional compact metrizable space, A a unital separable C ∗ -algebra and H aninfinite-dimensional separable Hilbert space. Denote U CP ( A, M n ) the space of unital, completely positivemaps from A to M n ( C ), equipped with the point-norm topology. We shall denote by L ( H ) ∗ s the algebra ofbounded linear operators on H equipped with the strong- ∗ topology. Proposition B.1 (Proposition 2.8 in [PPV:79]) . Consider an X -extension → C ( X, K ( H )) ֒ → B σ −→ A → with ideal symbol { I x } x ∈ X and Ψ : X → U CP ( A, M n ) be a continuous map such that Ψ( x ) | I x = 0 forall x ∈ X . Then, given ǫ > , V ⊂ H and ∈ W ⊂ B finite-dimensional subspaces, there exists anorm-continuous map U : X → L ( C n , H ) such that U ∗ ( x ) U ( x ) = id C n , U ( x )( C n ) ⊥ V, ∀ x ∈ X and || Ψ( x )( σ ( b )) − U ∗ ( x ) b ( x ) U ( x ) || ≤ ǫ || b || ∀ x ∈ X, b ∈ W. The linear span of { U ( x ) C n } x ∈ X in H is finite-dimensional. Using Proposition B.1, one gets the following:
Corollary B.2.
Consider an X -extension → C ( X, K ( H )) ֒ → B σ −→ A → with exact lsc ideal symbol { I x } x ∈ X and let Ψ : X → U CP ( A, M n ) be a continuous map such that Ψ( x ) | I x = 0 for all x ∈ X . Let V be a finite-dimensional subspace of H . Then there exists a sequence of norm-continuousmaps U k : X → L ( C n , H ) such that(1) U ∗ k ( x ) U k ( x ) = id C n U k ( x )( C n ) ⊥ V ∀ x ∈ X, ∀ k ∈ N (2) lim k →∞ sup x ∈ X || Ψ( x )( σ ( b )) − U ∗ k ( x ) b ( x ) U k ( x ) || = 0 ∀ b ∈ B , and(3) lim k →∞ sup x ∈ X || U ∗ k ( x ) η ( x ) || = 0 ∀ η ∈ C ( X, H ) .Proof. Fix a convergent sequence F = { b i } i ∈ N in B containing 1 ∈ B , such that F = F ∗ , || b i || ≤
1, for all i ,and the linear span of F is dense in B . Recall that B is separable here since it is an extension algebra. Since F is compact, for each k ∈ N , there exists N k ∈ N and a finite set F k := { b i m } N k m =1 which includes 1 ∈ B ,such that for any b i ∈ F there exists an index m ∈ { , , ..., N k } such that || b i m − b i || < / k . Let { e n } be an orthonormal basis for H . Denote by P j the linear span of { e , e , ..., e j } . For each k ∈ N ,we iteratively apply Proposition B.1 by taking V k = span { V, P k } and W k = F k and ǫ k = 1 / k . We thusobtain norm-continuous maps U k : X → L ( C n , H ) such that U ∗ k ( x ) U k ( x ) = id C n U k ( x )( C n ) ⊥ V k ∀ x ∈ X, ∀ k ∈ N which shows that (1) is satisfied. We also havesup x ∈ X || Ψ( x )( σ ( b )) − U ∗ k ( x ) b ( x ) U k ( x ) || ≤ / k ∀ b ∈ W k Now, for any b ∈ F , there exists an element b ′ ∈ W k such that || b − b ′ || < / k , then we get for any x ∈ X , || Ψ( x )( σ ( b )) − U ∗ k ( x ) b ( x ) U k ( x ) || ≤ || Ψ( x )( σ ( b ) − Ψ( x )( σ ( b ′ ) || + || Ψ( x )( σ ( b ′ )) − U ∗ k ( x )( b ′ ) U k ( x ) || + || U ∗ k ( x )( b − b ′ ) U k ( x ) ||≤ / k + 1 / k + 1 / k = 1 /k where we have used the fact that || Ψ || = 1 (since A is unital) and U k ( x ) U ∗ k ( x ) is an orthogonal projectionfor all x ∈ X , so || U k ( x ) U ∗ k ( x ) || =1. Thus (2) is established for all b ∈ F . Since F spans B , another densityargument then gives the result for all b ∈ B .To check (3), let ǫ > η ∈ P j for some j , then < η, U k ( x ) U ∗ k ( x ) η > = 0 for all k > j , sincerange of U k ( x ) is perpendicular to P k . Now let η = P ∞ i =1 α i e i , choose N such that || η − P N i =1 α i e i || < ǫ .Then for any k , we have || U ∗ k ( x ) η || ≤ || U ∗ k ( x )( η − N X i =1 α i e i ) || + || U ∗ k ( x )( N X i =1 α i e i ) || since P N i =1 α i e i ∈ P N , the second term above is zero for k > N for all x ∈ X . Therefore one getssup x ∈ X || U ∗ k ( x ) η || ≤ ǫ ∀ k > N , ∀ x ∈ X. which establishes (3) in the case when η ∈ C ( X, H ) is constant in the X -variable. To deal with the generalcase, let for each x ∈ X , W x be an open neighbourhood of x such that for any x ′ ∈ W x , we have: || η ( x ) − η ( x ′ ) || ≤ ǫ/ X is compact we get a finite collection { W x i } mi =1 of such open neighbourhoods with centers { x i } mi =1 .Choose N large enough such thatsup x ∈ X || U ∗ k ( x ) η ( x i ) || ≤ ǫ/ k ≥ N and for all i = 1 , , · · · , m. Then we have for any k ≥ N and x ∈ W x i for some i , || U ∗ k ( x ) η ( x ) || ≤ || U ∗ k ( x )( η ( x ) − η ( x i )) || + || U ∗ k ( x ) η ( x i ) || ≤ ǫ This proves (3). (cid:3)
Remark B.3.
In the proof above one can also take any countable approximate unit { a k } k ∈ N for C ( X, K ( H )) consisting of increasing sequence of finite-rank operators which are constant in X , and take V k = span { V, P k } where P k is the projection onto the range of a k . Using the above result, we can now give a strengthening of Proposition 2.9 in [PPV:79]. Denote by d x : C ( X, L ( H ) ∗ s ) → L ( H ) the evaluation map. We keep the notations used above. Theorem B.4.
Given a trivial X -extension by A with exact lsc ideal symbol { I x } x ∈ X : → C ( X, K ( H )) ֒ → B σ −→ A → QUIVARIANT PPV AND PASCHKE DUALITY 23 which is implemented by a unital ∗ -homomophism µ : A → C ( X, L ( H ) ∗ s ) and another arbitrary X -extensionwith same ideal symbol { I x } x ∈ X , whose extension algebra is B ⊆ C ( X, L ( H ) ∗ s ) for some infinite-dimensionalseparable Hilbert space H : → C ( X, K ( H )) ֒ → B σ −→ A → Let F be a compact subset of B such that F = F ∗ , ∈ F and the linear span of F is dense in B . Given ǫ > , there exists an isometry S ∈ C ( X, L ( H ) s ∗ ) such that(i) S ∗ bS − µ ( σ ( b )) ∈ C ( X, K ( H )) for all b ∈ B .(ii) ∀ b ∈ F , ∃ C, C ′ independent of ǫ such that || S ∗ bS − µ ( σ ( b )) || ≤ Cǫ and || Sµ ( σ ( b )) − bS || ≤ C ′ ǫ .Proof. Let B be the unital C ∗ -algebra generated by the image of µ and C ( X, K ( H )), and let { a k } ∞ k =0 bea quasi-central approximate unit for C ( X, K ( H )) consisisting of an increasing sequence of constant (in the X -variable) finite-rank operators 0 = a ≤ a ≤ a · · · , || a k || ≤
1, andlim k || a k l − l || = 0 , ∀ l ∈ C ( X, K ( H )) and lim k || [ a k , h ] || = 0 , ∀ h ∈ B . where [ x, y ] denotes the commutator xy − yx . Let F be a compact, self-adjoint subset of the unit ball of B whose span is B . Passing to a subsequence if necessary, we may assume that || [ µ ( σ ( b )) , ( a k − a k − ) / ] || ≤ ǫ/ k ∀ b ∈ F, k ≥ . Let Q k be the constant orthogonal projection onto the range of a k for each k ≥
1. Using Corollary B.2,we iteratively define a sequence of compact operators U k ∈ C ( X, K ( H )) , k ∈ N whose initial projections arethe range of a k and final projections are of uniformly finite rank, and an increasing sequence of finite-rankprojections R k , k ∈ N on H , converging strongly to the identity, such that we have for all k ≥ U ∗ k ( x ) U k ( x ) = Q k , for all x ∈ X .(2) Range( U k ( x )) ⊆ ( R k +1 − R k )( H ), for all x ∈ X .(3) Range( U k ( x )) ⊥ Range( U k ′ ( x ′ )) for all x, x ′ ∈ X for all k ′ < k .(4) || ( a k − a k − ) / µ ( σ ( b ))( x )( a k − a k − ) / − U ∗ k ( x ) b ( x ) U k ( x ) || ≤ ǫ/ k , for all x ∈ X , b ∈ F .(5) || U ∗ i ( x ) b ( x ) U j ( x ) || ≤ ǫ/ i + j , for all b ∈ F , x ∈ X , i = j .Some remarks are in order. The first property is clear from the construction in Corollary B.2; the existenceof the finite-rank operators R k in the property (2) also follows from the fact that the U k themselves are ofuniformly finite-rank. The third property can be obtained in the construction of U k by adding the ranges ofall the U k ′ for k ′ < k in the choice of the finite-dimensional space V in Corollary B.2. The fourth propertyis simply obtained by taking the completely positive map Ψ in Corollary B.2 to be ( a k − a k − ) / µ ( • )( a k − a k − ) / . The last property (5) can be obtained from item (3) in Corollary B.2, since the initial space ofeach U j for j < i is of uniformly finite-dimension.Define the operator S ∈ C ( X, L ( H ) ∗ s ) pointwise in the following way: S ( x ) := ∞ X k =1 U k ( x )( a k − a k − ) / Indeed, it suffices to use properties (1), (2), and (3) above to show that S ( x ) is uniformly convergent in X with respect to the strong- ∗ topology on L ( H ). It can also be verified directly that S ∗ ( x ) S ( x ) = id H , thus S ( x ) is an isometry, using the fact that Range( a k − a k − ) / ⊆ Range( Q k ) = Range( U ∗ k ( x ) U k ( x )) for all x ∈ X .Let f k = ( a k − a k − ) / . Using the fact that µ ( σ ( b )) = P ∞ k =1 µ ( σ ( b )) f k , where the series converges inthe strict topology, we get: µ ( σ ( b ))( x ) − ∞ X k =1 f k µ ( σ ( b ))( x ) f k = ∞ X k =1 [ µ ( σ ( b ))( x ) , f k ] f k Thus, by the assumptions on f k = ( a k − a k − ) / , we get || µ ( σ ( b ))( x ) − P ∞ k =1 f k µ ( σ ( b ))( x ) f k || ≤ ǫ , for all b ∈ F . Therefore, we finally get for all b ∈ F ,(B.1) || S ( x ) b ( x ) S ∗ ( x ) − µ ( σ ( b ))( x ) || ≤ || µ ( σ ( b ))( x ) − ∞ X k =1 f k µ ( σ ( b ))( x ) f k || + ∞ X k =1 || f k µ ( σ ( b ))( x ) f k − U ∗ k ( x ) b ( x ) U k ( x ) || + X i = j || U ∗ i ( x ) b ( x ) U j ( x ) || ≤ ǫ by properties (4) and (5) above. On the other hand, since the partial sums of S ( x ) are all compact, we alsoget µ ( σ ( b )) − S ∗ bS ∈ C ( X, K ( H )) for all b ∈ F . Using a density argument as before, one can finish theproof of ( i ) by establishing the desired properties for all b ∈ B .Notice also that we have the following relation for any b ∈ B ,(B.2) ( Sµ ( σ ( b )) − bS ) ∗ ( Sµ ( σ ( b )) − bS ) = ( S ∗ b ∗ bS − µ ( σ ( b ∗ b ))) + µ ( σ ( b ∗ ))( µ ( σ ( b ) − S ∗ bS )+ ( µ ( σ ( b ∗ )) − S ∗ b ∗ S ) µ ( σ ( b ))from which the claim follows. (cid:3) Remark B.5.
The operator S ∈ C ( X, L ( H ) ∗ s ) constructed in the proof above also satisfies || b (1 − SS ∗ ) || = || K ∗ S − K S ∗ || ≤ C ′ ǫ where K = bS − Sµ ( σ ( b )) . We may rewrite for b ∈ B , the relations in Theorem B.4, using as well the previous remark, as S ∗ bS − µ ( σ ( b )) ǫ ∼ , bS − Sµ ( σ ( b )) ǫ ∼ , b (1 − SS ∗ ) ǫ ∼ , [ b, SS ∗ ] ǫ ∼ . Notice that S ∗ bS − µ ( σ ( b )) satisfies conditions (1) and (2) in Theorem B.4. Corollary B.6.
Let H be a separable infinite-dimensional Hilbert space. Given a trivial X -extension by A with exact lsc ideal symbol { I x } x ∈ X : → C ( X, K ( H )) ֒ → B σ −→ A → which is implemented by a unital ∗ -homomophism µ : A → C ( X, L ( H ) ∗ s ) and another arbitrary X -extension with same ideal symbol { I x } x ∈ X , whose extension algebra is B ⊆ C ( X, L ( H ) ∗ s ) for some infinite-dimensional separable Hilbert space H : → C ( X, K ( H )) ֒ → B σ −→ A → , there exists a sequence of operators { S n } n ∈ N , S n ∈ C ( X, L ( H , H ) ∗ s ) for all n ∈ N , such that we have forall b ∈ B :(1) µ ( σ ( b )) − S ∗ n bS n ∈ C ( X, K ( H )) for any n ∈ N ,(2) lim n →∞ || µ ( σ ( b )) − S ∗ n bS n || = 0 , and(3) S ∗ n S n = id H ⊗ C ( X ) for all n ∈ N .Proof. This is an immediate application of Theorem B.4, by reducing to the case H = H via a unitaryisomorphism u : H → H . (cid:3) Corollary B.7.
Given a trivial X -extension by A with exact lsc ideal symbol { I x } x ∈ X : → C ( X, K ( H )) ֒ → B σ −→ A → which is implemented by a unital ∗ -homomophism µ : A → C ( X, L ( H ) ∗ s ) , and another trivial X -extensionwith the same ideal symbol { I x } x ∈ X , whose extension algebra is B ⊆ C ( X, L ( H ) ∗ s ) : → C ( X, K ( H )) ֒ → B σ −→ A → , there exists a sequence of unitary operators { S n } n ∈ N , S n ∈ C ( X, L ( H ) ∗ s ) for all n ∈ N , such that we havefor all b ∈ B :(1) µ ( σ ( b )) − S ∗ n bS n ∈ C ( X, K ( H )) for any n ∈ N ,(2) lim n →∞ || µ ( σ ( b )) − S ∗ n bS n || = 0 . QUIVARIANT PPV AND PASCHKE DUALITY 25
Moreover, given ǫ > , and a compact subset F in B such that F = F ∗ , ∈ F and whose linear span isdense in B , there exists a unitary S ∈ C ( X, L ( H ) ∗ s ) such that for all b ∈ F we have: µ ( σ ( b )) − S ∗ bS ǫ ∼ . Proof.
This is done by the usual PPV trick to pass from isometries to unitaries, as in Theorem 2.10 in[PPV:79]. It only remains to note that the condition (2) above is still valid; this can be easily verified by adirect inspection. (cid:3)
References [BC:00] P. Baum, A. Connes
Geometric K-theory for Lie groups and foliations , Enseign. Math. (2) 46 (2000), no. 1-2, 3-42.[BM:15] M.-T. Benameur and V. Mathai,
Spectral sections, twisted rho invariants and positive scalar curvature.
J. Noncommut.Geom. 9 (2015), no. 3, 821-850.[BM:20] M.-T. Benameur and V. Mathai,
Proof of the Magnetic Gap-Labelling conjecture for principal solenoidal tori.
Volume278, Issue 3, 1 February 2020[BP:09] M.-T. Benameur and P. Piazza,
Index, eta and rho invariants on foliated bundles.
Ast´erisque No. 327 (2009), 201-287(2010).[BR:15] M.-T. Benameur and I. Roy,
The Higson-Roe exact sequence and ℓ eta invariants. J. Funct. Anal. 268 (2015), no. 4,974-1031.[BR1:20] M.-T. Benameur and I. Roy,
The Higson-Roe sequence for ´etale groupoids. I. Dual algebras and compatibility with theBC map , To appear in the J. Noncommutative Geometry, 2020, arXiv preprint: https://arxiv.org/abs/1801.06040.[BR2:20] M.-T. Benameur and I. Roy,
The Higson-Roe sequence for ´etale groupoids. II. The universal sequence for equivariantfamilies , To appear in the J. Noncommutative Geometry, 2020. arXiv preprint: https://arxiv.org/abs/1812.04371.[B:71] I. D. Berg,
An extension of the Weyl-von Neumann theorem to normal operators . Transactions of the AmericanMathematical Society 160 (1971): 365-371.[BDF:73] L. G. Brown, R. G. Douglas and P. A. Fillmore,
Unitary equivlence modulo the compact operators and extensions of C ∗ -algebras , Proc. Conf. on Operator Theory, Springer Lecture Notes 345 (1973), 58-128.[BEKW:18] U. Bunke, A. Engel, D. Kasprowski and C. Winges, Equivariant coarse homotopy theory and coarse algebraicK-homology . arXiv preprint: arXiv:1710.04935[CC:00] A. Candel and L. Conlon
Foliations (Volume 1), Graduate studies in mathematics, American Mathematical Soc., 2000[CS:84] A. Connes , G. Skandalis,
The longitudinal index theorem for foliations.
Publ. Res. Inst. Math. Sci. 20 (1984), no. 6,1139–1183.[Dix:77] J. Dixmier,
Les C ∗ -algebres et leur representations , 1977.[GWY:16] E. Guentner, R. Willett and G. Yu, Dynamic asymptotic dimension and controlled operator K -theory ,arXiv:1609.02093, 2016[H:70] P. R. Halmos, Ten problems in Hilbert space.
Bull. Amer. Math. Soc. 76 (1970), 887-933.[Hig:95] N. Higson, C ∗ -algebra extension theory and duality , Journal of Functional Analysis, 129, 349-363, 1995[HPS:15] Hanke, B.; Pape, D.; Schick, T. Codimension two index obstructions to positive scalar curvature , Annales de l’InstitutFourier, Volume 65 (2015) no. 6, pp. 2681-2710.[HR:00] N. Higson and J. Roe,
Analytic K-homology , Oxford Mathematical Monographs, Oxford University Press, Oxford,2000.[HPR:97] N. Higson, E.K. Pedersen, J. Roe, C ∗ -algebras and controlled topology , K -theory 11 (1997) 209–239.[HR1:05] N. Higson and J. Roe, Mapping surgery to analysis. I. Analytic signatures.
K-Theory 33 (2005), no. 4, 277-299.[HR2:05] N. Higson and J. Roe,
Mapping surgery to analysis. II. Geometric signatures.
K-Theory 33 (2005), no. 4, 301-324.[HR3:05] N. Higson and J. Roe,
Mapping surgery to analysis. III. Exact sequences.
K-Theory 33 (2005), no. 4, 325-346.[HR:10] N. Higson and J. Roe,
K -homology, assembly and rigidity theorems for relative eta invariants.
Pure Appl. Math. Q.6 (2010), no. 2, Special Issue: In honor of Michael Atiyah and Isadore Singer, 555-601.[Ka:80] G. G. Kasparov,
Hilbert C*-modules: theorems of Stinespring and Voiculescu.
J. Operator Theory 4 (1980), no. 1,133–150.[Ka:81] G. G. Kasparov,
Operator K Functor and extension of C ∗ -algebras , Mathematics of the USSR-Izvestiya 16. 513(reprinted in English IOPScience, 2007)[Ka:88] G. G. Kasparov, Equivariant KK -theory and the Novikov conjecture , Invent. Math. 91, 147-201, 1988[La:95] E. Lance, Hilbert C ∗ -modules: a toolkit for operator algebraists , Lon. Math. Soc.Lec. Notes Series 210[LeGall:99] P.-Y. Le Gall, Theorie de Kasparov equivariante et groupoides I , K-theory, 16 (1999), 361-390.[Osin:16] D. Osin,
Acylindrically hyperbolic groups , Trans. Amer. Math. Soc. 368 (2016), no. 2, 851-888.[P:81] W. L. Paschke,
K-theory for commutants in the Calkin algebra.
Pacific J. Math. 95 (1981), no. 2, 427-434.[PPV:79] M. Pimsner, S. Popa and D. voiculescu,
Homogeneous C ∗ -extensions of C ( X ) ⊗ K ( H ) , Part I , J. Operator TheoryI (1979),55-108.[PS:13] P. Piazza and T. Schick, Rho-classes, index theory and Stolz’ positive scalar curvature sequence
Journal of Topology,7(4), 965-1004, 2013. [Roe:16] J. Roe,
Positive Curvature, Partial Vanishing Theorems and Coarse Indices . Proceedings of the Edinburgh Mathe-matical Society, 59(1), 223-233, 2016.[RS:81] J. Rosenberg and C. Schochet,
Comparing functors classifying extensions of C ∗ -algebras , J. Operator Theory, Vol. 5(1981), pp. 267-282.[STY:02] G. Skandalis, J.-L. Tu, and G. Yu, The coarse Baum–Connes conjecture and groupoids , Topology 41.4 (2002):807-834.[V:83] A. Valette,
A remark on the Kasparov groups Ext(A,B).
Pacific J. Math. 109 (1983), no. 1, 247-255.[Tu:99] J.-L. Tu,
La conjecture de Novikov pour les feuilletages hyperboliques , K theory 16: 129-184, 1999[V:76] D. Voiculescu,
A noncommutative Weyl-von Neumann theorem , Rev. Roum. Math. Pures Appl., 21 (1976), 97-113.[vN:35] J. von Neumann,
Charakterisierung des Spektrums eines Integraloperators , Actualit´es Sci. Indust., no. 229, Hermann,Paris, 1935.[W:09] H. Weyl, ¨Uber beschr¨ankte quadratischen Formen deren Differenz vollstetig ist , Rend. Circ. Mat. Palermo 27 (1909),373-392.[XY:14] Z. Xie, G. Yu
Positive scalar curvature, higher rho invariants and localization algebras , Advances in MathematicsVolume 262, 2014, Pages 823-866[Yu1:97] G. Yu,
The Novikov conjecture for groups with finite asymptotic dimension , Annals of Mathematics 147 (1998):325-355.[Yu2:00] G. Yu,
The coarse Baum–Connes conjecture for spaces which admit a uniform embedding into Hilbert space , Inven-tiones Mathematicae 139.1 (2000): 201-240.[Zen:16] R. Zeidler.
Positive scalar curvature and product formulas for secondary index invariants , J. Topol., 9(3):687–724,2016.[Z:19] V. F. Zenobi,
Adiabatic groupoid and secondary invariants in K-theory . Adv. Math. 347 (2019), 940-1001.
IMAG, Univ. Montpellier, CNRS, Montpellier, France
E-mail address : [email protected] Institute of Mathematical Sciences (HBNI), Chennai, India
E-mail address ::