Equivariant KK -theory for non-Hausdorff groupoids
aa r X i v : . [ m a t h . K T ] D ec Equivariant
K K -theory for non-Hausdorff groupoids
Lachlan E. MacDonaldDecember 2019
Abstract
We give a detailed and unified survey of equivariant KK -theory over locallycompact, second countable, locally Hausdorff groupoids. We indicate precisely howthe “classical” proofs relating to the Kasparov product can be used almost word-for-word in this setting, and give proofs for several results which do not currentlyappear in the literature. This article is intended as a detailed survey of the theory of groupoid-equivariant KK -theory, in the setting of locally compact, second countable, locally Hausdorff groupoids.Following a substantial period of innovation and development [18, 12, 35, 14, 19] inthe non-equivariant and group-equivariant cases, groupoid-equivariant KK -theory wasfirst treated, for Hausdorff groupoids, by Le Gall in his PhD thesis [24]. Since manyexamples of groupoids are only locally Hausdorff (for instance, the holonomy groupoidsof foliated manifolds [10]), it is desirable to have a groupoid-equivariant KK -theory for non-Hausdorff groupoids as well.Despite the demand arising from applications to foliation theory, the literature onequivariant KK -theory over non-Hausdorff groupoids has, up until this point, been rathersparse. The non-Hausdorff theory makes its first appearance in [36], in which, for detailson the inner workings of the theory, the reader is referred to the expositions given forHausdorff groupoids in [25, 37]. Since then the theory has been very rapidly expoundedupon, in an extremely general context that covers not only groupoids but Hopf algebras,in [1] for applications to singular foliations. Finally, the theory has found use in [27] inthe study of the Godbillon-Vey invariant of regular foliated manifolds.The common denominator in all of the treatments [36, 1, 27] of equivariant KK -theoryover non-Hausdorff groupoids is that each has been focussed on a particular applicationof equivariant KK -theory, rather than on a detailed development of the theory itself.As a consequence, most details (some of which are somewhat nontrivial) are skippedover, and it is difficult to glean a precise understanding of how the non-Hausdorffness ofthe underlying groupoid affects the “classical” definitions, statements of results, and theirproofs. The goal of the present paper is to rectify this state of the literature, by providing1 unified and detailed description of the theory. In particular, we include all the necessarydefinitions (of which there are many), and include detailed proofs for several results whichdo not yet appear in the literature. We will see, as remarked in [1], that much of thetheory can be proved in exactly the manner of the “classical” theory, provided one makessome conceptual substitutions.Let us briefly describe the content of the paper. The first three sections consistof the relevant definitions for groupoids, upper-semicontinuous bundles and actions ofgroupoids on algebras and Hilbert modules. Note that we do not go to the generality ofFell bundles, and we refer the reader to [37] and [36] for an indication of how the theoryworks at this level of generality. The bundle-theoretic approach we take to equivariant KK -theory is heavily influenced by [13, 38, 31], which are referenced throughout thepaper. The following two sections consist of definitions and results regarding equivariant KK -theory, its functoriality, and the Kasparov product. We have made a conscious choicenot to derive our proofs from those given by Le Gall [24, 25], whose thesis and paperon the topic can be relatively difficult to find. Instead we closely follow the “classical”theory as detailed in [18, 35, 14, 19], inspired by the very modern approach of [1]. Thefinal two sections are concerned with crossed products, both full and reduced, and theassociated descent map on equivariant KK -theory, for which we draw greatly on [21, 31].Throughout the exposition, we include a description of the unbounded picture [2] forfuture applications where a need to compute explicit representatives of Kasparov products[22, 26, 29, 17, 3, 30] and index formulae [11, 15, 6, 7, 8, 9, 4, 5] may arise. The resultsregarding the unbounded picture have already appeared, in the context of Lie groupoids,in [27], which itself draws on the theory outlined in [33] in the Hausdorff setting. I thank the Australian Federal Government for financing my doctoral research, of whichthe present survey forms a part, via a Research Training Program scholarship. Specialthanks go to A. Rennie, for his consistent support over the past several years, and to A.Carey, for encouraging me to write this article.
We begin with the basic definitions that we require regarding groupoids and their actions.
Definition 1.1. A groupoid ( G (0) , G , r, s, m, i ) consists of a set G , the total space , witha distinguished subset G (0) , the unit space , maps r, s : G → G (0) called the range and source respectively, a multiplication map m : G × s,r
G → G and an inversion map i : G → G such that . i satisfies i = id ,2. m is associative,3. r ◦ i = s .4. m ( u, x ) = u = m ( y, u ) for all u ∈ G with r ( u ) = y and s ( u ) = x .We say that G is locally compact, second-countable, and locally Hausdorff if itstotal space admits a locally compact, second-countable and locally Hausdorff topology forwhich all of the maps r, s, m, i are continuous, and for which r and s are open. We willalways assume in addition that the unit space G (0) is Hausdorff. A homomorphism ϕ : H → G of locally compact, second-countable, locally Hausdorff groupoids is a continuousmap of their total spaces which is compatible with the multiplication and inversion mapson each.
We will usually denote a groupoid ( G (0) , G , r, s, m, i ) by simply G , m ( u, v ) by simply uv , and i ( u ) by u − . Definition 1.2.
Let X be a topological space. A left action of G on X consists of acontinuous map p : X → G (0) called the anchor map , together with a continuous map α : G × s,p X → X , denoted α ( u, x ) α u ( x ) , such that1. p ( α u ( x )) = r ( u ) for all ( u, x ) ∈ G × s,p X ,2. α uv ( x ) = α u ( α v ( x )) for all ( v, x ) ∈ G × s,p X and ( u, v ) ∈ G (2) ,3. α p ( x ) ( x ) = x for all x ∈ X . We give a brief study of upper-semicontinuous bundles of spaces, algebras and modules.We refer primarily to the delightful books [13] and [38]. Concerning notation, for anymap f : Y → X of sets, we will for each x ∈ X denote by Y x the fibre f − { x } over x . Wewill moreover let K denote either C or R . Note that, just as in [19], all of what followsin this article is valid for both complex and real C ∗ -algebras and Hilbert modules. Thenotation C ( X ), defined for locally compact spaces X , is used to mean the continuous K -valued functions vanishing at infinity on X . Definition 2.1.
Suppose that X is a topological space. By an upper-semicontinuousBanach-bundle over X , we mean a topological space A together with an open surjectivemap p A : A → X and a K -Banach space structure on each fibre A x such that1. the map a
7→ k a k is upper semicontinuous from A to R (that is, for all ǫ > , { a ∈ A : k a k ≥ ǫ } is closed), . the map + : A × p A ,p A A → A given by ( a, b ) a + b is continuous,3. the map K × A → A given by ( λ, a ) λa is continuous,4. if { a i } is a net in A such that p A ( a i ) → x and k a i k → , then a i → x , where x isthe zero element in A x .A section of p A is a function β : X → A such that p A ◦ β = id X . The space of continuoussections of p A is denoted Γ( X ; A ) , the space of continuous sections that are bounded inthe norm topology denoted Γ b ( X ; A ) , and the space of continuous sections vanishing atinfinity in the norm topology on the fibres is denoted Γ ( X ; A ) .If Y is a topological space and φ : Y → X is a continuous map, we denote by φ ∗ A the pullback of A by φ to an upper-semicontinuous bundle over Y , whose fibre over y ∈ Y is A φ ( y ) , and for any continuous section a ∈ Γ( X ; A ) we denote by φ ∗ a its pullback to asection of φ ∗ A defined for y ∈ Y by the formula ( φ ∗ a )( y ) := a ( φ ( y )) . Some important special cases of upper-semicontinuous Banach bundes occur whenthe fibres are C ∗ -algebras, or Hilbert modules thereover. While upper-semicontinuousbundles of C ∗ -algebras are by now quite well-studied, their Hilbert module analogueshave not yet made an appearance in the literature. Definition 2.2.
An upper-semicontinuous Banach bundle p A : A → X is said to be an upper-semicontinuous C ∗ -bundle if each A x is a C ∗ -algebra, for which the multi-plication · : A × p A ,p A A → A is continuous. We say that A is Z -graded if it admits adirect sum decomposition A = A (0) ⊕ A (1) such that for each x ∈ X , A x = A (0) x ⊕ A (1) x is a Z -graded C ∗ -algebra (cf. [18, Section 2.1]).If p A : A → X is an upper-semicontinuous C ∗ -bundle, an upper-semicontinuousHilbert A -module bundle is an upper semicontinuous Banach bundle p E : E → X suchthat each E x is a Hilbert A x -module, with A x -valued inner product h· , ·i x , and for which themaps h· , ·i : E × p E ,p E E → A and E × p E ,p A A → E induced by the inner product and rightaction respectively are continuous. We say that such a bundle is Z -graded if it admitsa direct sum decomposition E = E (0) ⊕ E (1) such that for each x ∈ X , E x = E (0) x ⊕ E (1) x isa Z -graded Hilbert A x -module (cf. [18, Section 2.2]). Remark 2.3.
For the remainder of this paper, we will always assume our algebra andHilbert module bundles to be Z -graded. In particular, all C ∗ -algebras and Hilbert mod-ules are assumed to be Z -graded, and all tensor products and commutators are assumedto be Z -graded also (cf. [18, Section 2]). Remark 2.4.
For an upper-semicontinuous Banach bundle p B : B → X over an arbitrarytopological space X , there is no guarantee of a wealth of continuous sections of p B . When X is locally compact and Hausdorff, however, by results of Hofmann [16] (see also the4iscussion in [31, p. 16]), for any x ∈ X and b ∈ B x , one is guaranteed a section σ forwhich σ ( x ) = b . For the rest of this section we will always let X denote a locally compact,Hausdorff space .Observe that if p A : A → X is an upper-semicontinuous C ∗ -bundle, then the continu-ous sections vanishing at infinity Γ ( X ; A ) themselves form a C ∗ -algebra under pointwiseoperations and the supremum norm, and inherit a natural Z -grading from the fibres.Letting M ( A ) denote the associated upper-semicontinuous C ∗ -bundle whose fibre over x ∈ X is equal to M ( A x ), we can naturally identify M (Γ ( X ; A )) with the sectionalgebra Γ b ( X ; M ( A )). Note moreover that the C ∗ -algebra Γ ( X ; A ) associated to anyupper-semicontinuous C ∗ -bundle p A : A → X admits a nondegenerate homomorphism C ( X ) → Z M (Γ ( X ; A )) into the center of M ( A ), defined by pointwise scalar multipli-cation. That is, Γ ( X ; A ) is a C ( X ) -algebra in the following sense. Definition 2.5. A C ∗ -algebra A is called a C ( X ) -algebra if it admits a nondegeneratehomomorphism C ( X ) → Z M ( A ) . A C ∗ -homomorphism of C ( X ) -algebras is said to bea C ( X ) -homomorphism if it is in addition a homomorphism of C ( X ) -modules. Given any C ( X )-algebra A , for each x ∈ X one defines a C ∗ -algebra (and left A -module) A x := A/ ( I x · A ), where I x is the kernel of the evaluation functional f f ( x )on C ( X ). The resulting space A := F x ∈ X A x is topologised by [13, Theorem II.13.18]and, with the obvious projection p A : A → X , becomes an upper-semicontinuous C ∗ -bundle [38, Theorem C.25]. The C ( X )-algebra A is then C ( X )-isomorphic to Γ ( X ; A )[38, Theorem C.27]. Thus C ( X )-algebras are the same thing as sections of upper-semicontinuous C ∗ -bundles, and we will without further comment assume A to be iden-tified with its associated section algebra Γ ( X ; A ) for the remainder of the document.Notice that any C ( X )-homomorphism φ : A → B of C ( X )-algebras descends for each x ∈ X to a C ∗ -homomorphism φ x : A x → B x of the fibres.In a similar fashion, if A = Γ ( X ; A ) is a C ( X )-algebra and E is a Hilbert A -module,then for each x ∈ X one defines the Hilbert A x -module E x to be the balanced tensorproduct E ˆ ⊗ A A x . The space E := F x ∈ X E x may also be topologised by means of [13,Theorem II.13.18] to obtain an upper-semicontinuous Hilbert A -module bundle p E : E → X , where one proves the continuity of the A -valued inner products and of the right actionof A on E using the same estimates as in the verification of Axiom A3 in [38, TheoremC.25]. The continuous sections vanishing at infinity Γ ( X ; E ) of this bundle are equippedwith pointwise operations to furnish a Hilbert Γ ( X ; A )-module, to which E is canonicallyisomorphic as a Hilbert A -module. We will without further comment assume that E isidentified with its associated section space Γ ( X ; E ).Given such a Hilbert module E = Γ ( X ; E ), we have associated upper-semicontinuousbundles of C ∗ -algebras K ( E ) and L ( E ), whose fibres over x ∈ X are K ( E x ) and L ( E x )respectively. By the identification E = Γ ( X ; E ) we then have K ( E ) = Γ ( X ; K ( E ))5nd L ( E ) = Γ b ( X ; L ( E )). The natural C ( X )-module structures on these spaces definedby pointwise multiplication then agree with [19, Definition 1.5], and K ( E ) in particularbecomes a C ( X )-algebra when equipped with this structure.Let us now give some standard definitions, phrased in terms of upper-semicontinuousbundles. Definition 2.6.
Let A = Γ ( X ; A ) be a C ( X ) -algebra, and let E = Γ ( X ; E ) and E ′ =Γ ( X ; E ′ ) be Hilbert A -modules. The direct sum E ⊕ E ′ of E and E ′ is the Hilbert A -module Γ ( X ; E ⊕ E ′ ) , where E ⊕ E ′ is the upper-semicontinuous Hilbert A -module bundlewhose fibre over x ∈ X is the direct sum E x ⊕ E ′ x in the sense of [18, p. 518].If B = Γ ( X ; B ) is another C ( X ) -algebra and F = Γ ( X ; F ) is a Hilbert B -module,we say that a C ∗ -homomorphism π : A → L ( F ) is a C ( X ) -representation if it isadditionally a homomorphism of C ( X ) -modules. Given such a representation, the bal-anced tensor product E ˆ ⊗ A F is the Hilbert B -module Γ ( X ; E ˆ ⊗ A F ) , where E ˆ ⊗ A F isthe upper-semicontinuous Hilbert B -module bundle whose fibre over x ∈ X is the balancedtensor product E x ˆ ⊗ A x F x in the sense of [18, Section 2.8].Let Y be a locally compact Hausdorff space and let φ : Y → X be a continuousmap. The pullback of a C ( X ) -algebra A = Γ ( X ; A ) by φ is the C ( Y ) -algebra φ ∗ A :=Γ ( Y ; φ ∗ A ) . If E = Γ ( X ; E ) is a Hilbert A -module, then its pullback by φ is the Hilbert φ ∗ A -module φ ∗ E := Γ ( Y ; φ ∗ E ) . Remark 2.7.
The balanced tensor product and direct sum of Hilbert modules given inDefinition 2.6 are easily seen to coincide with the corresponding analytic notions. Forinstance, any element ξ ˆ ⊗ η of the balanced tensor product ( E ˆ ⊗ A F ) an in the sense of [18,Section 2.8] canonically determines an element ξ ˆ ⊗ η : x ξ ( x ) ˆ ⊗ η ( x ) of Γ ( X ; E ˆ ⊗ A F ),and the resulting map ( E ˆ ⊗ A F ) an → Γ ( X ; E ˆ ⊗ A F ) an isomorphism by [38, PropositionC.24].Regarding pullbacks, note that if φ : Y → X is a continuous map of locally compactHausdorff spaces, then C ( Y ) carries a C ( X )-module structure defined for f ∈ C ( X )and g ∈ C ( Y ) by ( f · g )( y ) := f ( φ ( y )) g ( y ) , y ∈ Y. As remarked in [21, Section 2.7 (d)], if A = Γ ( X ; A ) is a C ( X )-algebra, then thebalanced tensor product A ˆ ⊗ C ( X ) C ( Y ) (where C ( Y ) is of course assumed to be triviallygraded) identifies naturally with φ ∗ A via the map a ˆ ⊗ g g · φ ∗ a , where the · denotespointwise scalar multiplication. Thus Definition 2.6 agrees (up to isomorphism) with thedefinitions used in [24, 1]. In the same way, if E is a Hilbert A -module, then regarding φ ∗ A as a left A -module via multiplication, the balanced tensor product E ˆ ⊗ A φ ∗ A isisomorphic as a Hilbert φ ∗ A -module to φ ∗ E via the map sending ξ ˆ ⊗ ( g · φ ∗ a ) ∈ E ˆ ⊗ A φ ∗ A to g · φ ∗ ( ξ · a ) ∈ φ ∗ E (where now the · inside the brackets denotes the right action of A on E ). 6he final notion we will need is the pullback of an operator on a Hilbert module, whichis outlined in the generality of unbounded operators in [27, Section 2]. Let A = Γ ( X ; A )be a C ( X )-algebra, and let E = Γ ( X ; E ) be a Hilbert A -module. Let T : dom( T ) → E bean A -linear operator on E . For each x ∈ X , define dom ( T ) x := { ξ ( x ) : ξ ∈ dom( T ) } ⊂ E x and let T ( x ) : dom ( T ) x → E x be the A x -linear operator defined by T ( x ) ξ ( x ) := ( T ξ )( x ) , ξ ∈ dom( T ) . Defining dom ( T ) := S x ∈ X dom ( T ) x ⊂ E , we see that dom( T ) identifies with the subspaceΓ ( X ; dom ( T )) of Γ ( X ; E ) consisting of sections whose value at each x ∈ X is in dom ( T ) x .The pullback of T by a continuous map of locally compact Hausdorff spaces is then definedin the obvious way. Definition 2.8.
Let φ : Y → X be a continuous map of locally compact Hausdorffspaces, let A = Γ ( X ; A ) be a C ( X ) -algebra, and let E = Γ ( X ; E ) be a Hilbert A -module. If T : dom( T ) → E is an A -linear operator, then the pullback of T by φ isthe operator φ ∗ T : dom( φ ∗ T ) → φ ∗ E = Γ ( Y ; φ ∗ E ) defined on the domain dom( φ ∗ T ) :=Γ ( Y ; φ ∗ dom ( T )) ⊂ φ ∗ E by the formula (cid:0) φ ∗ T η (cid:1) ( y ) := T ( φ ( y )) η ( y ) ∈ E φ ( y ) , η ∈ dom( φ ∗ T ) . In particular, if T ∈ L ( E ) , then φ ∗ T ∈ L ( φ ∗ E ) . Finally, let us point out if T ∈ L ( E ) as in Definition 2.8, then φ ∗ T ∈ L ( φ ∗ E ) identifiesvia the map considered in Remark 2.7 with the operator T ˆ ⊗ E ˆ ⊗ A φ ∗ A that is usedin [24]. We will assume for the rest of the paper that G is a locally compact, second countable,locally Hausdorff groupoid with locally compact, Hausdorff unit space G (0) . The theoryof G -algebras was originally developed by Le Gall in [24] for Hausdorff G . In the locallyHausdorff setting it has been expounded upon in [21, 36, 31, 1]. We primarily follow thebundle-theoretic picture adopted in [31]. Definition 3.1.
Let A = Γ ( G (0) ; A ) be a C ( G (0) ) -algebra. An action α of G on A consists of a family { α u } u ∈G such that1. for each u ∈ G , α u : A s ( u ) → A r ( u ) is a degree 0 isomorphism of C ∗ -algebras,2. the map G × s,p A A → A defined by ( u, a ) α u ( a ) defines a continuous action of G on A . he triple ( A , G , α ) is called a groupoid dynamical system , and we say that A is a G -algebra and that it admits a G -structure . Remark 3.2.
By [31, Lemma 4.3], Definition 3.1 agrees with the definition given in[21, Section 3.2] using pullbacks over Hausdorff open subsets of G , and agrees with [24,Definition 3.1.1] when G is Hausdorff.The action of G on a Hilbert module over a G -algebra is defined in a similar way. Definition 3.3.
Let ( A , G , α ) be a groupoid dynamical system, and let E = Γ ( G (0) ; E ) bea Hilbert A -module. An action W of G on E consists of a family { W u } u ∈G such that1. for each u ∈ G , W u : E s ( u ) → E r ( u ) is a degree 0, isometric isomorphism of Banachspaces such that h W u e, W u f i r ( u ) = α u (cid:0) h e, f i s ( u ) (cid:1) for all e, f ∈ E s ( u ) , and2. the map G × s,p E E → E defined by ( u, e ) W u e defines a continuous action of G on E .The tuple ( E , A , G , W, α ) is called a Hilbert module representation . We then say that E is a G -Hilbert module and that E admits a G -structure . Conjugation by W givesrise to a continuous action ε : G × s,p L ( E ) L ( E ) → L ( E ) of G on the upper semicontinuousbundle L ( E ) , which in particular makes ( K ( E ) , G , ε ) a G -algebra. Remark 3.4.
The arguments of [31, Lemma 4.3] also show that Definition 3.3 agreeswith the notion given in [1, Section 4.2.4].
Definition 3.5.
Let A and B be G -algebras, with G -actions α and β respectively. Wesay that a homomorphism φ : A → B is a G -homomorphism if φ r ( u ) ◦ α u = β u ◦ φ s ( u ) for all u ∈ G . Let E be a G -Hilbert B -module, with action ε of G on L ( E ) . We say thata C ( G (0) ) -representation is a G -representation if for all u ∈ G we have ε u ◦ π s ( u ) = π r ( u ) ◦ α u . Such a representation makes E into a G -equivariant A - B -correspondence . Let us end this section by noting that if A is a G -algebra, and E and E ′ are two G -Hilbert A -modules with G -structures W and W ′ respectively, then the formula( W ⊕ W ′ ) u := W u ⊕ W ′ u : E s ( u ) ⊕ E ′ s ( u ) → E r ( u ) ⊕ E ′ r ( u ) , u ∈ G G -structure W ⊕ W ′ on their direct sum E ⊕ E ′ . Similarly, if B is another G -algebra and F is an equivariant A - B -correspondence with G -structure V , then theformula ( W ˆ ⊗ V ) u := W u ˆ ⊗ V u : E s ( u ) ˆ ⊗ A s ( u ) F s ( u ) → E r ( u ) ˆ ⊗ A r ( u ) F r ( u ) , u ∈ G defines a G -structure W ˆ ⊗ V on the balanced tensor product E ˆ ⊗ A F . When consideringdirect sums and balanced tensor products of Hilbert modules in what follows, we willalways consider them to be equipped with these G -structures without further comment. K K G -theory We assume from here on that all G -algebras and G -Hilbert modules are Z -graded, andcarry G -actions that are of degree 0 with respect to the Z -grading. We will moreoverassume all Hilbert modules to be countably generated. Given G -algebras A , B , thecorresponding G -actions will be denoted by the corresponding lower-case Greek letters α , β . Given any G -Hilbert module E , the corresponding action on L ( E ) given by conjugationwill be denoted by ε .Let us for the rest of this paper fix a countable base { U i } for the topology of G consisting of locally compact, Hausdorff open subsets, and define G ⊔ := G i U i . For each i let ι i : U i ֒ → G ⊔ denote the canonical (continuous) inclusion. Then G ⊔ is alocally compact Hausdorff space. We equip G ⊔ with the continuous maps r ⊔ : G ⊔ → G (0) and s ⊔ : G ⊔ → G (0) defined by r ⊔ ( u, i ) := r ( u ) , s ⊔ ( u, i ) := s ( u ) , ( u, i ) ∈ G ⊔ . We will use pullbacks over G ⊔ to treat pullbacks over all Hausdorff open subsets of G simultaneously. Note that the action α of G on any G -algebra A induces an isomorphism α ⊔ : s ∗⊔ A → r ∗⊔ A of C ∗ -algebras defined by the formula α ⊔ ( a ′ ( u, i )) := α u (cid:0) ( ι ∗ i a ′ )( u ) (cid:1) ∈ A r ( u ) , a ′ ∈ s ∗⊔ A. Similarly, if E is a G -Hilbert module, then the action ε of G by conjugation on L ( E ) inducesan isomorphism ε ⊔ : L ( s ∗⊔ E ) → L ( r ∗⊔ E ) of C ∗ -algebras defined by a similar formula. Definition 4.1.
Let A and B be G -algebras. A G -equivariant Kasparov A - B -module is a tuple ( E , F ) , where E is a G -equivariant A - B correspondence, and where F ∈ L ( E )9 s homogeneous of degree 1 such that for all a ∈ A one has1. a ( F − F ∗ ) ∈ K ( E ) ,2. a ( F − ∈ K ( E ) ,3. [ F, a ] ∈ K ( E ) , and4. a ′ (cid:0) ε ⊔ ( s ∗⊔ F ) − r ∗⊔ F (cid:1) ∈ r ∗⊔ K ( E ) for all a ′ ∈ r ∗⊔ A .We will sometimes refer to item 4. by saying that F is almost-equivariant . We saythat two G -equivariant Kasparov A - B -modules ( E , F ) and ( E ′ , F ′ ) are unitarily equiv-alent if there exists a degree 0 unitary isomorphism V : E → E ′ of G -equivaraint A - B -correspondences for which V F = F ′ V . We denote by E G ( A, B ) the set of all unitaryequivalence classes of G -equivariant Kasparov A - B -modules. Remark 4.2.
Notice of course that in the case where G is simply a point, E G ( A, B )coincides with the set E ( A, B ) of non-equivariant KK -theory [18, Definition 1]. Remark 4.3.
Definition 4.1 finds its origin in [1, Section 4.3.4], although the idea ofusing a disjoint union to treat locally Hausdorff groupoids dates back at least as earlyas [20]. It differs from that usually seen in the literature (for instance, [36, Definition10.10], [1, Definition 4.6], or [27, Definition 2.7]) in which one replaces item 4. with therequirement that for each Hausdorff open subset U of G , one has a ′ (cid:0) ε U ( s | ∗ U F ) − r | ∗ U F (cid:1) ∈ r | ∗ U K ( E ) (1)for all a ′ ∈ r | ∗ U A . It is clear that the usual requirement (1) implies item 4. in Definition4.1. On the other hand, by pulling back via the inclusions ι i : U i ֒ → G ⊔ , it is easily checkedthat any G -equivariant Kasparov module in the sense of Definition 4.1 is an equivariantKasparov module in the sense of (1). The advantage of Definition 4.1 is that it allows usto treat all Hausdorff open subsets of G at once, resulting in much cleaner notation andeasy extension of the “classical” proofs to the equivariant context.Notice that if B is a G -algebra, then the tensor product B ˆ ⊗ C ([0 , C ([0 , G -algebra - as a C ( G (0) ) algebra, it has fibre B x ˆ ⊗ C ([0 , x ∈ G (0) , and the action β of G on B extends trivially to the C ([0 , B ˆ ⊗ C ([0 , t ∈ [0 , t : C ([0 , → C denote theevaluation functional ev t ( f ) := f ( t ). Then to each t ∈ [0 ,
1] and each G -equivariant Kas-parov A - B ˆ ⊗ C ([0 , E , F ) is associated the G -equivariant Kasparov A - B mod-ule (ev t ) ∗ ( E , F ) := ( E ˆ ⊗ B ˆ ⊗ C ([0 , B, F ˆ ⊗ B is regarded as a left B ˆ ⊗ C ([0 , b ˆ ⊗ f ) · b ′ := ev t ( f ) bb ′ .10 efinition 4.4. Two G -equivariant Kasparov A - B -modules ( E , F ) and ( E , F ) are saidto be G - homotopic , written ( E , F ) ∼ h ( E , F ) , if there exists a G -equivariant Kasparov A - B ˆ ⊗ C ([0 , module ( E , F ) such that (ev i ) ∗ ( E , F ) = ( E i , F i ) for i = 0 , . The relation ∼ h is an equivalence relation, and we denote by KK G ( A, B ) the set of ∼ h -equivalenceclasses of elements in E G ( A, B ) . The following analogue of [35, Lemma 11] is essential for the uniqueness and associa-tivity of the Kasparov product at the level of the KK G groups. Its proof in the equivariantsetting does not currently appear in the literature. Lemma 4.5.
Let A and B be G -algebras, and let ( E , F ) , ( E , F ′ ) ∈ E G ( A, B ) . If for all a ∈ A one has a [ F, F ′ ] a ∗ ≥ modulo K ( E ) , then ( E , F ) and ( E , F ′ ) are G -homotopic.Proof. By the arguments of [35, Lemma 11], we obtain a positive operator P ≥ P, a ] ∈ K ( E ) for all a ∈ A , and an operator K for which Ka ∈ K ( E ) for all a ∈ A ,such that [ F, F ′ ] = P + K. As in [35, Lemma 11], for each t ∈ [0 , π/
2] we define the operator F t := (1 + cos( t ) sin( t ) P ) − (cos( t ) F + sin( t ) F ′ ) , and the pair ( E , F t ) satisfies items 1., 2., and 3. of Definition 4.1. All that remains tocheck to complete the proof of the lemma is that each F t is almost-equivariant.For t ∈ [0 , π/ F t := (1 + cos( t ) sin( t ) P ) − , and F t = (cos( t ) F + sin( t ) F ′ ).Note then that for a ∈ r ∗⊔ A , we can write aε ⊔ ( s ∗⊔ F t ) − r ∗⊔ F t = a (cid:0) ε ⊔ ( s ∗⊔ F t ) − r ∗⊔ F t (cid:1) ε ⊔ ( s ∗⊔ F t ) + ar ∗⊔ ( F t ) (cid:0) ε ⊔ ( s ∗⊔ F t ) − r ∗⊔ F t (cid:1) . (2)The second of these terms is contained in K ( E ) by the almost-equivariance of F and F ,together with the fact that P commutes with A up to K ( E ).To show that the first term is contained in K ( E ), observe first that, modulo K ( E ), wehave a ( ε ⊔ ( s ∗⊔ P ) − r ∗⊔ P ) = a ( ε ⊔ ( s ∗⊔ [ F, F ′ ]) − r ∗⊔ [ F, F ′ ])= a (cid:0) ( ε ⊔ ( s ∗⊔ F ) − r ∗⊔ F ) ε ⊔ ( s ∗⊔ F ′ ) + r ∗⊔ F ( ε ⊔ ( s ∗⊔ F ′ ) − r ∗⊔ F ′ )+ ( ε ⊔ ( s ∗⊔ F ′ ) − r ∗⊔ F ′ ) ε ⊔ ( s ∗⊔ F ) + r ∗⊔ F ′ ( ε ⊔ ( s ∗⊔ F ) − r ∗⊔ F ) (cid:1) which is contained in K ( E ) by the equivariance properties of F and F ′ together with thefact that both F and F ′ commute up to K ( E ) with A . That this implies that the firstterm in Equation (2) is contained in K ( E ) now follows from an elegant integral argument,which was brought to our attention by A. Rennie.11enote cos( t ) sin( t ) by c ( t ). Using the Laplace transform, we have the norm-convergentintegral formula ε ⊔ ( s ∗⊔ F t ) − r ∗⊔ F t = 1 √ π Z ∞ λ − (cid:0) exp (cid:0) − λ (1 + c ( t ) ε ⊔ ( s ∗⊔ P )) (cid:1) − exp (cid:0) − λ (cid:0) c ( t ) r ∗⊔ P ) (cid:1) dλ. By the fundamental theorem of calculus, this becomes1 √ π Z ∞ λ − Z dds exp (cid:0) − λ (cid:0) c ( t )( sε ⊔ ( s ∗⊔ P ) + (1 − s ) r ∗⊔ P )) (cid:1) ds dλ, which we easily compute to be − c ( t ) √ π ( ε ⊔ ( s ∗⊔ P ) − r ∗⊔ P ) Z ∞ λ Z exp (cid:0) − λ (1 + c ( t )( sε ⊔ ( s ∗⊔ P ) + (1 − s ) r ∗⊔ P ) (cid:1) ds dλ. Since a ( ε ⊔ ( s ∗⊔ P ) − r ∗⊔ P ) ∈ K ( E ), it follows that the first term in Equation (2) is containedin K ( E ), hence that ( E , F t ) ∈ E G ( A, B ) as required.As we would expect, KK G ( A, B ) is indeed an abelian group for all G -algebras A and B . In order to prove this, we must define the appropriate notion of degeneracy for oursetting. Definition 4.6.
An element ( E , F ) of E G ( A, B ) is said to be degenerate if for all a ∈ A one has1. a ( F −
1) = 0 ,2. [ F, a ] = 0 , and3. ε ⊔ ( s ∗⊔ F ) − r ∗⊔ F = 0 . For G -algebras A and B , we define the sum ( E ⊕ E ′ , F ⊕ F ′ ) of two elements ( E , F )and ( E ′ , F ′ ) in E G ( A, B ) in the same way as in [18, Definition 3]. The natural operator F ⊕ F ′ := F F ′ ! ∈ L ( E ⊕ E ′ )satisfies item 4. of Definition 4.1 by the corresponding properties of F and F ′ , so that( E ⊕ E ′ , F ⊕ F ′ ) ∈ E G ( A, B ). Proposition 4.7.
Under the sum of equivariant Kasparov modules, KK G ( A, B ) is anabelian group.Proof. For ( E , F ) ∈ E G ( A, B ), the explicit homotopy given in [18, Theorem 1] from(
E ⊕ ( − E ) , F ⊕ ( − F )) to a degenerate module is (up to a reparameterisation of [0 , π/ , G -homotopy by the almost-equivariance of F .12unctoriality of the equivariant Kasparov groups goes through just as in the classicalcase. Proposition 4.8.
Let A , B , and C be G -algebras.1. any G -homomorphism φ : A → C determines a homomorphism φ ∗ : KK G ( C, B ) → KK G ( A, B ) , of abelian groups,2. any G -homomorphism ψ : B → C determines a homomorphism ψ ∗ : KK G ( A, B ) → KK G ( A, C ) of abelian groups, and3. any homomorphism ϕ : H → G of locally compact, second-countable, locally Haus-dorff groupoids determines a homomorphism ϕ ∗ : KK G ( A, B ) → KK H ( ϕ ∗ A, ϕ ∗ B ) of abelian groups, where ϕ ∗ A = Γ ( H (0) ; ϕ ∗ A ) and ϕ ∗ B = Γ ( H (0) ; ϕ ∗ B ) are re-garded as H -algebras using ϕ in the obvious way.Proof. The maps φ ∗ and ψ ∗ are defined at the level of Kasparov modules just as in [18,Definition 4]. That they send equivariant Kasparov modules to equivariant Kasparovmodules follows in a routine manner from the equivariance of the maps φ and ψ . For thethird item, given ( E , F ) ∈ E G ( A, B ) we note that ϕ ∗ ( E , F ) := (Γ ( H (0) ; ϕ ∗ E ) , ϕ ∗ F ) definesan element of E H ( ϕ ∗ A, ϕ ∗ B ) because ϕ is a homomorphism, and ( E , F ) ϕ ∗ ( E , F )determines the stated map ϕ ∗ on KK -groups.We end the section with a final definition and result on unbounded representatives,which are slight modifications of those found in [33]. For a G -algebra A , we say that a(not necessarily norm-closed) subalgebra A ⊂ A is a ∗ - G -subalgebra if there exists somesubspace A ⊂ A which is preserved by the action of G , for which A x is a subalgebraof A x , and for which A can be written as a subalgebra of Γ ( G (0) ; A ) whose elementstake values in A . For instance, when G is a Lie groupoid and P → G (0) is a smoothsubmersion carrying a smooth G -action, one sees that A = C ( P ) is a G -algebra, whosefibre over x ∈ G (0) is C ( P x ). In this case, A := C ∞ ( P ) is a ∗ - G -subalgebra of A , withcorresponding fibre A x := C ∞ ( P x ) ⊂ C ( P x ) for each x ∈ X .13 efinition 4.9. Let A and B be G -algebras. An unbounded G -equivariant Kasparov A - B -module is a triple ( A , E , D ) , where A is a dense ∗ - G -subalgebra of A , E is a G -equivariant A - B -correspondence, and where D is a densely defined, B -linear, self-adjointand regular operator on E of degree 1 such that1. a (1 + D ) − ∈ K ( E ) for all a ∈ A ,2. for all a ∈ A , the operator [ D, a ] extends to an element of L ( E ) , and for f ∈ C c ( G ⊔ ) one has f r ∗⊔ a (cid:0) ε ⊔ ( s ∗⊔ D ) − r ∗⊔ D (cid:1) ∈ L ( r ∗⊔ E ) , and3. for all f ∈ C c ( G ⊔ ) , one has dom( r ∗⊔ D f ) = W ⊔ dom( s ∗⊔ D f ) , where W ⊔ : s ∗⊔ E → r ∗⊔ E is the isometric isomorphism of Banach spaces induced by the action of G on E . Proposition 4.10.
Let A and B be G -algebras. Every unbounded equivariant Kas-parov A - B -module ( A , E , D ) determines a G -equivariant Kasparov A - B -module ( E , D (1 + D ) − ) , defining a class [ D ] ∈ KK G ( A, B ) .Proof. By the same arguments as in the non-equivariant case [2], the pair ( E , D (1+ D ) − )satisfies items 1., 2. and 3. of Definition 4.1. The final item 4. in Definition 4.1 followsfrom [33, Th´eor`eme 6], with Pierrot’s G replaced with our G ⊔ . The characteristic feature of KK -theory is the Kasparov product. In [24], it is provedthat when G is Hausdorff, for all G -algebras A , D , B there is an associative and non-trivialproduct KK G ( A, D ) ⊗ KK G ( D, B ) → KK G ( A, B ) . The same is true when G is locally Hausdorff. In fact, using the ideas of [1, Section 4.3.4],this can be seen by direct substitution of concepts into the proofs used in [19]. We firstgive the groupoid-equivariant version of the the lemma [19, Lemma 1.4] on the existenceof quasi-invariant, quasi-central approximate units. Lemma 5.1 (Existence of quasi-central approximate units) . Let B be a G -algebra, A a σ -unital G -subalgebra of B , Y a σ -compact, locally compact Hausdorff space, and ϕ : Y → B a function satisfying:1. [ ϕ ( y ) , a ] ∈ A for all a ∈ A and y ∈ Y , and2. for all a ∈ A , the functions y [ ϕ ( y ) , a ] are norm-continuous on Y . hen there is a countable approximate unit { e i } for A with the following properties:1. lim i →∞ k [ e i , ϕ ( y )] k = 0 for all y ∈ Y , and2. lim i →∞ k α ⊔ ( s ∗⊔ e i ) − r ∗⊔ e i k = 0 in r ∗⊔ A .Proof. The proof given in [19, Lemma 1.4] applies immediately. One need only replaceKasparov’s G with our G ⊔ , and Kasparov’s g ( u i ) − u i , defined for g ∈ G , with our α u ( e i ( s ( u ))) − e i ( r ( u )), defined for u ∈ G .Kasparov’s Technical Theorem below now follows essentially “classically”. Theorem 5.2 (Kasparov’s Technical Theorem) . Let J be a σ -unital G -algebra, A and A σ -unital subalgebras of M ( J ) , with A a G -algebra. Let ∆ be a subset of M ( J ) which isseparable in the norm topology and whose commutators with A act as derivations of A ,and let ϕ ∈ M ( r ∗⊔ J ) . Assume that A · A ⊂ J , r ∗⊔ A · ϕ ⊂ r ∗⊔ J , and that ϕ · r ∗⊔ A ⊂ r ∗⊔ J .Then there are M , M ∈ M ( J ) of degree 0 such that M + M = 1 , and for which1. M i a i ∈ J for all a i ∈ A i and i = 1 , ,2. [ M i , d ] ∈ J for all d ∈ ∆ , and3. r ∗⊔ ( M ) · ϕ , ϕ · r ∗⊔ ( M ) and α ⊔ ( s ∗⊔ ( M i )) − r ∗⊔ ( M i ) are all contained in r ∗⊔ ( J ) , for i = 1 , .Proof. After making the same replacements as in the proof of Lemma 5.1, the proof of[19, Theorem 1.4] (which in this form is due to Higson [14]) applies without change.
Remark 5.3.
Note that by hypothesis, our ϕ ∈ M ( r ∗⊔ ( J )) in Theorem 5.2 alreadygives, for each Hausdorff open subset U of G , a section ϕ U ∈ M ( r | ∗ U J ) which is norm-continuous over U . This is to be contrasted with Kasparov’s weaker hypotheses on his ϕ : G → M ( J ) in [19, Theorem 1.4], where he requires only that ϕ be bounded andinduce norm-continuous functions g aϕ ( g ) and g ϕ ( g ) a for all a ∈ A + J . Aswe will see, our stronger hypothesis does not impact the proof of the existence of theKasparov product in our setting.We recall here the notion of a connection given by Connes and Skandalis [12]. Let A and B be C ∗ -algebras, and suppose that E is a Hilbert A -module, E is a Hilbert B -module, and that π : A → L ( E ) is a representation. Let E = E ˆ ⊗ A E , and for each ξ ∈ E we denote by T ξ ∈ L ( E , E ) defined by T ξ η := ξ ˆ ⊗ η, η ∈ E . The adjoint of T ξ is given on η ˆ ⊗ ζ ∈ E by T ∗ ξ ( η ˆ ⊗ ζ ) = π ( h ξ, η i ) ζ . F ∈ L ( E ), we say that an operator F ∈ L ( E ) is an F -connection for E if for all ξ ∈ E , one has T ξ F − ( − deg( ξ ) deg( F ) F T ξ ∈ K ( E , E ) , and F T ∗ ξ − ( − deg( ξ ) deg( F ) T ∗ ξ F ∈ K ( E , E ) . If E is countably generated and [ F , π ( A )] ⊂ K ( E ), then the algebra L ( E ) contains an F -connection for E [12, Proposition A.2].We can now define the Kasparov product of two equivariant Kasparov modules in ananalogous fashion to the non-equivariant case. Definition 5.4.
Let A , B and D be G -algebras, with A separable, ( E , F ) ∈ E G ( A, D ) ,and ( E , F ) ∈ E G ( D, B ) . Denote by E the bimodule E ˆ ⊗ D E . A pair ( E , F ) , where F ∈ L ( E ) , is called a Kasparov product of ( E , F ) and ( E , F ) , written F ∈ F ♯ D F ,if and only if1. ( E , F ) ∈ E G ( A, B ) ,2. F is an F -connection, and3. for all a ∈ A , a [ F ˆ ⊗ , F ] a ∗ ≥ modulo K ( E ) . Theorem 5.5.
Let A , B and D be G -algebras, with A separable, and suppose ( E , F ) ∈ E G ( A, D ) and ( E , F ) ∈ E G ( D, B ) . Let E := E ˆ ⊗ D E . Then F ♯ D F is nonempty,and path connected. Moreover the class in KK G ( A, B ) determined by any F ∈ F ♯ D F depends only on the G -homotopy classes of ( E , F ) and ( E , F ) in KK G ( A, D ) and KK G ( D, B ) respectively, and the Kasparov product descends to an associative, bilinearproduct KK G ( A, D ) ⊗ KK G ( D, B ) → KK G ( A, B ) .Proof. Choose an F -connection ˜ F for E . Let J denote the algebra K ( E ), A the G -algebra J + K ( E ) ˆ ⊗ A the algebra generated by ˜ F − ˜ F ∗ , ˜ F −
1, [ ˜ F , F ˆ ⊗
1] and[ ˜ F , A ], and let ∆ denote the subspace of L ( E ) generated by F ˆ ⊗
1, ˜ F and A . Finallydenote by ϕ the element α ⊔ ( s ∗⊔ ˜ F ) − r ∗⊔ ˜ F of L ( r ∗⊔ E ). Apply Theorem 5.2 to obtain M and M with the stated properties, and define F := M ( F ˆ ⊗
1) + M ˜ F . Exactly as in the non-equivariant case, the pair ( E , F ) satisfies properties 1., 2., and 3.,in Definition 4.1, and all that remains to verify that ( E , F ) ∈ E G ( A, B ) is to show that F is almost-equivariant. For any a ∈ r ∗⊔ A , however, we can write a · ( α ⊔ ( s ∗⊔ F ) − r ∗⊔ F ) ∈ ( r ∗⊔ E ) as the sum a · ( α ⊔ ( s ∗⊔ ( M )) − r ∗⊔ ( M )) α ⊔ ( s ∗⊔ ( F ˆ ⊗ a · r ∗⊔ ( M )( α ⊔ ( s ∗⊔ ( F ˆ ⊗ − r ∗⊔ ( F ˆ ⊗ a · ( α ⊔ ( s ∗⊔ ( M )) − r ∗⊔ ( M )) α ⊔ ( s ∗⊔ ( ˜ F )) + a · r ∗⊔ ( M )( α ⊔ ( s ∗⊔ ( ˜ F )) − r ∗⊔ ( ˜ F )) (3)The first, third and final terms in Equation (3) are contained in K ( E ) by item 3.of Theorem 5.2, while the second term in Equation (3) is contained in K ( E ) by theequivariance of the Kasparov module ( E , F ) together with items 1. and 2. of Theorem5.2. Thus ( E , F ) ∈ E G ( A, B ).Items b. and c. of Definition 5.4 hold by the same arguments as in the non-equivariantcase [35, Theorem 12] (cf. [12, Theorem A.5]). This gives existence. Path connectednessof F ♯ D F follows from the same argument given in [35, Theorem 12], using Lemma 4.5in the place of Skandalis’ [35, Lemma 11], giving uniqueness up to G -homotopy. That theKasparov product descends to a well-defined bilinear product on the KK G groups alsofollows from the arguments of [35, Theorem 12]. Associativity of this product followsfrom [35, Lemma 22], again using Lemma 4.5 in the place of [35, Lemma 11]. Remark 5.6.
With Theorem 5.5 in hand, one can now define the external product KK G ( A , B ˆ ⊗ D ) ⊗ KK G ( D ˆ ⊗ A , B ) → KK G ( A ˆ ⊗ A , B ˆ ⊗ B ) in the same manneras [19, Definition 2.12]. We now discuss crossed products of G -algebras as well as their relationship to KK G -theory. We will assume for this that G is equipped with a Haar system , whose definitionwe recall for the reader’s convenience.
Definition 6.1. A Haar system on G is a family { λ x } x ∈G (0) of measures on G suchthat each λ x is supported on G x , and is a regular Borel measure thereon, and such that1. for all v ∈ G and f ∈ C c ( G ) one has Z G f ( vu ) dλ s ( v ) ( u ) = Z G f ( u ) dλ r ( v ) ( u ) and,2. for each f ∈ C c ( G ) , the map x Z G f ( u ) dλ x ( u ) is continuous with compact support on G (0) . G admits a Haar system is a nontrivial requirement in general. Indeed, if G admits a Haar system then its range and source maps must be open [32, Proposition 2.2.1],and it is not difficult to find examples of locally compact (even Hausdorff) groupoids forwhich this is not the case [34, Section 3]. In the context of a foliated manifold, any choiceof leafwise half-density (see [10, Section 5]) determines a Haar system on the associatedholonomy groupoid.If p A : A → G (0) is an upper-semicontinuous Banach bundle, then for any Hausdorffopen subset U of G , we denote by Γ c ( U ; r ∗ A ) the space of compactly supported continuoussections of the bundle r ∗ A over U , extended by zero outside of U to a (not necessarilycontinuous) function on G . We define Γ c ( G ; r ∗ A ) to be the subspace of sections of r ∗ A over G that are spanned by elements of Γ c ( U ; r ∗ A ) as U varies over all Hausdorff opensubsets of G . Remark 6.2. If G is a Lie groupoid (such as the holonomy groupoid of a foliation), thenone will usually take smooth sections, denoted Γ ∞ c , instead of continuous sections. Proposition 6.3. [31, Proposition 4.4] If ( A , G , α ) is a groupoid dynamical system, thespace Γ c ( G ; r ∗ A ) is a ∗ -algebra with respect to the convolution product and adjoint givenrespectively by f ∗ g ( u ) := Z G f ( v ) α v (cid:0) g ( v − u ) (cid:1) dλ r ( u ) ( v ) f ∗ ( u ) := α u ( f ( u − ) ∗ ) for all f, g ∈ Γ c ( G ; r ∗ A ) and u ∈ G . Remark 6.4.
Note that one could also define a ∗ -algebra structure on the space Γ c ( G ; s ∗ A )obtained by pulling back A via the source map. In this case, the multiplication and in-volution are given by f ∗ g ( u ) := Z G α w (cid:0) f ( uw ) (cid:1) g ( w − ) dλ s ( u ) ( w ) , f ∗ ( u ) := α u − (cid:0) f ( u − ) ∗ (cid:1) . Moreover, the map f α ◦ f defines an isomorphism Γ c ( G ; s ∗ A ) → Γ c ( G ; r ∗ A ) of ∗ -algebras. While Γ c ( G ; s ∗ A ) is in a certain sense the more natural object to consider for left actions of groupoids, we choose to work with pullbacks over the range in accordancewith [27].We now obtain the full crossed product of a G -algebra as follows (cf. [21, Section 3.5],[31, p. 23]). Definition 6.5. If ( A , G , α ) is a groupoid dynamical system, one defines the I -norm ofan element f ∈ Γ c ( G ; r ∗ A ) by k f k I := max (cid:26) sup x ∈G (0) Z G k f ( u ) k x dλ x ( u ) , sup x ∈G (0) Z G k f ( u ) k r ( u ) dλ x ( u ) (cid:27) . he I -norm makes Γ c ( G ; r ∗ A ) into a normed ∗ -algebra, and the full crossed product A ⋊ G is the associated enveloping C ∗ -algebra. More specifically, A ⋊ G is the completionof Γ c ( G ; r ∗ A ) in the norm k f k A ⋊ G := sup {k π ( f ) k : π is a k · k I -decreasing ∗ -representation of Γ c ( G ; r ∗ A ) } . The reduced C ∗ -algebra of G is obtained from the algebra C c ( G ) by completing itwith respect to the norm obtained from the canonical family of ∗ -representations π x : C c ( G ) → L ( L ( G x )), x ∈ G (0) , called the regular representation. For a groupoid dynamicalsystem ( A , G , α ) the situation is slightly more complicated - namely, we must completethe convolution algebra Γ c ( G ; r ∗ A ) with respect to the norm obtained from a particularfamily Hilbert modules constructed from A . Proposition 6.6.
Let ( A , G , α ) be a groupoid dynamical system. Then for each x ∈ G (0) ,the completion L ( G x ; r ∗ A ) of Γ c ( G x ; r ∗ A ) in the A x -valued inner product h ξ, η i x := ( ξ ∗ ∗ η )( x ) = Z G α u (cid:0) ξ ( u − ) ∗ η ( u − ) (cid:1) dλ x ( u ) , defined for ξ, η ∈ Γ c ( G x ; r ∗ A ) , is a Hilbert A x -module, with right A x -action given by ( ξ · a )( u ) := ξ ( u ) α u ( a ) , u ∈ G x for all ξ ∈ Γ c ( G x ; r ∗ A ) and a ∈ A x .Moreover, for each x ∈ G (0) , a representation π x : Γ c ( G ; r ∗ A ) → L (cid:0) L ( G x ; r ∗ A ) (cid:1) isdefined by the formula π x ( f ) ξ ( u ) := f ∗ ξ ( u ) = Z G f ( v ) α v (cid:0) ξ ( v − u ) (cid:1) dλ r ( u ) ( v ) , u ∈ G x for f ∈ Γ c ( G ; r ∗ A ) and ξ ∈ Γ c ( G x ; r ∗ A ) . The family { π x } x ∈G (0) is called the regularrepresentation associated to ( A , G , α ) .Proof. It is clear from inspection that for x ∈ G (0) , ξ, η ∈ Γ c ( G x ; r ∗ A ) and a ∈ A x , theformulae defining h ξ, η i x and ( ξ · a ) make sense. Moreover, we have h ξ, η i ∗ x = ( ξ ∗ ∗ η ) ∗ ( x ) = ( η ∗ ∗ ξ )( x ) = h η, ξ i x since ∗ is an involution, and h ξ, η · a i x = Z G α u (cid:0) ξ ( u − ) ∗ ( η · a )( u − ) (cid:1) dλ x ( u )= Z G α u (cid:0) ξ ( u − ) ∗ η ( u − ) α u − ( a ) (cid:1) dλ x ( u ) = h ξ, η i x a α u − = α − u as C ∗ -isomorphisms. Consequently [23, p. 4], the completion L ( G x ; r ∗ A )is indeed a Hilbert A x -module.The extension of π x to a representation π x : Γ c ( G ; r ∗ A ) → L (cid:0) L ( G x ; r ∗ A ) (cid:1) followsfrom the argument of Khoshkam and Skandalis [21, 3.6]. Specifically, Khoshkam andSkandalis show that for ˜ f ∈ Γ c ( G ; s ∗ A ) (see Remark 6.4) and ˜ ξ ∈ Γ c ( G x ; A x ), the formula˜ π x ( ˜ f ) ξ ( u ) := Z G α w (cid:0) ˜ f ( uw ) (cid:1) ˜ ξ ( w − ) dλ s ( u ) ( w ) , u ∈ G x extends to a representation ˜ π x : Γ c ( G ; s ∗ A ) → L (cid:0) L ( G x ; A x ) (cid:1) , where L ( G x ; A x ) is theHilbert A x -module that is the completion of Γ c ( G x ; A x ) in the A x -valued inner product h ˜ ξ, ˜ η i x := R G ˜ ξ ( u − ) ∗ ˜ η ( u − ) dλ x ( u ). Observe that the map U x : Γ c ( G x ; A x ) → Γ c ( G x ; r ∗ A )defined by (cid:0) U x ˜ ξ (cid:1) ( u ) := α u (cid:0) ˜ ξ ( u ) (cid:1) satisfies (cid:0) U x ( ˜ ξ · a ) (cid:1) ( u ) = α u ( ˜ ξ ( u )) α u ( a ) = (cid:0) U x ˜ ξ (cid:1) · a ( u ) for all a ∈ A x , ˜ ξ ∈ Γ c ( G x ; A x ) and u ∈ G x , and satisfies h U x ˜ ξ, η i x = Z G α u (cid:0) ( U x ˜ ξ )( u − ) ∗ η ( u − ) (cid:1) dλ x ( u ) = Z G ˜ ξ ( u − ) ∗ α u (cid:0) η ( u − ) (cid:1) dλ x ( u )= Z G ˜ ξ ( u − ) ∗ (cid:0) U − x η (cid:1) ( u − ) dλ x ( u ) = h ˜ ξ, U − x η i x for all ˜ ξ ∈ Γ c ( G x ; A x ) and η ∈ Γ c ( G x ; r ∗ A ). Thus it extends to a unitary isomorphism U x : L ( G x ; A x ) → L ( G x ; r ∗ A ) of Hilbert A x -modules. Now we observe that (cid:0) U x ◦ ˜ π x ( ˜ f ) ◦ U ∗ x (cid:1) ξ ( u ) = α u (cid:18)(cid:0) ˜ π x ( ˜ f ) ◦ U ∗ x (cid:1) ξ ( u ) (cid:19) = α u (cid:18) Z G α w (cid:0) ˜ f ( uw ) (cid:1)(cid:0) U ∗ x ξ (cid:1) ( w − ) dλ x ( w ) (cid:19) = α u (cid:18) Z G α w (cid:0) ˜ f ( uw ) (cid:1) α w (cid:0) ξ ( w − ) (cid:1) dλ x ( w ) (cid:19) = Z G α uw (cid:0) ˜ f ( uw ) (cid:1) α uw (cid:0) ξ ( w − ) (cid:1) dλ x ( w )= Z G α v (cid:0) ˜ f ( v ) (cid:1) α v (cid:0) ξ ( v − u ) (cid:1) dλ r ( u ) ( v ) = π x (cid:0) α ◦ ˜ f (cid:1) ξ ( u )for any ξ ∈ Γ c ( G x ; r ∗ A ), so that π x is unitarily equivalent to the representation ˜ π x and isconsequently itself a representation by [21, Section 3.6].Completing in the norm obtained from the Hilbert module representations of Propo-sition 6.6 we obtain the reduced crossed product algebra.20 efinition 6.7. The completion A ⋊ r G of Γ c ( G ; r ∗ A ) in the norm k f k A ⋊ r G := sup x ∈G (0) k π x ( f ) k L ( G x ; r ∗ A ) is a C ∗ -algebra called the reduced crossed product algebra associated to the dynamicalsystem ( A , G , α ) . Remark 6.8.
The proof of Proposition 6.6 together with [21, Section 3.7] implies thatthe C ∗ -algebra defined in Definition 6.7 coincides with that given in [21]. Remark 6.9.
It is important to note that, for any G -algebra A , the calculations of [21,Section 3.6] show that k · k A ⋊ r G ≤ k · k I . Consequently, we have that k · k A ⋊ r G ≤ k · k A ⋊ G (cf. Definition 6.5). Remark 6.10.
In doing calculations with crossed product C ∗ -algebras, one usually mustwork with the algebra Γ c ( G ; r ∗ A ) associated to the upper-semicontinuous C ∗ -bundle A .In working with this algebra, it is frequently necessary to utilise a finer topology than the(reduced or full) norm topology. If p B : B → G (0) is an upper-semicontinuous Banachbundle, one says that a net z λ in Γ c ( G ; r ∗ B ) converges in the inductive limit topology to z ∈ Γ c ( G ; r ∗ B ) if z λ → z uniformly, and if there is a compact set C in G such that z and all of the z λ vanish off C [31, Remark 3.9]. It is then clear that if z λ converges to z in the inductive limit topology, then z λ also converges to z in the I -norm topology.Consequently z λ → z in both the full and reduced C ∗ -norm topologies. We now prove that Kasparov’s descent map [19] exists and continues to function asexpected in the equivariant setting for locally Hausdorff groupoids. While the relevantdefinitions in the Hausdorff case [24] are relatively simple using balanced tensor products,in the locally Hausdorff case one must use the bundle picture, which requires slightly morework.Suppose that we are given a G -algebra ( B = Γ ( G ; r ∗ B ) , β ) and a G -Hilbert B -module ( E = Γ ( G (0) ; r ∗ E ) , W ). Observe that Γ c ( G ; r ∗ E ) admits the Γ c ( G ; r ∗ B )-valuedinner product h· , ·i G defined for ξ, ξ ′ ∈ Γ c ( G ; r ∗ E ) h ξ, ξ ′ i G ( u ) := Z G β v (cid:0) h ξ ( v − ) , ξ ′ ( v − u ) i s ( v ) (cid:1) dλ r ( u ) ( v ) , u ∈ G , which is positive by [31, Proposition 6.8] and the argument given in [38, p. 116]. Thespace Γ c ( G ; r ∗ E ) also admits a right action of Γ c ( G ; r ∗ B ) given by( ξ · f )( u ) := Z G ξ ( v ) · β v (cid:0) f ( v − u ) (cid:1) dλ r ( u ) ( v ) , u ∈ G ξ ∈ Γ c ( G ; r ∗ E ) and f ∈ Γ c ( G ; r ∗ B ), with · denoting the (fibrewise) right action of B on E . By completing in the norm attained from either the reduced or full crossed productwe obtain [23, p. 4] a Hilbert module. Definition 7.1.
Given a G -algebra ( B = Γ ( G (0) ; B ) , β ) and a G -Hilbert B -module ( E =Γ ( G (0) ; r ∗ E ) , W ) , we define E ⋊ G and E ⋊ r G to be the completions of Γ c ( G ; r ∗ E ) in thenorms k ξ k E ⋊ G := kh ξ, ξ i G k B ⋊ G , k ξ k E ⋊ r G := kh ξ, ξ i G k B ⋊ r G respectively. The space E ⋊ G is a Hilbert B ⋊ G -module, while the space E ⋊ r G is aHilbert B ⋊ r G -module, which we will refer to respectively as the full and reducedcrossed products of E by G . Remark 7.2. If B is a G -algebra and E a Hilbert B -module, then since the reduced C ∗ -norm on Γ c ( G ; r ∗ B ) is bounded by the full norm, we see that the reduced norm onΓ c ( G ; r ∗ E ) is bounded by the full norm also.We pull back operators on equivariant modules to their crossed products as follows. Proposition 7.3.
Let ( B = Γ ( G (0) ; B ) , β ) be a G -algebra and let ( E = Γ ( G (0) ; E ) , W ) be a G -Hilbert B -module. Given T ∈ L ( E ) = Γ b ( X ; L ( E )) , the operator r ∗ ( T ) defined on Γ c ( G ; r ∗ E ) defined by (cid:0) r ∗ ( T ) ξ (cid:1) ( u ) := T ( r ( u )) ξ ( u ) extends to an operator r ∗ ( T ) ∈ L ( E ⋊ r G ) , and consequently to an operator r ∗ ( T ) ∈L ( E ⋊ G ) also.Proof. That r ∗ ( T ) is Γ c ( G ; r ∗ B )-linear is clear by the B -linearity of T . Moreover, for any ξ ∈ Γ c ( G ; r ∗ E ) we have h r ∗ ( T ) ξ, r ∗ ( T ) ξ i G ( u ) = Z G β v (cid:0) h T ( s ( v )) ξ ( v − ) , T ( s ( v )) ξ ( v − u ) i s ( v ) (cid:1) dλ r ( u ) ( v ) ≤ Z G k T ( s ( v )) k E s ( v ) β v (cid:0) h ξ ( v − ) , ξ ( v − u ) i s ( v ) (cid:1) dλ r ( u ) ( v ) ≤ sup x ∈G (0) k T ( x ) k E x Z G β v (cid:0) h ξ ( v − ) , ξ ( v − u ) (cid:1) dλ r ( u ) ( v ) , from which we deduce that k r ∗ ( T ) ξ k E ⋊ r G ≤ k T k L ( E ) k ξ k E ⋊ r G . Consequently, r ∗ ( T ) ex-tends to a bounded, B ⋊ r G -linear operator on all of E ⋊ r G . Remark 7.2 then tells usthat r ∗ ( T ) also extends to a bounded B ⋊ G -linear operator on E ⋊ G . It is then easilychecked in both cases that r ∗ ( T ) is adjointable, with adjoint r ∗ ( T ∗ ). Remark 7.4.
The identifications of Remark 2.7 show that when G is Hausdorff, our r ∗ ( T )defined on E ⋊ r G and E ⋊ G agrees with T ˆ ⊗ E ˆ ⊗ B ( B ⋊ r G ) and E ˆ ⊗ B ( B ⋊ G )respectively. 22 emark 7.5. If, in the notation of Proposition 7.3, one has an unbounded B -linearoperator T : dom( T ) → E , one defines dom( r ∗ ( T )) := Γ c ( G ; r ∗ dom ( T )), where dom ( T )is exactly as in Definition 2.8, and obtains r ∗ ( T ) : dom( r ∗ ( T )) → E ⋊ r G and r ∗ ( T ) :dom( r ∗ ( T )) → E ⋊ G using the same formula as in Proposition 7.3.That the crossed product of an equivariant A - B -correspondence is an A ⋊ r G - B ⋊ r G -correspondence turns out to require some mildly nontrivial argumentation, which doesnot currently appear in the literature. We give a full proof below. Proposition 7.6.
Let ( A = Γ ( G (0) ; A ) , α ) and ( B = Γ ( G (0) ; B ) , β ) be G -algebras andlet ( E = Γ ( G (0) ; E ) , W ) be a G -Hilbert module. Then if π : A → L ( E ) is an equivariantrepresentation, the formula ( f · ξ )( u ) := Z G π r ( v ) ( f ( v )) W v (cid:0) ξ ( v − u ) (cid:1) dλ r ( u ) ( v ) , u ∈ G defined for f ∈ Γ c ( G ; r ∗ A ) and ξ ∈ Γ c ( G ; r ∗ E ) determines representations π ⋊ r G : A ⋊ r G → L ( E ⋊ r G ) and π ⋊ G : A ⋊ G → L ( E ⋊ G ) .Proof. First observe that π ⋊ r G is ∗ -preserving in the sense that for any f ∈ Γ c ( G ; r ∗ A ),and ξ, ξ ′ ∈ Γ c ( G ; r ∗ E ), the element h f · ξ, ξ ′ i G ( u ) of B r ( u ) is equal to Z G Z G β v (cid:0) h π s ( v ) ( f ( w )) W w (cid:0) ξ ( w − v − ) (cid:1) , ξ ′ ( v − u ) i s ( v ) (cid:1) dλ s ( v ) ( w ) dλ r ( u ) ( v )= Z G Z G β v (cid:0) h W w (cid:0) ξ ( w − v − ) (cid:1) , π s ( v ) ( f ( w ) ∗ ) ξ ′ ( v − u ) i s ( v ) (cid:1) dλ s ( v ) ( w ) dλ r ( u ) ( v )= Z G Z G β vw (cid:0) h ξ ( w − v − ) , ε w − (cid:0) π s ( v ) ( f ( w ) ∗ ) (cid:1) W w − (cid:0) ξ ′ ( v − u ) (cid:1) i s ( w ) (cid:1) dλ s ( v ) ( w ) dλ r ( u ) ( v )= Z G Z G β v (cid:0) h ξ ( v − ) , π s ( v ) ( f ∗ ( w )) W w (cid:0) ξ ′ ( w − v − u ) (cid:1) i s ( v ) (cid:1) dλ s ( v ) ( w ) dλ r ( u ) ( v )= h ξ, ( f ∗ · ξ ′ ) i G ( u )for all u ∈ G . Here we have used the equivariance of the representation π and thesubstitutions v := vw and w := w − in going from the third line to the fourth.To prove that f f · extends to a homomorphism A ⋊ r G → L ( E ⋊ r G ) we must showthat k f · ξ k ≤ k f k A ⋊ r G k ξ k E ⋊ r G for all f ∈ Γ c ( G ; r ∗ A ) and ξ ∈ Γ c ( G ; r ∗ E ). Because E , A and B are in general different bundles, we cannot use the techniques of [28, Theorem1.4] or its analogue in the non Hausdorff case [36]; nor can we use the techniques of[31, Section 8] since we are working with reduced crossed products and not full crossedproducts. Observe, however, that if p : E → π ( A ) E denotes the family of projections p x : E x → π x ( A x ) E x , x ∈ G (0) , π x ( a ) e = π x ( a )( p x e ) for all x ∈ G (0) , a ∈ A x and e ∈ E x . Then for any f ∈ Γ c ( G ; r ∗ A )and ξ ∈ Γ c ( G ; r ∗ E ) we compute( f · ξ )( u ) = Z G π r ( v ) ( f ( v )) W v (cid:0) ξ ( v − u ) (cid:1) dλ r ( u ) ( v )= Z G W v (cid:18) ε v − (cid:0) π r ( v ) ( f ( v )) (cid:1)(cid:0) ξ ( v − u ) (cid:1)(cid:19) dλ r ( u ) ( v ) . Using the equivariance of π , we then see that( f · ξ )( u ) = Z G W v (cid:0) π s ( v ) ( α v − ( f ( v ))) ξ ( v − u ) (cid:1) dλ r ( u ) ( v )= Z G W v (cid:0) π s ( v ) ( α v − ( f ( v ))) p s ( v ) ξ ( v − u ) (cid:1) dλ r ( u ) ( v )= Z G π r ( v ) ( f ( v )) W v (cid:0) p s ( v ) ξ ( v − u ) (cid:1) dλ r ( u ) ( v )=( f · ( r ∗ ( p ) ξ ))( u ) , so we can assume without loss of generality that ξ ∈ Γ c ( G ; r ∗ ( p E )). Since the rep-resentation π of A on p E is (fibrewise) nondegenerate, by [31, Proposition 6.8] thereexists a sequence ( e i ) i ∈ N in Γ c ( G ; r ∗ A ) such that for any ξ ∈ Γ c ( G ; r ∗ ( p E )), the sequence( π ( e i ) ξ ) i ∈ N converges to ξ in the inductive limit topology on Γ c ( G ; r ∗ E ). Thus we canassume without loss of generality that ξ is of the form g · ξ ′ for g ∈ Γ c ( G ; r ∗ A ) and ξ ′ ∈ Γ c ( G ; r ∗ ( p E )). For such ξ we estimate h f · ξ, f · ξ i G = h ξ ′ , ( g ∗ ∗ f ∗ ∗ f ∗ g ) · ξ ′ i G ≤k f k A ⋊ r G h ξ ′ , ( g ∗ ∗ g ) · ξ ′ i G = k f k A ⋊ r G h ξ, ξ i G . Hence k f · ξ k E ⋊ r G ≤ k f k A ⋊ r G k ξ k E ⋊ r G , so f f · does indeed define a homomorphism π ⋊ r G : A ⋊ r G → L ( E ⋊ r G ). By Remark7.2, f f · also extends to a homomorphism π ⋊ G : A ⋊ G → L ( E ⋊ G ).Observe now that if ( E , F ) is a G -equivariant Kasparov A - B -module, then we canform the pairs ( E ⋊ G , r ∗ ( F )) and ( E ⋊ r G , r ∗ ( F )). It is in this way that we obtain thefollowing theorem. Theorem 7.7.
Let A and B be G -algebras. For any G -equivariant Kasparov A - B -module ( E , F ) , we have ( E ⋊ G , r ∗ ( F )) ∈ E ( A ⋊ G , B ⋊ G ) and ( E ⋊ r G , r ∗ ( F )) ∈ E ( A ⋊ r G , B ⋊ r G )24 see Remark 4.2). The induced maps j G : KK G ( A, B ) → KK ( A ⋊ G , B ⋊ G ) , j G r : K G ( A, B ) → KK ( A ⋊ r G , B ⋊ r G ) are homomorphisms of abelian groups, and are compatible with the Kasparov product inthe sense that if x ∈ KK G ( A, C ) and y ∈ KK G ( C, B ) , then we have j G ( x ⊗ C y ) = j G ( x ) ⊗ C ⋊ G j G ( y ) , j G r ( x ⊗ y ) = j G r ( x ) ⊗ C ⋊ r G j G r ( y ) . Proof.
That E ⋊ G and E ⋊ r G are countably generated can be seen by taking a countableapproximate identity { f i } i ∈ N for C c ( G ) in the inductive limit topology and a countablegenerating set { e j } j ∈ N for Γ ( X ; E ). Then, letting · denote pointwise multiplication, thecountable collection { f i · r ∗ e j } i,j ∈ N is a generating set for both E ⋊ G and E ⋊ r G (notethat since each f i is a finite sum of functions that are continuous and supported inHausdorff open subsets, so too is each f i · r ∗ e j ). We have already proved in Proposition7.3 that the operator r ∗ ( F ) makes sense in both L ( E ⋊ G ) and L ( E ⋊ r G ). The resultis otherwise proved in the same manner as in [19, Theorem 3.11], once one replacesKasparov’s C c ( G, A ), C c ( G, E ) and C c ( G, K ( E )) with our Γ c ( G ; r ∗ A ), Γ c ( G ; r ∗ E ) andΓ c ( G ; r ∗ K ( E )) respectively.We end the paper by noting that the descent map can also be applied to an unboundedequivariant Kasparov module, to yield an unbounded Kasparov module for the associatedcrossed product algebras. Proposition 7.8.
Let A and B be G -algebras, and let ( A , E , D ) be a G -equivariant un-bounded Kasparov A - B -module. Let A ⊂ A be as in Definition 4.9. Then (Γ c ( G ; r ∗ A ) , E ⋊ G , r ∗ ( D )) , (Γ c ( G ; r ∗ A ) , E ⋊ r G , r ∗ ( D )) are, respectively, an unbounded Kasparov A ⋊ G - B ⋊ G -module and an unbounded Kasparov A ⋊ r G - B ⋊ r G -module.Proof. Here r ∗ ( D ) means the closure of the operator r ∗ ( D ) defined in Remark 7.5. Thesame arguments used in [27, Proposition 2.11] now apply, provided one switches out thehalf-densities therein for a choice of Haar system on G . References [1] I. Androulidakis and G. Skandalis,
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