Structure and K-theory of crossed products by proper actions

STR UCTURE AND K-THEOR Y OF CR OSSED PR ODUCTS BY PR OPER ACTI ONS SIEGFRIED ECHTERHOFF AND HEA TH EMERSON Abstract. W e study the C*-algebra crossed product C 0 ( X ) ⋊ G of a locally compact group G a cting properl y on a lo cally compact Hausdorﬀ space X . Under some mild extra conditions, which are automat ic if G is discrete or a Lie group, we describ e in detail, and in terms of the action, t he pr imitive i deal space of such crossed pro ducts as a topological space, i n particular with r espect to its ﬁbring ov er the quotien t space G \ X . W e also give some r esults on the K-theory of such C*-algebras. These more or less compute the K-theory in the case of isolated orbits w i th non-trivial (ﬁnite) stabilizers. W e also give a purely K-theoretic proof of a result due to Pau l Baum and Alain Connes on K-theory with co mplex co eﬃcient s of crossed pro ducts b y ﬁnite groups. 0. I ntroduction The C*-a lg ebra crossed pro ducts C 0 ( X ) ⋊ G asso ciated to ﬁnite group a ctions on smo o th co mpact manifolds give the simplest non-trivia l examples of noncom- m utative spaces. Such g roup actions also play a role in v arious other parts of mathematics a nd physics, e.g. the linea r a ction of a W eyl group o n a complex torus in r epresentation theor y . As s o on as the actio n o f G on X is no t free, the primitive idea l s pace of the crossed pr o duct C 0 ( X ) ⋊ G is non-Hausdor ﬀ, although the quotient space G \ X is Hausdorﬀ. In fact Prim  C 0 ( X ) ⋊ G  ﬁbres ov er G \ X , in a canonica l wa y , with ﬁnite ﬁbres. As a ﬁbration of s ets this is ea sy enough to describ e in direct, geometric terms, and it is ‘well-kno wn to exp erts’: the primitive ideal s pace is in a natura l set bijection with G \ Stab( X ) b , where Stab( X ) b = { ( x, π ) | x ∈ X , π ∈ b G x } , and the ﬁrst c o ordinate pro jection deﬁnes the requir ed ﬁbration; the ﬁbre ov er Gx is th us the unitary dual b G x of the stabilizer G x . How ever, for purpos es, for e x ample, of K- theo ry computation, what is wan ted here is a description o f the open sets of G \ Stab( X ) b which c orresp ond to the op en subsets of the primitive ideal space equipp e d with the F ell top ology , sinc e t his des crib es the space of ideals of C 0 ( X ) ⋊ G and p otentially leads to a metho d of K-theo ry computation using excis io n. This description is the main contribution of this article. Prop er actions of gener al loca lly compact gr oups naturally generalize actions o f compact, or ﬁnite, groups – b e c ause every pr op er action is ‘lo cally induced’ from actions by compact s ubg roups. Such cros s ed pro ducts are impo rtant in oper ator algebras , b eca use of the Baum-Co nnes conjecture. F or an amenable, loc a lly com- pact gr oup G with G -co mpact universal prop er G -spa ce E G , K asparov’s F redholm representation ring R( G ) := KK G ( C , C ) is canonically is o morphic to the K -theory of C 0 ( E G ) ⋊ G . This is one statemen t of the Baum-Connes conjecture (and is due to Hig s on and Ka sparov). In fact ther e a re several po ssible statemen ts of the Baum-Connes co njecture, but the general idea is that computation of K-theory of the C* -algebra cro ssed pro ducts in volving arbitr ary group actions can b e, in certain The researc h for this paper w as partiall y s upported by the German Researc h F oundat ion (SFB 478 and SFB 878) and the EU-Netw ork Quan tum SpacesNoncomm utativ e Geometry (Con tract No. HPRN-CT - 2002-00280) . 1 2 ECHTERHOFF AND EMERSON circumstances, be r educed to the case where the a ction is prop er. Thus the imp or- tance of co mputing K -theory for prop er actions . Since the analysis of the str ucture of the crosse d pro duct runs along similar lines in the case of pro p er actions of lo- cally compact groups as for actions of compact (or ﬁnite) groups, we trea t the mor e general problem in this article. If G is a lo cally compact group acting prop erly on X , we show tha t C 0 ( X ) ⋊ G is isomorphic to a certa in gener alized ﬁxed- p oint a lgebra, denoted C 0 ( X × G K ), with resp ect to the diagona l action of G on C 0 ( X ) ⊗ K ( L 2 ( G )), with action of G on the seco nd factor given b y the adjoint o f the right regular repr esentation ρ o f G (in fact, we sho w a more general result along these lines for crossed pro ducts B ⋊ G where B is a ‘ﬁbred’ o ver some pro pe r G -spa ce X ). It follows tha t C 0 ( X ) ⋊ G is the algebra of C 0 -sections of a contin uous bundle of C*-alg ebras o ver the orbit space G \ X with ﬁb er ov er Gx isomorphic to the ﬁxed point alge bra K ( L 2 ( G )) G x , where G x denotes the (compact) stabilizer at x whic h acts via conjugation b y the restriction o f r ight reg ula r representation o f G to G x . This result has b een shown by Bruce E v ans in [25] for co mpact gro up actions, but we a re not aw are of any refer ence for the more genera l c lass of prop er actions. The Peter-W eyl theor em implies that the ﬁxed-p oint algebras K ( L 2 ( G )) G x de- comp ose into direct sums of a lgebras o f compact op erator s indexed o ver the unitary duals b G x of G x . It follows that the primitive ideal space of A = C 0 ( X ) ⋊ G is in a natural set bijection with G \ Stab( X ) b , where Stab( X ) b = { ( x, π ) | x ∈ X , π ∈ b G x } , which is a bundle, by the ﬁrst co ordinate pro jection, ov er the space G \ X of orbits, the ﬁbre ov er Gx b eing the unitary dual b G x of the stabilizer G x ( c.f. the ﬁrst parag raph o f this Intro duction). It is po ssible to describ e the top olo g y on G \ Stab( X ) b corresp onding to the F ell top ology on Prim( A ), in direct terms of the action. W e do this in the case where the action of G on X sa tisﬁes Palais’s slice prop erty , which means that X is lo ca lly induced fro m the st abilizer subgro ups of G (se e § 1 for this notion). By a famous theorem of Palais, this prop erty is alwa ys satisﬁed if G is a Lie gro up. In g eneral, G \ X is alwa ys an o pen subset of G \ Stab( X ) b , and therefor e cor re- sp onds to an ideal of C 0 ( X ) ⋊ G , a s r emarked ab ov e. Therefore to compute the K-theory of the cros sed product, it suﬃces t o compute the K-theory of the quotient space G \ X together with the b oundary maps in the asso ciated six- term sequence. In the case of isolated ﬁxed p oints and discr ete G this is fairly str a ightforw ard, at least up to torsio n; in general, it is non- trivial, but we show in examples how the knowledge of the ideal structure of C 0 ( X ) ⋊ G ca n s till help in K -theory computa- tions, even when ﬁxed p oints a re not iso lated. The problem of computing the K-theory in gener al do es not have an o b vious so- lution. Indeed, it may not hav e any s o lution at all. How ever, if one ignores torsion, the problem g e ts m uch e asier, at lea st for compact gr oup actions. The r eason is that if G is compact, the K-theory of C 0 ( X ) ⋊ G is a mo dule ov e r the represent ation ring Rep( G ), and similarly the K-theory tensored b y C is a mo dule over Rep( G ) ⊗ Z C . F or man y gro ups G of interest, like ﬁnite groups, or connected groups, the complex representation ring Rep( G ) ⊗ Z C is quite a tractable r ing, and the mo dule struc- ture of the complex K-theo r y of C 0 ( X ) ⋊ G gives signiﬁcant additional information when used in conjunction with a ‘lo ca lization principal’ dev elop ed mainly by A tiyah and Segal in the 196 0’s in connection with the Index Theore m. Thes e ideas were exploited b y Paul Baum and Ala in Connes in the 198 0’s to give a very b eautiful formula for the K-theory (t ensor e d by C ) o f the crossed product in the case of ﬁnite group actio ns. W e give a full pro of of the theorem of Ba um and Connes in this article, without attempting to generaliz e it to pro per actions of loc a lly compact groups. The formula of Baum and Connes means that the diﬃcult y in c omputing CR OSSED PRODUCT S BY PR OPER ACTIONS 3 K ∗ ( C 0 ( X ) ⋊ G  for ﬁnite group actions, is concen trated in the pro blem o f computing the torsio n s ubgroup. W e do not shed m uch light on this pr oblem. This pap er is to some extent expo sitory . O ur go al is to provide a readable sy n- opsis of wha t is known, and what c a n be proved without too muc h diﬃculty , ab out crossed pro ducts by prop er actions, and their K-theory . The paper is organized as follows: after giving some preliminaries on proper actions in § 1 we give a detailed discussion of the bundle structure of C 0 ( X ) ⋊ G in § 2 . In § 3 we discuss the natural bijection betw een Prim( C 0 ( X ) ⋊ G ) and the quotient space G \ Stab( X ) b and in § 4 we show tha t this bijection is a homeomorphism for a quite naturally deﬁned top ology on Stab( X ) b if the actio n satisﬁes the slice prop erty . All K -theoretic discussions can be found in § 5. All spaces co nsidered in this pap er with the o b vious exceptions of primitive ideal spaces of C*- algebras and the like, are ass umed lo c ally compact Hausdo rﬀ. A go o d part o f this pa p er has b een written while the ﬁrs t named author visited the Univ ersity of Victor ia in Summer 2008. He is very g rateful to the second named author a nd his collea gues for their warm hospitality during that stay! The authors are also grateful for so me useful con versations with W o lfg ang L¨ uck and Jan Spakula. 1. P reliminaries on proper actions Assume that G is a loca lly compact gr oup. Suppose that G acts on the (locally compact Hausdor ﬀ ) spa ce X . The action is pr op er if the map G × X → X × X : ( g , x ) 7→ ( gx, x ) is prop er – i.e. if inv erse images of co mpact sets are co mpact. Since X is loca lly compact, this is equiv a lent to the co nditio n that for ev ery compa c t subset K ⊆ X the set { g ∈ G : g − 1 K ∩ K 6 = ∅} is compact in G . In particula r, any action of a compact gro up o n a lo cally compact Hausdorﬀ space is pr op er. W e use to Palais’s fundamen tal paper [4 0] as a bas ic reference. It is immediately clea r that if G a cts proper ly on X then the sta biliz e rs G x := { g ∈ G : g x = x } are compa c t. More over, as one can show (e.g. see [4 0, Theorem 1.2.9]) without much diﬃcult y , the quo tien t space G \ X endow ed with the quotient top ology is a locally compact Hausdorﬀ space. If H ⊆ G is a closed subgroup of G whic h acts on some space Y , then the induced G -space G × H Y is deﬁned as the quotient s pace ( G × Y ) /H with resp ect to the diagonal a ction h · ( g , y ) = ( g h − 1 , hy ). The action of H on G × Y is ob viously free; it is a go o d exercis e to prov e that it is also prop er (s e e [40, § 1.3]). Hence G × H Y is a loc a lly compa ct Hausdorﬀ space. It carr ies a natural G -a ction by left transla tion on the ﬁrst factor. The proce s s of going from H acting on Y to G acting on G × H Y is called ‘inductio n.’ Prop ositio n 1 . 1 (c.f. [14, Coro llary]) . Supp ose that X is a G -sp ac e and H is a close d sub gr oup of G . Th en the fol lowing ar e e quivalent: (1) Ther e exists an H -s p ac e Y such that X ∼ = G × H Y as G -sp ac es. (2) Ther e exists a c ontinuous G -map ϕ : X → G/H . In c ase of (1), the c orr esp onding G -map ϕ : G × H Y → G/H is given by ϕ ([ g , y ]) = g H and in c ase of (2), the c orr esp onding H -s p ac e Y is the close d subset Y := ϕ − 1 ( { eH } ) of X . The G -home omorphism fr om Φ : G × H Y → X is then given by Φ([ g , y ]) = g y . F ollowing Palais [40], we shall call a closed subset Y ⊆ X a glob al H -slic e if there exists a map ϕ : X → G/H as in part (2) of the ab ov e theor e m with Y = ϕ − 1 ( { eH } ), i.e., Y ⊆ X is a global H -slice if and o nly if Y is H -inv ariant and X ∼ = G × H Y as a G -spa ce. If U is a G -inv ariant o pe n subset o f X , then w e say 4 ECHTERHOFF AND EMERSON that Y ⊂ U is a lo c al H -slic e of X , if Y is a global H - slice for the G - s pace U . The following obser v ation is well kno wn, but by lac k of a direct reference , we give the pro of. Lemma 1.2. Supp ose that H is a close d sub gr oup of G and that Y is an H -sp ac e. Then G × H Y is a pr op er G -sp ac e if and only if Y is a pr op er H -sp ac e. Pr o of. Supp ose ﬁrst that X = G × H Y is a proper G -space. Then it is also a proper H -space, and since ϕ : Y → G × H Y ; ϕ ( y ) = [ e , y ] includes Y as an H -inv a riant closed subset of X , it m ust b e a proper H -s pa ce, too . Conv ersely , if Y is a pr op er H -space and K ⊆ G × H Y is any compact set, then we may ch o ose compact sets C ⊆ G and D ⊆ Y such that K ⊆ C × H D := { [ c, d ] ∈ G × H Y : c ∈ C, d ∈ D } . Let F := { h ∈ H : h − 1 D ∩ D 6 = ∅} . Since H acts prop erly on Y , F is a compact subset of H . Supp ose no w that g ∈ G such g − 1 ( C × H D ) ∩ ( C × H D ) 6 = ∅ . Then ther e exist c 1 , c 2 ∈ C and d 1 , d 2 ∈ D such that [ g − 1 c 1 , d 1 ] = [ c 2 , d 2 ], which in turn implies that there exis ts en element h ∈ H such that ( g − 1 c 1 h, h − 1 d 1 ) = ( c 2 , d 2 ). Then h ∈ F and g − 1 c 1 h = c 2 implies that g = c 1 hc − 1 2 ∈ C F C − 1 , which is compact in G .  The a bove lemma s hows in particular that every G -spa ce whic h is induced fro m some co mpact subgroup L of G m ust be pro pe r. This obser v ation has a partial conv erse, a s the following theorem of Abels ([1]) shows: Theorem 1.3 (Abels) . Supp ose that X is a pr op er G -sp ac e. Then for e ach x ∈ X ther e exist a G -invari ant op en neighb orho o d U x of x , a c omp act sub gr oup L x of G , and a c ontinuous G - m ap ϕ x : U x → G/L x . Th us Y x := ϕ − 1 ( { L x } ) is a lo c al L x -slic e for X and U x ∼ = G × L x Y x . Thu s, combining the Theo rem with Prop osition 1.1, we see that every prop er G -space is lo ca lly induced from compact subgr oups. R emark 1 .4 . W e make tw o minor remarks about slices whose easy v eriﬁcation w e leav e to the reader : • W e may always choose L x and ϕ x in the theorem in s uc h a w ay that ϕ x ( x ) = eL x , and hence x ∈ Y x (at the exp ense of c hanging the s ubg roup L x to a conjugate subgr oup). • If Y ⊂ U is a slice for the subgro up L x then the is otropy subgr oups of G in U ar e all sub conjugate to L x , i.e. a re conjugate to s ubg roups of L x . • If U ⊆ X is a G -in v ar iant subset with L x -slice Y x then the intersection Y x ∩ V is a L x -slice for an y given G -inv ariant subset V ⊆ U . Example 1 .5 . Let G := Q n ∈ N Z / 2, realize Z / 2 as {± 1 } ⊂ T , and let X := Q n ∈ N T , with action of G on X given by tra nslation. This is a free and prop er action of a compact, totally disco nnected gr o up. It is easy to construct many lo cal s lices. Let I b e a small in terv al neighbourho o d of 1 ∈ T , J = I ∪ − I . A G -neigh bo urho o d bas is (in the sens e of giving a neighbourho o d in the quotient space) of x := (1 , 1 , 1 , · · · ) ∈ X is s upplied by bas ic pro duct sets of the form U = J × · · · × J × T × T × · · · . Thu s any op en G -inv ariant neighbourho o d of x must contain one o f these. E a ch of these ope n subsets ha s 2 k comp onents, wher e k is the num ber of facto rs of J o ccurring, and under any G -map from U to a totally disconnected s pace like G itself m ust thus map the a bove open s ubset to a ﬁnite s ubset o f the target. A slice at x pr o duces a G -map from any suﬃcien tly small one o f these G -neighbour ho o ds , with targ e t some G/L x . Since this is also a totally disco nnected spac e, the ma p m ust have image in a ﬁnite subset, since the subs et must be G -inv ariant, it must be that G/ L x is itself ﬁnite. Hence a ll slices must use closed subgro ups L x of ﬁnite index. CR OSSED PRODUCT S BY PR OPER ACTIONS 5 In pa rticular there is no slice through x with L x the trivial subgroup of G , i.e. no slices throug h x with gr oup L x exactly equal to the isotropy gr oup G x . If L x is co- ﬁnite, it must be a subgro up of o ne of the ‘o b vious’ one s L x = { ( x i ) | x i = 1 if i ≤ n } , the quo tien t G/L x G -equiv ar iantly iden tiﬁes with a ﬁnite pro duct of Z / 2’s and it is eas y to pro duce a slice, i.e. a G -map U := J × J × · · · J × T × T × · · · to G/L x , by identif ying J ∼ = I × Z / 2 and using Y x := I × · · · × I × T × T × · · · . In particula r, whenever one has a slice, the stabilize r G x of x is a clos ed subgr o up of L x , but G x ⊆ L x may b e strict. Howev e r , this can a lways be a voided when G is a Lie gr oup . This is the co nten t of the follo wing well-kno wn result of Palais. Theorem 1.6 (Palais’s Slice Theorem) . S upp ose that the Lie gr oup G acts pr op erly on the lo c al ly c omp act G -sp ac e X . Then for every x ∈ X ther e exists an op en G - invariant neighb orho o d U x of X which admits a G x -slic e Y x ⊆ U x with x ∈ Y x . W e should p oint out that P alais’s or iginal theo rem (see [40, Prop osition 2.3.1]) is stated for c ompletely regular prop er spaces, and therefore is actually slightly more general; we will not need the extra generality here. Motiv ated by Palais’s theor em we give the following Deﬁnition 1.7. Let G b e a loca lly compact group and let X be a prop er G - space. W e say that ( G, X ) satisﬁes Palais’s slic e pr op ert y (SP) if the conclusion of Theorem 1.6 holds for ( G, X ), i.e., if X is lo ca lly induced from the stabilizers G x . R emark 1.8 . W e emphasise that pr op erty (SP) for ( G, X ) implies that every po in t x ∈ X has a G -in v ar iant neighbo rho o d U x such that the stabilizer s G y for all y ∈ U x are sub-co njugate to G x ( c.f. Remar k 1.4.) 2. Proper a ctions and C*-algebra bundles If X is a lo cally co mpact G -space, then there is a corr esp onding action on the C*-algebr a C 0 ( X ) of all functions on X whic h v anish at ∞ g iven by ( g · f )( g ′ ) = f ( g − 1 g ′ ). The main o b ject of this p ap er is the study o f the cro ssed pro duct C 0 ( X ) ⋊ G in cas e wher e X is a prop er G -space. F or the ge ne r al theor y of cro ssed pro ducts we r efer to Dana Williams’s b o ok [48]. The co nstruction of cross ed pro ducts for gro up actions o n spaces go es ba ck to early work of Glimm (see [27]). Consider the space C c ( G × X ) of contin uous func- tions with compact supports o n G × X equipped with c o nv o lution and inv olution given by the formulas ϕ ∗ ψ ( g , x ) = Z G ϕ ( t, x ) ψ ( t − 1 g , t − 1 x ) dt and ϕ ∗ ( g , x ) = ∆( g − 1 ) ϕ ( g − 1 , g − 1 x ) . Let L 1 ( G, X ) deno te the completion of C c ( G × X ) with re s pec t to the norm k f k = R G k f ( g , · ) k ∞ dg . Then C 0 ( X ) ⋊ G is the en velopping C ∗ -algebra of the Bana ch -*- algebra L 1 ( G, X ). It enjoys the universal prop erty for cov ariant representations o f the pair ( C 0 ( X ) , G ) as expla ined in detail in [48, Prop osition 2.40 ]. It follows from [49] that the crossed pr o duct C 0 ( X ) ⋊ G for any prop e r G -spa ce X has a ca nonical structur e a s an algebr a of sections of a contin uo us C*-a lgebra bundle ov er G \ X . In this section w e wan t to g ive a more detailed description of this bundle. Recall that if Z is any lo cally c o mpact s pace, then a C*-algebr a A is called a C 0 ( Z ) -algebr a (o r an upp er semi-c ontinuous bund le of C*-algebr as over Z ) if there exists a ∗ -homomorphism φ : C 0 ( Z ) → Z M ( A ), the cent er of the m ultiplier a lgebra of A , such that φ ( C 0 ( Z )) A = A . F or z ∈ Z put I z := φ ( C 0 ( Z r { z } )) A and A z = A/I z . Then A z is called the ﬁbr e of A at z . Then every a ∈ A can be viewed as a section of the bundle o f C*- algebras { A z : z ∈ Z } via a : z 7→ a z := a + I z . 6 ECHTERHOFF AND EMERSON The resulting pos itiv e function z 7→ k a z k is alw ays upp er semi contin uo us and we hav e k a k = sup z ∈ Z k a z k for all a ∈ A . W e s ay that A is a c ontinu ou s bund le of C*-algebr as over Z if in addition all functions z 7→ k a z k are con tinuous. W e refer to [48, C.2] for the g eneral prop erties of C 0 ( Z )-a lgebras. In what follows we shall usually suppress the name of the structure map φ : C 0 ( Z ) → Z M ( A ) and we shall simply write f a for φ ( f ) a if f ∈ C 0 ( Z ) and a ∈ A . A ∗ -homomorphism Ψ : A → B b etw een tw o C 0 ( Z )-a lgebras A and B is ca lle d C 0 ( Z )-linea r, if it comm utes with the C 0 ( Z )-a ctions, that is if Ψ( f a ) = f Ψ( a ) for all f ∈ C 0 ( Z ), a ∈ A . A C 0 ( Z )-linea r homomorphism Ψ induces ∗ -homomorphis ms Ψ z : A z → B z for all z ∈ Z by deﬁning Ψ z ( a + I A z ) = Ψ ( a ) + I B z for all z ∈ Z , a ∈ A . If A is a C 0 ( Z )-a lgebra a nd G a cts on A by C 0 ( Z )-linea r a utomorphisms, then the full crossed pro duct A ⋊ G is again a C 0 ( Z )-a lgebra with resp ect to the compo sition C 0 ( Z ) Φ − → Z M ( A ) i M ( A ) − → Z M ( A ⋊ G ) , where i M ( A ) : M ( A ) → M ( A ⋊ G ) denotes the extensio n to M ( A ) of the canonica l inclusion i A : A → M ( A ⋊ G ) (see [48, P rop osition 2.3 .4] for the deﬁnition of i A ). The action α then induces actions α z of G o n ea ch ﬁber A z and it fo llows then from the exactness of the maximal cross ed pro duct functor that the ﬁbre ( A ⋊ G ) z of the cr ossed pro duct at a p oint z ∈ Z coincides with A z ⋊ G (e.g., s ee [48, Theor e m 8.4]). The fo llowing well-known lemma is o ften useful. Lemma 2.1. Supp ose that Φ : A → B is a C 0 ( Z ) -line ar ∗ -homomorphi sm b etwe en the C 0 ( Z ) -algebr as A and B . Then Φ is inje ctive (r esp. surje ctive, r esp. bije ctive) if and only if every ﬁbr e map Φ z is inje ctive (r esp. surje ctive, r esp. bije ctive). Pr o of. Since k a k = sup z ∈ Z k a z k for all a ∈ A , it is clear that Φ is injectiv e if Φ z is injectiv e for all z ∈ Z . Conv ersely a ssume Φ is injective. Then B ′ := Φ( A ) ⊆ B is a C 0 ( X )-subalgebra of B and there ex ists a C 0 ( X )-linear in verse Φ − 1 : B ′ → A which induces ﬁbre-wise inv e r ses Φ − 1 z for Φ z : A z → B ′ z ⊆ B z . Surjectivity of Φ clearly implies surjectivit y of Φ z for all z ∈ Z . Conv ersely , if all Φ z are surjective, then Φ( A ) ⊆ B satisﬁes the conditions of [48, Prop os ition C.2 4], which then implies that Φ( A ) = B .  Assume now tha t X is a pro pe r G -space. W e pro ceed with some genera l con- structions of bundles over G \ X : F or this assume tha t B is an y C* - algebra equipp ed with an action β : G → Aut( B ) of G . W e deﬁne the a lgebra C 0 ( X × G,β B ) (we shall simply w r ite C 0 ( X × G B ) if there is no doubt a bo ut the given action) a s the set of all b ounded cont inuous functions F : X → B such that F ( g x ) = β g ( F ( x )) for a ll x ∈ X and g ∈ G and such that the function x 7→ k F ( x ) k (whic h is co nstant on G -orbits) v anishes at inﬁnity on G \ X . It is easily chec ked that C 0 ( X × G B ) bec omes a C*-algebr a when equipp ed with p oint wis e a ddition, multiplication, in- volution and the sup-nor m. Note that this construction is well known under the name of the gener alize d ﬁxe d p oint algebr as for the pr o pe r action o f G on C 0 ( X, B ) (e.g. see [41, 35, 39]). Lemma 2 . 2. C 0 ( X × G B ) is t he se ction alg ebr a of a c ontinu ous bund le of C*- algebr as over G \ X with ﬁbr e over the orbit Gx isomorphi c to the ﬁxe d p oint algebr a B G x , wher e G x = { g ∈ G : g x = x } is the stabilizer of x in G . Pr o of. Ass ume ﬁrst that the action of G on X is tra nsitive, i.e., X = Gx for some x ∈ X . Then it is s tr aightforw ard to ch eck that ev alua tion at x induces an isomorphism C 0 ( Gx × G B ) → B G x ; F 7→ F ( x ). CR OSSED PRODUCT S BY PR OPER ACTIONS 7 F or the gener a l case w e ﬁrst note that multiplication with functions in C 0 ( G \ X ) provides C 0 ( X × G B ) with the s tructure of a C 0 ( G \ X )-algebra . The ideal I Gx = C 0 (( G \ X ) r { Gx } ) C 0 ( X × G B ) then coincides with the set of functions F ∈ C 0 ( X × G B ) which v anish on Gx , a nd henc e with the kernel of the restriction map F 7→ F | Gx from C 0 ( X × G B ) into C 0 ( Gx × G B ). If we co mpo s e this with the ev aluation at x we now see that the map F 7→ F ( x ) factors through an injective ∗ -homomor phism of the ﬁb er C 0 ( X × G B ) Gx int o B G x . W e need to show that this map is sur jectiv e. Since images of ∗ -homo morphism b etw een C*-algebra s a re c lo sed, it suﬃces to sho w that the ev aluation ma p has dense image. F o r this ﬁx b ∈ B G x . F or any neigh bo rho o d U o f x choos e a p ositive function f U ∈ C c ( X ) such that supp f U ⊆ U and R G f U ( g − 1 x ) dg = 1. Then deﬁne F U ∈ C 0 ( X × G B ) by F U ( y ) := R G f U ( g − 1 y ) β g ( b ) dg for all y ∈ X . O ne chec ks that F U ∈ C 0 ( X × G B ) and that F U ( x ) → b as U shrink s to x . This shows the desir ed density re s ult. Finally , the fact tha t the C 0 ( X × G B ) is a contin uo us bundle follows from the fact that the contin uo us function x 7→ k F ( x ) k is consta nt on G -orbits in X , and hence factor s through a contin uous function o n G \ X .  F or an induced prop er G -spa ce X = G × H Y w e get the follo wing result. Lemma 2. 3 . Supp ose that H is a close d sub gr oup of G and B is a G -algebr a. Then ther e is a c anonic al isomorphism Φ : C 0  ( G × H Y ) × G B  → C 0 ( Y × H B ) given by Φ( F )( y ) = F ([ e, y ]) for F ∈ C 0  ( G × H Y ) × G B  and y ∈ Y . Pr o of. It is s traightforw ard to check that Φ is a well deﬁned ∗ -homomorphism with inv er se Φ − 1 : C 0 ( Y × H B ) → C 0  ( G × H Y ) × G B  given by Φ − 1 ( F )([ g , y ]) = β g ( F ( y )), whe r e β : G → Aut ( B ) denotes the given action on B .  In some cases the algebra C 0 ( X × G B ) has a muc h easier description. Deﬁnition 2. 4. Supp ose that G acts on the lo cally compa ct space X . A closed subspace Z ⊆ X is called a t op olo gic al fundamental domain for the action of G on X if the mapping Z → G \ X ; z 7→ Gz is a homeomor phism. Of course, a topolo gical fundamen tal domain as in the deﬁnition, do es not exist in most cases, but the fo llowing examples show that there are at least some in teresting cases where they do exist: Example 2.5 . F or the ﬁrst example we consider the obvious a ction of SO( n ) on R n . Then the set Z = { ( x, 0 , . . . , 0 ) : x ≥ 0 } is a topo logical fundamental domain for this action. Example 2.6 . F or the se c ond exa mple let G (the dihedr al g roup D 4 ) be genera ted by a rotation R around the o rigin in R 2 by π 2 radians, and the reﬂection S through the line l S := { ( x, 0) : x ∈ R } . W riting R =  0 − 1 1 0  and S =  1 0 0 − 1  , the group G has the ele men ts { E , R, R 2 , R 3 , S, S R , S R 2 , S R 3 } , where E denotes the unit matrix. Since G ⊆ GL(2 , Z ), it’s action on R 2 ﬁxes Z 2 , and ther efore factor s thro ugh an ac tio n on T 2 ∼ = R 2 / Z 2 . If we study this action on the fundamen tal do main ( − 1 2 , 1 2 ] 2 ⊆ R 2 for the translatio n action of Z 2 on R 2 , it is a n easy e x ercise to chec k that the s et Z := { ( e 2 π is , e 2 π it ) : 0 ≤ t ≤ 1 2 , 0 ≤ s ≤ t } is a to p olo gical fundament al domain fo r the action o f G on T 2 . Of course, if we restrict the a bove ac tio n to the subgr oup H := h R i ⊆ G , we obtain an e xample of a gr oup action with no topolog ical fundamen tal do main. 8 ECHTERHOFF AND EMERSON Prop ositio n 2 . 7. Supp ose that Z ⊆ X is a top olo gic al fundamental domain for the pr op er action of G on X . Then, for any G -algebr a B , ther e is an isomorphi sm C 0 ( X × G B ) ∼ = { f : Z → B : f ( z ) ∈ B G z } given by F 7→ F | Z . Pr o of. T his is an eas y co nsequence of (the pr o of of ) Lemma 2.2 together with Lemma 2.1.  Assume no w that we hav e t wo comm uting actions α, β : G → Aut( B ) of G on the same C*-algebra B . Then β induces a n a ction ˜ β on the crossed pr o duct B ⋊ α G in the cano nical wa y . On the o ther hand, we also obtain an action ˜ α : G → Aut( C 0 ( X × G,β B )) via  ˜ α g ( F )  ( x ) = α g ( F ( x )) . W e w ant to show the following Prop ositio n 2.8. In the ab ove situation we have a c anonic al isomorphism C 0 ( X × G,β B ) ⋊ ˜ α G ∼ = C 0  X × G, ˜ β ( B ⋊ α G )  . F or the pro of we ﬁrs t need Lemma 2. 9. Le t K b e a c omp act gr oup and G a lo c al ly c omp act gr oup such that β : K → Aut( B ) , α : G → Aut( B ) ar e c ommu ting actions of K and G on the C*- algebr a B . Then the ﬁxe d-p oint algebr a B K for the action of K on B is G -invariant and the inclus ion ι : B K → B induc es an isomorphism ι ⋊ G : B K ⋊ G ∼ = − → ( B ⋊ G ) K , wher e the ﬁx e d-p oint algebr a on the right hand side is taken with r esp e ct to the action ˜ β of K on B ⋊ G induc e d by β in the c anonic al way. Pr o of. No te that the lemma is not ob vious, since there might exist G -inv ariant sub- algebras D ⊆ B s uc h that the ful l cros s ed pro duct D ⋊ G do es not include faithfully int o B ⋊ G (while this would alwa ys be true for the reduced cr ossed pro ducts). F or the pro of we use the fa ct that B K ident iﬁes with the compact op erator s of a B ⋊ K -Hilb ert mo dule deﬁned as follo ws: we make B in to a pre - B ⋊ K Hilb ert mo dule, with completion denoted by X B , by deﬁning the B ⋊ K -v alued inner pro duct h a, b i B ⋊ K =  k 7→ β k ( a ∗ ) b  ∈ C ( K , B ) for a, b ∈ X B , and rig h t action of C ( K, B ) ⊆ B ⋊ K on B ⊆ X B by a · f = Z K aβ k ( f ( k − 1 )) dk , a ∈ X B , f ∈ C ( K, B ) . W e can also chec k that X B carries a s tr ucture of a ful l left Hilb ert- B K -mo dule given by B K h a, b i = R K β k ( ab ∗ ) dk and a left action of B K on B ⊆ X B given by m ultiplication in B . O ne easily chec ks that these Hilb ert-mo dule structures on X B are compatible in the sense that B K h a, b i c = a h b, c i B ⋊ K for all a, b, c ∈ X B . Thus, the left a ction of B K on X B ident iﬁes B K with K ( X B ). W e can then consider the desc en t mo dule X B ⋊ G , whic h is a ( B ⋊ K ) ⋊ G -Hilber t mo dule with K ( X B ⋊ G ) ∼ = B K ⋊ G (e.g., see [1 1] for the for m ulas for the actions and inner pr o ducts). Similarly , via th e a ction of K on B ⋊ G we obtain a ( B ⋊ G ) ⋊ K -Hilber t bimo dule X B ⋊ G with compact o per ators isomor phic to ( B ⋊ G ) K . Since the actions of K and G on B commut e, we can identify ( B ⋊ G ) ⋊ K ∼ = B ⋊ ( G × K ) ∼ = ( B ⋊ K ) ⋊ G. CR OSSED PRODUCT S BY PR OPER ACTIONS 9 W e now c heck that under this iden tiﬁcation w e obta in an is omorphism of right Hilber t B ⋊ ( G × K )-mo dules b etw een X B ⋊ G and X B ⋊ G , such that ι ⋊ G intert wines the left actions of B K ⋊ G and ( B ⋊ G ) K . Since isomorphisms b etw een Hilb ert mo dules induce isomorphisms between their compact operator s, this will imply that ι ⋊ G is a n isomorphism. Note that b y construction bo th modules X B ⋊ G and X B ⋊ G contain C c ( G, B ) as a dense C c ( G × K, B )-submo dule, where w e view C c ( G × K, B ) as a dense subalgebra of B ⋊ ( G × K ). One then chec ks that the iden tity map on C c ( B ⋊ G ) induces the desired mo dule isomorphism such that ι ⋊ G commutes with the left actions of C c ( G, B K ) = C c ( G, B ) K sitting as dense subalg ebra in B K ⋊ G and ( B ⋊ G ) K , resp ectively . Th us the identit y o n C c ( G, B ) extends to the desired isomorphism X B ⋊ G ∼ = X B ⋊ G .  Pr o of of Pr op osition 2.8. W e shall describ e the iso morphism via a c ov a riant pair (Φ , U ). F or this let ( i B , i G ) : ( B , G ) → M ( B ⋊ α G ) denote the canonical inclusions (see [48, Pr op osition 2.3.4]). W e then deﬁne Φ : C 0 ( X × G,β B ) → M  C 0  X × G, ˜ β ( B ⋊ α G )  by s e nding F ∈ C 0 ( X × G,β B ) to the multiplier of C 0  X × G, ˜ β ( B ⋊ α G )  given by p oint wise m ultiplication with x 7→ i B ( F ( x )). Similarly , for g ∈ G w e deﬁne U g ∈ M  C 0  X × G, ˜ β ( B ⋊ α G )  by the po in twise applicatio n o f i G ( g ). O ne checks that (Φ , U ) gives a w ell deﬁned cov ariant homomor phism o f ( C 0 ( X ⋊ G,β B ) , G ) into M  C 0  X × G, ˜ β ( B ⋊ α G )  whose integrated form Φ ⋊ U is C 0 ( G \ X )-linear, since Φ is C 0 ( G \ X )-linear. Thu s it suﬃces to chec k that Φ ⋊ U induces isomo rphisms of the ﬁbres. F or this ﬁx any x ∈ X . W e then obta in a comm utative dia g ram C 0 ( X × G,β B ) ⋊ ˜ α G Φ × U − − − − → C 0  X × G, ˜ β ( B ⋊ α G )  ǫ x   y   y ǫ x B G x ⋊ G − − − − → i B × i G ( B ⋊ G ) G x and the res ult follows fro m Le mma 2.9.  W e are now go ing to descr ibe the crossed pro duct C 0 ( X ) ⋊ G in terms of sec - tion algebras of suitable C*-alge bra bundles. Consider the a lgebra K = K ( L 2 ( G )) equipp e d with the action Ad ρ : G → Aut( K ), where ρ : G → U ( L 2 ( G )) de no tes th e right regular repre sent ation of G given by ρ ( g ) ξ ( t ) = p ∆ G ( g ) ξ ( tg ) for ξ ∈ L 2 ( G ). Let lt , rt : G → Aut( C 0 ( G )) denote the actions giv en b y left a nd right translation on G , resp ectively . It then follows from the extended version of the Stone- von Neumann theorem (see [48, Theorem 4.24 ], but see [46] or a more direct pro o f ) that (2.1) M × λ : C 0 ( G ) ⋊ lt G ∼ = − → K ( L 2 ( G )) , where M × λ is the in tegrated form of the cov aria n t pa ir ( M , λ ) with M : C 0 ( G ) → B ( L 2 ( G )) being the r epresentation b y m ultiplication o pe r - ators and λ : G → U ( L 2 ( G )) the left regula r representation of G . Let ˜ rt : G → Aut( C 0 ( G ) ⋊ lt G ) denote the action induced fro m the right translation ac tion r t : G → Aut( C 0 ( G )). F or f ∈ C 0 ( G ) one checks that M (rt g ( f )) = ρ ( g ) M ( f ) ρ ( g − 1 ). F rom this a nd the fact that ρ commutes with λ it follows tha t M × λ ( ˜ rt g ( ϕ )) = ρ ( g )  M × λ ( ϕ ) ) ρ ( g − 1 ) for a ll g ∈ G and ϕ ∈ C 0 ( G ) ⋊ lt G . W e sha ll use all this for the pro o f of following result, where we write K for K ( L 2 ( G )): 10 ECHTERHOFF AND EMERSON Theorem 2.10 . L et G b e a lo c al ly c omp act gr oup acting pr op erly on t he lo c al ly c omp act sp ac e X with c orr esp onding action τ : G → Aut ( C 0 ( X )) and let β : G → Aut( B ) b e any action of G on a C*-algebr a B . Then ther e is a c anonic al isomor- phism ( C 0 ( X ) ⊗ B ) ⋊ τ ⊗ β G ∼ = C 0  X × G, Ad ρ ⊗ β ( K ⊗ B )  . Pr o of. W e cons ider the commuting actions rt ⊗ id B and lt ⊗ β of G on C 0 ( G, B ) = C 0 ( G ) ⊗ B . It follows from Prop osition 2.8 that there is a canonical is omorphism C 0  X × G, ^ rt ⊗ id B ( C 0 ( G, B ) ⋊ lt ⊗ β G )  ∼ = C 0 ( X × G, rt ⊗ id B C 0 ( G, B )) ⋊ ] lt ⊗ β G. If w e deﬁne X × G, rt G = G \ ( X × G ) with resp ect to the action g ( x, h ) = ( g x, hg − 1 ), and if we equip this space with the G -ac tion given by left translatio n o n the second factor, called ˜ lt b elow, we obtain a ˜ lt ⊗ β − ] lt ⊗ β -equiv ar iant isomorphism C 0 ( X × G, rt G ) ⊗ B ∼ = C 0 ( X × G, rt ⊗ id B C 0 ( G, B )) , which induces an isomo rphism of the r e s pec tive crossed-pr o ducts. W e further ob- serve that the map X × G, rt G → X ; [ x, g ] 7→ g x is a homeo morphism (with inv erse given b y x 7→ [ x, e ]) which transforms ˜ lt into the given action τ on X . Com bining all this, we obtain a ca nonical isomorphism C 0  X × G, rt ⊗ id B C 0 ( G, B )  ⋊ ] lt ⊗ β G ∼ = C 0 ( X, B ) ⋊ τ ⊗ β G. Using the isomorphism C 0  X × G, rt ⊗ id B C 0 ( G, B )  ⋊ ] lt ⊗ β G ∼ = C 0  X × G, ^ rt ⊗ id B ( C 0 ( G, B ) ⋊ lt ⊗ β G )  , all remains to do is to identify C 0 ( G, B ) ⋊ lt ⊗ β G with K ( L 2 ( G )) ⊗ B equiv a r iantly with res pect to the action ^ rt ⊗ id B and Ad ρ ⊗ β . But s uc h is o morphism is well known from T akesaki-T ak ai duality: it is straight- forward to check that the isomor phism Φ : C 0 ( G, B ) → C 0 ( G, B ) given by Φ( f )( g ) = β g − 1 ( f ( g )) transfor ms the action lt ⊗ β to the ac tion lt ⊗ id B and the action rt ⊗ id B to the action rt ⊗ β . This induces an is omorphism C 0  X × G, ^ rt ⊗ id B ( C 0 ( G, B ) ⋊ lt ⊗ β G )  ∼ = C 0  X × G, ˜ rt ⊗ β ( C 0 ( G ) ⋊ lt G ) ⊗ B )  ∼ = C 0  X × G, Ad ρ ⊗ β ( K ⊗ B )  .  Corollary 2.11. Le t G b e a lo c al ly c omp act gr oup acting pr op erly on the lo c al ly c omp act sp ac e X and let K = K ( L 2 ( G )) . Then C 0 ( X ) ⋊ τ G ∼ = C 0 ( X × G, Ad ρ K ) . R emark 2.1 2 . Explicitly , for any orbit Gx ∈ G \ X the ev aluation map q x : C 0 ( X × G K ) → K G x can b e describ ed o n the level of C 0 ( X ) ⋊ G b y the co mpo s ition of maps C 0 ( X ) ⋊ G q Gx − → C 0 ( Gx ) ⋊ G ∼ = C 0 ( G/G x ) ⋊ G M × λ − → K G x , where the ﬁrst map is induced by the G - e quiv aria n t restriction map C 0 ( X ) → C 0 ( Gx ) and the second is induced by the G -homeomorphism G/G x → Gx ; g G x 7→ g x . Example 2.13 . Le t X := T , G := Z / 2 a cting by conjugation on the circle. Then C ( T ) ⋊ Z / 2 ∼ = { f ∈ C  [0 , 1] , M 2 ( C )  | f (0) and f (1 ) are diag onal } . This is immediate fr om Corolla ry 2.11, using the basis { 1 √ 2 (1 , 1) , 1 √ 2 (1 , − 1) } for ℓ 2 ( Z / 2) to ident ify it with C 2 (this dia gonalizes the Z / 2-action.) A con tinuous function f : T → K  ℓ 2 ( Z / 2)) suc h that f ( g x ) = Ad g  f ( x )  is determined by its restriction to { z ∈ T | Im( z ) ≥ 0 } , where, us ing the ab ov e basis, w e c a n identify it CR OSSED PRODUCT S BY PR OPER ACTIONS 11 with a map f : [0 , 1 ] → M 2 ( C ) such that f (0 ) and f (1) commute with the matrix  1 0 0 − 1  . The comm utant of this matr ix consists of the diagonal matrices . Theorem 2.10 can be e xtended to the case of prop er actions on general C*- algebras B , i.e. s uch tha t B is an X ⋊ G -a lgebra for some prop er G -spa ce X . Thu s, B is a C 0 ( X )-algebra equipped with a G -actio n β : G → Aut( B ) such that the structure map φ : C 0 ( X ) → Z M ( B ) is G -equiv ariant. In this situation the generalized ﬁxed p oint algebra B G,β can be constructed as follows: we co ns ider the algebra C 0 ( X × G B ) as studied ab ov e. If b ∈ B , we write b ( y ) for the ev aluation of B in the ﬁb er B y , y ∈ X . Similarly , for F ∈ C 0 ( X × G B ) we write F ( x, y ) fo r the ev aluation o f the element F ( x ) ∈ B in the ﬁber B y . Then C 0 ( X × G B ) becomes a C 0 ( G \ ( X × X ))-algebra via the structur e map Φ : C 0 ( G \ ( X × X )) → Z M ( C 0 ( X × GB ));  Φ( ϕ ) F  ( x, y ) = ϕ ([ x, y ]) F ( x, y ) . W e then deﬁne B G,β (or just B G if β is understo o d) as the restr ic tion of C 0 ( X × G B ) to G \ ∆( X ) ∼ = G \ X , where ∆( X ) = { ( x, x ) : x ∈ X } . Note that with this notation we ha ve C 0 ( X × G,β B ) ∼ =  C 0 ( X ) ⊗ B  G,τ ⊗ β (if τ denotes the corresp onding action on C 0 ( X )). Moreov er, if G is co mpact, this notation coincides with the usual notation of the ﬁxed-p o int algebr a B G . With this nota tion we get Theorem 2. 14. S upp ose that B is an X ⋊ G -algebr a for the pr op er G -sp ac e X via some action β : G → Aut( B ) . Then B ⋊ G is isomorphic to ( K ⊗ B ) G, Ad ρ ⊗ β . Pr o of. C o nsider the C 0 ( X × X )-alge br a C 0 ( X, B ). B y Theorem 2.10 w e hav e C 0 ( X, B ) ⋊ τ ⊗ β G ∼ = C 0 ( X × G, Ad ρ ⊗ β ( K ⊗ B )). The cros s ed pro duct C 0 ( X, B ) ⋊ G carries a canonica l str uc tur e a s a C 0 ( G \ ( X × X ))-alg ebra which is induced from the C 0 ( X × X )-structure of C 0 ( X, B ). A car eful lo ok at the pr o of of the isomor- phism C 0 ( X, B ) ⋊ G ∼ = C 0 ( X × G ( K ⊗ B )) r eveals that this isomorphism pre- serves the C 0 ( G \ ( X × X ))-structures on b oth algebras. Using the G -isomorphism C 0 ( X, B ) | ∆( X ) ∼ = B which is induced from the ∗ -homo morphism C 0 ( X, B ) ∼ = C 0 ( X ) ⊗ B → B ; ( f ⊗ b ) 7→ f b , we no w obtain a chain of iso morphisms B ⋊ G ∼ =  C 0 ( X, B ) | ∆( X )  ⋊ G ∼ =  C 0 ( X, B ) ⋊ G ) | G \ ∆( X ) ∼ =  C 0 ( X × G ( K ⊗ B ))  | G \ ∆( X ) = ( K ⊗ B ) G, Ad ρ ⊗ β .  R emark 2.15 . Using this, it is not diﬃcult to chec k that fo r any X ⋊ G -algebra B for some prop er G -space X the cr ossed pro duct B ⋊ G is a C 0 ( G \ X )-algebra with ﬁber at an orbit Gx iso morphic to ( K ⊗ B x ) G x , Ad ρ ⊗ β x , where β x : G x → Aut( B x ) is the actio n induced from β in the canonical w ay . The r esults o f this section ﬁt into the framework of gener alized ﬁxed-p oint al- gebras (see the w ork of Ma rc Rieﬀel and a lso Ralf Meyer in [43, 4 4, 39]); our aim here is no t generality , but explicitnes s, and we hav e taken a direct a ppr oach. 3. The Mackey-Rie ffel-Green theorem f or pr oper actions The Mack ey-Rieﬀel-Gr e en theorem (or Mackey-Rieﬀel-Green mac hine) supplies, under some suitable conditions on a given C*-dynamica l system ( A, G, α ), a sys- tematic wa y o f describing the irreducible r epresentations (or primitive ideals) of the cros sed pro duct A ⋊ α G in ter ms of the asso ciated actio n of G on Prim( A ) by 12 ECHTERHOFF AND EMERSON inducing representations (or ideals) from the s tabilizers for this action. W e refer to [48, 20 ] for re c e n t discuss ions of this genera l machinery , and to [3 0, 31] for some impo rtant contributions tow ards this theory . In this section we wan t to give a self-contained exposition of the Mack ey -Rieﬀel- Green machine in the sp ecial case of cross ed pro ducts b y prop er a c tions on spaces, in which the result will follow easily from the bundle descr iption of the crossed pro duct C 0 ( X ) ⋊ G as obtained in the previous section and the following explicit description of the ﬁb ers. This explicit de s cription o f the ﬁb ers will also play an impo rtant rˆ ole in our description of the F ell to po logy o n ( C 0 ( X ) ⋊ G ) b as given in § 4 b elow. F ro m Coro llary 2.11 and Lemma 2 .2, if G acts prop erly on X , then the ﬁbre ( C 0 ( X ) ⋊ G ) G ( x ) of the crossed pro duct C 0 ( X ) ⋊ G ∼ = C 0  X × G, Ad ρ K ( L 2 ( G )  at the orbit G ( x ) is isomor phic to C 0 ( G/G x ) ⋊ G , equiv a lent ly , to the ﬁxed-p oint algebr a K ( L 2 ( G )) G x , Ad ρ for the compact stabilize r G x at x , where ρ : G → U ( L 2 ( G )) denotes the rig h t regula r re presentation of G . W e now analyse the structure o f this ﬁbre, using the Peter-W eyl theorem. Let us recall some basic constructions in r epresentation theory . If H is any Hilber t space we denote by H ∗ its adjoint Hilb ert sp ac e , that is H ∗ = { ξ ∗ : ξ ∈ H} with the linear op erations λξ ∗ + µη ∗ = ( ¯ λξ + ¯ µη ) ∗ and the inner pro duct h ξ ∗ , η ∗ i = h η , ξ i . Note that H ∗ ident iﬁes canonically with the space of contin uous linear functionals o n H . If σ : G → U ( H ) is a representation o f the group G on the Hilber t space H , then its ad joint re pr esent ation σ ∗ : G → U ( H ∗ ) is giv en b y σ ∗ ( g ) ξ ∗ := ( σ ( g ) ξ ) ∗ . Assume now that K is a c ompact subgro up o f G and let σ : K → U ( V σ ) b e a unitary r epresentation of K . W e then deﬁne a repres en tation π σ = P σ × U σ of the crossed pro duct C 0 ( G/K ) ⋊ G as follows: we deﬁne (3.1) H U σ := { ξ ∈ L 2 ( G, V σ ) : ξ ( g k ) = σ ( k − 1 ) ξ ( g ) ∀ g ∈ G, k ∈ K } . Then deﬁne the cov a riant representation ( P σ , U σ ) of C 0 ( G/K ) ⋊ G on H U σ by (3.2)  P σ ( ϕ ) ξ  ( g ) = ϕ ( gK ) ξ ( g ) and ( U σ ( t ) ξ )( g ) = ξ ( t − 1 g ) . Note t hat in classical representation theory of lo ca lly compact g roups the c ov a ri- ant pair ( P σ , U σ ) is often called the “system of imprimitivity” induced from the representation σ of K (e.g. s ee [10]). In what follo ws, if σ ∈ b K , we denote b y p σ ∈ C ∗ ( K ) the central pro jection corres p onding to σ , i.e., we ha ve p σ ∗ = d σ χ σ ∗ , where χ σ ∗ ( k ) = trace σ ∗ ( k ) denotes the character of the adjoint σ ∗ of σ and d σ denotes the dimension of V σ . Note that it follows from the Peter-W ey l theorem (e.g. see [1 3, Chapter 7]) that σ ( p σ ) = 1 V σ and τ ( p σ ) = 0 for all τ ∈ b K not equiv alent to σ , and tha t P σ ∈ b K p σ conv erges strictly to the unit 1 ∈ M ( C ∗ ( K )). Lemma 3.1. L et G b e a lo c al ly c omp act gr oup and let K b e a c omp act sub gr oup of G acting on K = K ( L 2 ( G )) via k 7→ Ad ρ ( k ) . F or e ach σ ∈ b K let L 2 ( G ) σ := ρ ( p σ ∗ ) L 2 ( G ) . Then t he following ar e true: (i) L 2 ( G ) = L σ ∈ b K L 2 ( G ) σ ; (ii) e ach sp ac e L 2 ( G ) σ is ρ ( K ) -invariant and de c omp oses into a tensor pr o duct H U σ ⊗ V ∗ σ such that ρ ( k ) | L 2 ( G ) σ = 1 H U σ ⊗ σ ∗ ( k ) for al l k ∈ K ; (iii) K K ∼ = L σ ∈ b K K ( H U σ ) , wher e the isomorphism is given by sending an op er- ator T ∈ K ( H U σ ) to t he op er ator T ⊗ 1 V ∗ σ ∈ K ( L 2 ( G ) σ ) under the de c om- p osition of (ii) ; CR OSSED PRODUCT S BY PR OPER ACTIONS 13 (iv) the pr oje ction of C 0 ( G/K ) ⋊ G ∼ = K K onto the factor K ( H U σ ) in t he de- c omp osition in (iii) is e qual to the r epr esentation π σ = P σ × U σ c onstru cte d ab ove in (3.1) and (3.2) . Pr o of. T he pro o f is basically a consequence of the Peter-W eyl Theorem for the compact group K . Item (i) follows fro m the fa ct that the cent ral pro jections p σ ∗ add up to the unit in M ( C ∗ ( K )) with resp ect to the strict top ology . F or the pr o of of (ii) w e ﬁrst consider the induced G -representation U λ K , whe r e λ K denotes the left regular representation o f K . It acts o n the Hilb ert space H U λ := { ξ ∈ L 2 ( G, L 2 ( K )) : ξ ( g k , l ) = ξ ( g , k l ) ∀ g ∈ G, k , l ∈ K } . But a s hort computation sho ws that U λ K ∼ = λ G , the left regular representation of G , where a unitary in tertwining o pe rator is g iven by Φ λ : H U λ ∼ = − → L 2 ( G ); Φ λ ( ξ )( g ) = ξ ( g , e ) . By the Peter-W eyl theorem we know that L 2 ( K ) decomp oses into the direct sum L σ ∈ b K V σ ⊗ V ∗ σ in such a wa y that the left regular representation deco mpos es as λ K ∼ = L σ σ ⊗ 1 V ∗ σ and the right reg ular repre sent ation decompo ses as ρ K ∼ = L σ 1 V σ ⊗ σ ∗ . (e.g. see [13, Theor em 7 .2.3] tog ether with the obvious isomor phism V σ ⊗ V ∗ σ ∼ = End( V σ )) This induces a decomp osition (3.3) L 2 ( G ) ∼ = H U λ ∼ = M σ ∈ b K H U σ ⊗ V ∗ σ . T o see how the isomo rphism Φ λ restricts to the direct summand H U σ ⊗ V ∗ σ we should note that the inclusio n of V σ ⊗ V ∗ σ int o L 2 ( K ) is given by sending an e lemen tary vector v ⊗ w ∗ to the function k 7→ √ d σ h σ ( k − 1 ) v , w i . The corr esp onding inclusion of H U σ ⊗ V ∗ σ int o L 2 ( G ) is therefore given b y sending an elementary vector ξ ⊗ w ∗ to the L 2 -function g 7→ √ d σ h ξ ( g ) , w i . One can ea sily check directly , using the orthogo nality relations for ma trix co eﬃcients on K , that this deﬁnes an isometry Φ σ from H U σ ⊗ V ∗ σ int o L 2 ( G ). T o show tha t the image lies in L 2 ( G ) σ we compute  ρ ( p σ ∗ )Φ σ ( ξ ⊗ v ∗ )  ( g ) = Z K p σ ∗ ( k ) p d σ h ξ ( g k ) , v i dk = Z K p σ ∗ ( k ) p d σ h ξ ( g ) , σ ( k ) v i dk = p d σ h ξ ( g ) , v i = Φ σ ( ξ ⊗ v ∗ )( g ) which proves the claim. Using (i) and (3.3 ), this als o implies tha t the image is all of L 2 ( G ) σ . One easily checks that Φ σ int ertwines the representation 1 ⊗ σ ∗ on H U σ ⊗ V ∗ σ with the r estriction of ρ to K . F or the pro of of (iii) we ﬁrst o bserve that T ∈ K ( L 2 ( G )) K if and only if T commutes with ρ ( k ) for a ll k ∈ K , which then implies, via int egra tion, that T commutes with ρ ( p σ ) for all σ ∈ b K . It follows that K K lies in L σ ∈ b K K ( L 2 ( G ) σ ) ⊆ K ( L 2 ( G )). No w, using the decomp os itio n L 2 ( G ) σ = H U σ ⊗ V ∗ σ as in (ii) w e get K ( L 2 ( G ) σ ) Ad ρ ( K ) ∼ =  K ( H U σ ) ⊗ K ( V ∗ σ )  Id ⊗ Ad σ ∗ ( K ) = K ( H U σ ) ⊗ C 1 V ∗ σ . Finally , item (iv) now follows from the fact that the res triction of the represen- tation M × λ : C 0 ( G/K ) ⋊ G ∼ = → K ( L 2 ( G )) K to the subspa c e H U σ ⊗ V ∗ σ clearly coincides with ( P σ × U σ ) ⊗ 1 V ∗ σ .  Example 3.2 . The above decomp o sition b ecomes easie r in the case where the co m- pact subgroup K of G is abe lia n, s ince in this case the irreducible representations σ of K are one- dimensional. W e therefore get p σ = ¯ σ view ed as an elemen t of 14 ECHTERHOFF AND EMERSON C ( K ) ⊆ C ∗ ( K ) and then, for ξ ∈ L 2 ( G ) we hav e ξ ∈ L 2 ( G ) σ = ρ ( p σ ) L 2 ( G ) if and only if ξ ( g ) = ( ρ ( p σ ) ξ )( g ) = Z K ¯ σ ( k ) ξ ( g k ) dk for almost a ll g ∈ G . F or l ∈ K we then get ξ ( g l ) = Z K ¯ σ ( k ) ξ ( g l k ) dk k 7→ l − 1 k = Z K ¯ σ ( l − 1 k ) ξ ( g k ) dk = σ ( l ) ξ ( g ) , which s hows that in this situation w e hav e L 2 ( G ) σ = H U σ for a ll σ ∈ b K . Thus we get the direct decompo sitions L 2 ( G ) = M σ ∈ b K H U σ and K ( L 2 ( G )) K, Ad ρ = M σ ∈ b K K ( H U σ ) . This pictur e bec omes even more tra ns parent if G happ ens also to b e abelia n. In that case one chec ks that the F ourier transform F : L 2 ( G ) → L 2 ( b G ) maps the subspace L 2 ( G ) σ of L 2 ( G ) to the subspace L 2 ( b G σ ) of L 2 ( b G ) in whic h b G σ := { χ ∈ b G : χ | K = σ } (w e leav e the deta ils as an exercise to the reader). In the sp ecial case G = T and K = C ( n ), the group o f a ll n th r o ots o f unity , we get [ C ( n ) ∼ = Z /n Z and, using F our ie r tra nsform, the co mpo s ition of Lemma 3 .1 b ecomes L 2 ( T ) ∼ = ℓ 2 ( Z ) ∼ = M [ l ] ∈ Z /n Z ℓ 2 ( l Z ) a nd K ( L 2 ( T )) C ( n ) ∼ = M [ l ] ∈ Z /n Z K ( ℓ 2 ( l Z )) . If L ⊆ K a re t wo co mpact subgroups of the lo cally compact group G , then w e certainly have K ( L 2 ( G )) Ad ρ ( K ) ⊆ K ( L 2 ( G )) Ad ρ ( L ) . F or la ter us e, it is impor tant for us to hav e a precise understanding of this inclus ion. In the fo llowing lemma we denote by Rep( L ) the equiv alenc e classes o f all unita r y represent ations of a group L and Rep( A ) denotes the equiv alence c la sses of all non-degenerate ∗ - representation of a C* -algebra A . In this notation we o btain a map Ind G L : Rep( L ) → Rep( C 0 ( G/L ) ⋊ G ); σ 7→ Ind G L σ := P σ × U σ , and similarly for K , with P σ and U σ deﬁned a s in (3.1) and (3.2). Mor eov er, induction of unitary representations gives a mapping Ind K L : Rep( L ) → Rep( K ) and the inclusion C 0 ( G/K ) ⋊ G in to C 0 ( G/K ) ⋊ G indu ced b y the o b vious inclusion of C 0 ( G/K ) in to C 0 ( G/L ) induces a mapping Res G/K G/L : Rep( C 0 ( G/L ) ⋊ G ) → Rep( C 0 ( G/K ) ⋊ G ) . Lemma 3.3. Supp ose that L ⊆ K and G ar e as ab ove. Then the di agr am (3.4) Rep( K ) Ind G K − − − − → Rep( C 0 ( G/K ) ⋊ G ) Ind K L x   x   Res C 0 ( G/K ) C 0 ( G/L ) Rep( L ) − − − − → Ind G L Rep( C 0 ( G/L ) ⋊ G ) c ommut es. Pr o of. L e t σ ∈ Rep( L ). Recall from (3.1) that the Hilb ert space H U σ for the induced repr e sent ation Ind G L σ = P σ × U σ is deﬁned as H U σ = { ξ ∈ L 2 ( G, V σ ) : ξ ( g l ) = σ ( l − 1 ) ξ ( g ) ∀ g ∈ G, l ∈ L } , CR OSSED PRODUCT S BY PR OPER ACTIONS 15 and similar constructions giv e the Hilb ert spaces for the repr e sent ations Ind K L σ and Ind G K τ for some τ ∈ Rep( K ). In particular , for τ := Ind K L σ , we deduce the form ula H U τ = { η ∈ L 2 ( G, L 2 ( K, V σ )) : η ( g k , h ) = η ( g , k h ) and η ( g , hl ) = σ ( l − 1 ) η ( g , h ) ∀ g ∈ G, k , h ∈ K , l ∈ L } . It is then s traightforw ard to chec k that the op era tor V : H U σ → H U τ ; ( V ξ )( g , h ) = ξ ( g h ) is a unitar y with inv erse V − 1 given b y ( V − 1 η )( g ) = η ( g , e ) suc h that V in tertwines the G -repr e s ent ations U σ and U τ . More ov e r, for a ny ϕ ∈ C 0 ( G/K ) w e get ( P τ ( ϕ ) V ξ )( g , h ) = ϕ ( g K )( V ξ )( g , h ) = ϕ ( g K ) ξ ( g h ) = ϕ ( ghK ) ξ ( g h ) = ( P σ ( ϕ ) ξ )( g h ) = ( V P σ ( ϕ ) ξ )( g , h ) . This prov es that that V intert wines Res G/K G/L ◦ Ind G L σ with Ind L K τ = Ind G K ◦ Ind K L σ .  By Lemma 3.1 we hav e isomorphisms C 0 ( G/K ) ⋊ G M ⋊ λ ∼ = K ( L 2 ( G )) Ad ρ ( K ) = M τ ∈ b K K ( H U τ ) ⊗ 1 V ∗ τ where the r ight equation is induced b y the decompo sition L 2 ( G ) = M τ ∈ b K H U τ ⊗ V ∗ τ . Thu s w e we see that, as a subalg ebra of K ( L 2 ( G )), the alge bra K ( L 2 ( G )) Ad ρ ( K ) decomp oses in blo cks of co mpact op era tors K ( H U τ ) s uch that each block K ( H U τ ) app ears with the multiplicit y dim V τ in this decomp osition. If L ⊆ K , w e g et a similar decomp osition of K ( L 2 ( G )) Ad ρ ( L ) indexed ov er all σ ∈ b L with mult iplicities dim V σ . Since L ⊆ K we get K ( L 2 ( G )) Ad ρ ( K ) ⊆ K ( L 2 ( G )) Ad ρ ( L ) . T o unders tand this inclusion, we need to know how many copies of each blo ck K ( H U τ ), τ ∈ b K , app ear in any given blo ck K ( H U τ ), σ ∈ b L , o f the a lg ebra K ( L 2 ( G )) Ad ρ ( L ) . This m ultiplicity num b er m σ τ clearly co incides with the multiplicit y of the represe nta- tion π τ = P τ × U τ of C 0 ( G/K ) ⋊ G ∼ = K ( L 2 ( G )) Ad ρ ( K ) in the re s triction of the representation π σ = P σ × U σ of C 0 ( G/L ) ⋊ G ∼ = K ( L 2 ( G )) Ad ρ ( L ) to the subalg e br a C 0 ( G/K ) ⋊ G . By the abov e lemma, this m ultiplicity coincides with the m ultiplicit y of τ in the induced representation Ind K L σ , and by F rob enius recipro city , this equals the multiplicit y of σ in the r estriction τ | L . Thus w e conclude Prop ositio n 3.4. Supp ose that L ⊆ K ar e two c omp act sub gr oups of the lo c al ly c omp act gr oup G . F or e ach τ ∈ b K and σ ∈ b L let m σ τ denote the multiplicity of σ in the r estriction τ | L . Then, u nder the inclusion K ( L 2 ( G )) Ad ρ ( K ) ⊆ K ( L 2 ( G )) Ad ρ ( L ) e ach blo ck K ( H U τ ) of K ( L 2 ( G )) Ad ρ ( K ) app e ars with multiplicity m σ τ in e ach blo ck K ( H U σ ) of K ( L 2 ( G )) Ad ρ ( L ) . W e should note that since eac h blo ck K ( H U σ ) app ears with multiplicit y dim V σ in the representation of K ( L 2 ( G )) Ad ρ ( L ) on L 2 ( G ), and similarly fo r τ ∈ b K , we get the equation dim V τ = X σ ∈ b L m σ τ dim V σ , for the total m ultiplicity dim V τ of K ( H U τ ) in K ( L 2 ( G )) Ad ρ ( K ) . Le t us illustrate the ab ov e results in a concrete example: 16 ECHTERHOFF AND EMERSON Example 3 .5 . Let us consider the a c tion of the ﬁnite group G = D 4 = h R, S i ⊆ GL(2 , Z ) with R =  0 − 1 1 0  and S =  1 0 0 − 1  on T 2 as des c rib ed in Ex ample 2.6. It was show in that exa mple that we have the topolo gical fundament al domain Z := { ( e 2 π is , e 2 π it ) : 0 ≤ t ≤ 1 2 , 0 ≤ s ≤ t } and it follows then from P rop osition 2.7 that the cross e d pro duct C 0 ( T 2 ) ⋊ G is isomorphic to the sub-homogeneous algebra A := { f ∈ C ( Z, K ( ℓ 2 ( G ))) : f ( z , w ) ∈ K ( ℓ 2 ( G ))) Ad ρ ( G ( z,w ) ) } , where G ( z ,w ) denotes the stabilizer of the point ( z , w ) under the action o f G . In what follows we will iden tify Z with the tria ngle { ( s, t ) ∈ R 2 : 0 ≤ t ≤ 1 2 , 0 ≤ s ≤ t } and w e write G ( s,t ) for the corresp onding stabilizers of the p oints ( e 2 π is , e 2 π it ). A straightforward computation shows that • G ( s,t ) = { E } if 0 < t < 1 2 , 0 < s < t , • G ( s,s ) = h RS i =: K 1 if 0 < s < 1 2 , • G (0 ,t ) = h R 2 S i =: K 2 if 0 < t < 1 2 , • G ( s, 1 2 ) = h S i =: K 3 if 0 < s < 1 2 , • G (0 , 1 2 ) = h S, R 2 i =: H , and • G (0 , 0) = G ( 1 2 , 1 2 ) = G . It follows that K ( ℓ 2 ( G )) G ( s,t ) = K ( ℓ 2 ( G )) ∼ = M 8 ( C ) whenever 0 < t < 1 2 , 0 < s < t . In the three cases where G ( s,t ) = K i , i = 1 , 2 , 3, is a subgro up of order tw o, we get t wo one-dimensional repr esentations { 1 K i , ǫ K i } , i = 1 , 2 , 3 , s o by c ho osing s uita ble bases of ℓ 2 ( G ), in each case the a lgebra K ( ℓ 2 ( G )) G ( s,t ) has the for m  A 1 K i 0 0 A ǫ K i  A 1 K i , A ǫ K i ∈ M 4 ( C ) , where the 4 × 4-blo cks act on the four-dimensiona l subspace s ρ ( p 1 K i ) ℓ 2 ( G ) and ρ ( p ǫ K i ) ℓ 2 ( G ), resp ectively , where ρ deno tes the right r e g ular repr esentation of G restricted to the resp ective stabilizer. W e r efer to the discussion b efor e Lemma 3.1 for the deﬁnition of p 1 K i and p ǫ K i . A t the cor ne r (0 , 1 2 ) we hav e the stabilizer H = h S, R 2 i ∼ = Z / 2 × Z / 2 , s o we get four one-dimensio na l r epresentations 1 , µ 1 , µ 2 , µ 3 of this gr oup given by µ 1 ( R 2 ) = − µ 1 ( S ) = 1 , µ 2 ( R 2 ) = µ 2 ( S ) = − 1 and µ 3 ( R 2 ) = − µ 3 ( S ) = − 1 . Therefore K ( ℓ 2 ( G )) G (0 , 1 2 ) decomp oses as     B 1 B µ 1 B µ 2 B µ 3     B 1 , B µ 1 , B µ 2 , B µ 3 ∈ M 2 ( C ) with corr esp onding rank-tw o pro jections ρ ( p µ ), µ ∈ b H . The representation theory of G , the stabilizer o f the remaining co rners (0 , 0 ) and ( 1 2 , 1 2 ) o f Z , is a s follows: there is the ‘s tandard’ r epresentation λ : G → O (2 , R ) ⊂ U (2) (this is irreducible). The other ir reducible repres en tations are one-dimensio nal and corre spo nd to the repres e n tations of the quo tien t group G/ h R 2 i ∼ = Z / 2 × Z / 2. They are listed as { 1 G , χ 1 , χ 2 , χ 3 } with χ 1 ( R ) = − χ 1 ( S ) = 1 , χ 2 ( R ) = χ 2 ( S ) = − 1 and χ 3 ( R ) = − χ 3 ( S ) = − 1 . Therefore the set of irreducible repres e n tations of G is { 1 , χ 1 , χ 2 , χ 3 , λ } . Re pr e- senting C ∗ ( G ) ∼ = K ( ℓ 2 ( G )) G as a subalgebra of K ( ℓ 2 ( G )) ∼ = M 8 ( C ) g ives one blo ck CR OSSED PRODUCT S BY PR OPER ACTIONS 17 M 2 ( C ) with mult iplicity 2 and four one- dmensional blo cks. With resp ect to a suit- able c hosen base of ℓ 2 ( G ) ∼ = C 8 , w e obtain a repres en tation as matr ices of the form         C λ C λ d 1 d χ 1 d χ 2 d χ 3         C λ ∈ M 2 ( C ) , d 1 , d χ 1 , d χ 2 , d χ 3 ∈ C , where the low er diag onal entries a ct on the imag es o f the pro jections ρ ( p χ ) for χ ∈ { 1 G , χ 1 , χ 2 , χ 3 } and the blo ck  C λ C λ  acts on the four-dimensio nal space ρ ( p λ ) ℓ 2 ( G ). T o under stand the structur e of the algebr a C ( T 2 ) ⋊ G ∼ = { f ∈ C ( Z , K ( ℓ 2 ( G ))) : f ( s, t ) ∈ K ( ℓ 2 ( G )) G ( s,t ) } , we need to under stand what ha pp ens at the three corner s (0 , 0) , ( 1 2 , 1 2 ) , (0 , 1 2 ) o f the fundamental do main Z when approached on the bor der lines of Z . So assume that f ∈ C ( T 2 ) ⋊ G is represented as a function f : Z → M 8 ( C ) ∼ = K ( ℓ 2 ( G )). Since the stabilizer s of the co r ners contain the stabilizers of the adjacent b or der lines, w e see from the ab ov e discussion that the ﬁbers K ( ℓ 2 ( G )) G ( s,t ) at the corners m ust b e contained in the intersections of the ﬁber s at the a djacent b order lines. Prop ositio n 3.4 ab ove tells us, how these inclusions lo ok like: Let us consider the corner (0 , 0 ). The adjacent bo rder lines have stabilizers K 1 = G ( s,s ) = h RS i a nd K 2 = G (0 ,t ) = h R 2 S i res pectively . A short co mputation shows that the restrictio n of λ to K 1 and K 2 decomp oses into the direct sum 1 K i L ǫ K i for i = 1 , 2. So each of the tw o 4 × 4-blo cks in the decomp ositio n K ( ℓ 2 ( G )) K i =  A 1 K i 0 0 A ǫ K i  : A 1 K i , A ǫ K i ∈ M 4 ( C )  , contains exactly one copy of the tw o by tw o blocks C λ in the decompo sition of K ( ℓ 2 ( G )) G . Consider no w the one-dimensional repr esentations 1 G , χ 1 , χ 2 , χ 3 of G . If w e re strict these repr esentations to K 1 we see that 1 G and χ 2 restrict to trivia l character 1 K 1 and χ 1 , χ 3 restrict to the no n-trivial character ǫ K 1 . Th us, the 4 × 4 blo ck A 1 K 1 contains the diagonal e ntries d 1 , d χ 2 and the blo ck A ǫ K 1 contains the diagonal entries d χ 1 , d χ 3 . On the other side, if we r estrict 1 G , χ 1 , χ 2 , χ 3 to th e subgroup K 2 = G (0 ,t ) we see that 1 G , χ 3 restrict to 1 K 2 and χ 1 , χ 2 restrict to the non-trivial character ǫ K 2 . W e therefore see tha t, diﬀerent from the c ase K 1 = G ( s,s ) , A 1 K 2 contains the diagonal ent ries corresp onding to the c haracters d 1 , d χ 3 and the blo ck A ǫ K 2 contains the diagonal en tries corr esp onding to d χ 1 , d χ 2 . So, even in this simple example, we g et a quite in tricate str uc tur e o f the a lg ebra C ( T 2 ) ⋊ G at the ﬁbe rs with nontrivial stabilizers. W e shall r evisit this exa mple in the following section. W e no w pro ceed with the general theory: Deﬁnition 3.6. Let X b e a prop er G -space. W e deﬁne the st abilizer gr oup bund le Stab( X ) as Stab( X ) = { ( x, g ) : x ∈ X , g ∈ G x } . and we deﬁne Stab( X ) b := { ( x, σ ) : x ∈ X, σ ∈ b G x } . 18 ECHTERHOFF AND EMERSON Note that G a cts on Stab( X ) b b y g ( x, σ ) = ( g x, g σ ) with g σ := σ ◦ C − 1 g , where C g : G x → G gx is the isomor phism given by conjugation with g . F or ea ch ( x, σ ) ∈ Stab( X ) b consider the induced representation π σ x = P σ x × U σ x acting o n the Hilb e rt space H U σ as deﬁned in (3.1) a nd (3.2). One easily checks that π σ x is unitarily equiv a lent to π gσ gx , the eq uiv alence b eing giv en by the unitary W : H σ → H σ ◦ C − 1 g ;  W ξ  ( t ) = p ∆( g ) ξ ( tg ) . In what f ollows, we shall also write Ind G G x ( x, σ ) for the repr esentation π σ x , indicating that it is the induced r epresentation in the clas s ical sense of Mack ey , Glimm and others. Thus, as a corollar y of the a bove lemma, we obtain a pr o of of the following theorem, which is a well-known sp ecial ca se of the general Mack ey -Green-Rieﬀel machine for crossed pro ducts. Theorem 3.7 (Mack ey-Rieﬀel-Green) . The map Ind : Stab( X ) b → ( C 0 ( X ) ⋊ G ) b ; ( x, σ ) 7→ Ind G G x ( x, σ ) = π σ x factors thr ough a set bije ction (which we also denote Ind ) b etwe en the orbit sp ac e G \ Stab( X ) b and the sp ac e  C 0 ( X ) ⋊ G  b of e quivalenc e classes of irr e ducible ∗ - r epr esentations of C 0 ( X ) × G . Pr o of. T he pr o of is an easy combin ation of the bundle structure of C 0 ( X ) ⋊ G ∼ = C 0 ( X × G, Ad ρ K ) and the description of the ﬁbre K G x as given in the previous lemma.  If we loo k at the trivial represe ntation 1 G x : G x → { 1 } , it follows from Lemma 3.1 that the cor resp onding summand of the ﬁbre K G x of C 0 ( X ) ⋊ G is given by K ( L 2 ( G ) 1 G x ) with (3.5) L 2 ( G ) 1 G x = { ξ ∈ L 2 ( G ) : ξ ( g k ) = ξ ( g ) ∀ k ∈ G x } ∼ = L 2 ( G/G x ) . Let c : X → [0 , 1] b e a cut -oﬀ function for the prop er G -space X , which means that c is a contin uo us function with compact supp o rt on any G -compact subs e t of X (a closed subset Y of X is called G -c omp act if it is G -in v a r iant with G \ Y compact) such that R G c ( g − 1 x ) 2 dg = 1 for all x ∈ X . Then (3.6) p X ( g , x ) = p ∆( g − 1 ) c ( g − 1 x ) c ( x ) determines a pro jection p X ∈ M ( C 0 ( X ) ⋊ G ) via co n volution given by the same formula as for the multiplication o n C c ( G × X ) ⊂ C 0 ( X ) ⋊ G . Note t hat p X ∈ C 0 ( X ) ⋊ G if and only if X is G -c omp act , while in gene r al, f · p X ∈ C 0 ( X ) ⋊ G for every f ∈ C 0 ( G \ X ) ⊆ Z M ( C 0 ( X ) ⋊ G ). R emark 3.8 . W e make the following remarks ab out cut-oﬀ functions and their asso ciated pro jectio ns. • If X is compact G m ust be compact to o, and in this c a se with resp ect to normalised Haar mea sure on G , the constant function c ( x ) := 1 for all x ∈ X is a cut-oﬀ function. In th is case C ∗ ( G ) is a subalgebra of C ( X ) ⋊ G , and the pro jection p X is in the subalge bra: it is the constant function 1 on G and pro jects to the spac e of G -ﬁxed vectors in an y repr esentation of G . • F or any prop er actio n, the spac e of square ro o ts of cut-oﬀ functions is conv ex, so the space of cut-oﬀ functions is contractible and hence any t wo pro jections p X are homotopic. CR OSSED PRODUCT S BY PR OPER ACTIONS 19 Lemma 3 . 9. L et c : X → [0 , 1] and p X ∈ M ( C 0 ( X ) ⋊ G ) b e as ab ove and let Φ : C 0 ( X ) ⋊ G ∼ = − → C 0 ( X × G K ) denote the isomorphism of Cor ol lary 2.11. F or e ach x ∈ X let c x denote the unit ve ct or in L 2 ( G ) given by c x ( g ) = p ∆( g − 1 ) c ( g x ) and let p x ∈ K ( L 2 ( G )) denote the image of p X under the evaluation map q x : C 0 ( X ) ⋊ G → K G x as describ e d in R emark 2.12. Then c x ∈ L 2 ( G ) 1 G x and p x is the ortho gonal pr oje ction ont o C · c x . Pr o of. T he ﬁrst assertion fo llows from the deﬁnition of c x together with the fact that the mo dular function v anishes on c o mpact subgr oups of G . F or the second asser tio n observe that it follows fr o m Remark 2.12 that p x acts on L 2 ( G ) via conv olution with the function p x ∈ C c ( G × G/G x ) ⊆ C 0 ( G/G x ) ⋊ G given b y p x ( g , hG x ) = ∆( g − 1 h ) c x ( g − 1 h ) c x ( h ) . If ξ ∈ L 2 ( G ) is a r bitrary , w e get ( p x ξ )( h ) = Z G p x ( g , hG x ) ξ ( g − 1 h ) dg = Z G ∆( g − 1 h ) c x ( g − 1 h ) c x ( h ) ξ ( g − 1 h ) dg =  Z G c x ( g ) ξ ( g ) dg  c x ( h ) =  h ξ , c x i c x  ( h ) , where the se c ond to last e q uation follows fro m the transfor ma tion g − 1 h 7→ g .  It follows fro m the ab ove lemma that the pro jection p X ∈ M ( C 0 ( X ) ⋊ G ) co n- structed ab ov e is a c o nt inuous ﬁeld of rank- one pro jections on G \ X such that under the decompo s ition of ea ch ﬁbre K G x ∼ = L σ ∈ b G x K ( H σ ) as in part (iii) o f Lemma 3.1, the res tr iction of p X to that ﬁbre lies in the co mpo nen t K ( H 1 G x ). It follows in particula r that C 0 ( G \ X ) is isomorphic to the corner p X  C 0 ( X ) ⋊ G  p X via f 7→ f · p X , and thus C 0 ( G \ X ) ∼ = p X  C 0 ( X ) ⋊ G  p X is Morita equiv alent to the ideal I X =  C 0 ( X ) ⋊ G  p X  C 0 ( X ) ⋊ G  of C 0 ( X ) ⋊ G gener ated b y p X (this is a g eneral fact ab out co r ners.) Lemma 3.1 implies that under the iso morphism C 0 ( X ) ⋊ G ∼ = C 0 ( X × G K ) we get (3.7) I X = { F ∈ C 0 ( X × G K ) : F ( x ) ∈ K ( L 2 ( G ) 1 G x ) ∀ x ∈ X } . It follows in particular that the idea l I X do es not dep end o n the particular c hoice of the cut-oﬀ function c : X → [0 , 1] and the corresp onding pro jection p X . If the ac tio n of G on X is fr e e and pr op er , then it is immediate from (3.7) and the description of C 0 ( X ) ⋊ G in Coro llary 2 .11 and the following Remark 2.12, that I X = C 0 ( X ) ⋊ G . W e th us recov er the well-known theorem, due to Phillip Green (see [29]) that C 0 ( G \ X ) ∼ M C 0 ( X ) ⋊ G for a free and prop er actio n of a lo c a lly compact gro up G . Example 3.10 . The ab ov e can b e made rather e xplicit in the case of ﬁnite gr oup actions. F or deﬁnitenes s, w e let G = Z / 2, X is compact. By Remark 3.8, we may take p X ∈ C ( X ) ⋊ G to b e p X ( x, g ) = 1 | G | , while C ( X ) ⋊ G ∼ = C  X , K ( ℓ 2 ( G ))  G . W e can co nsider K ( ℓ 2 ( G )) as 2 × 2-matrices , and the G -inv ar iance says the element s in C  X , K ( ℓ 2 ( G ))  G m ust have the for m a =  f g σ ( g ) σ ( f )  20 ECHTERHOFF AND EMERSON for some f , g ∈ C ( X ), wher e σ is the underlying o rder tw o automorphism of C ( X ). The pr o jection p X corres p onds to the matrix 1 | G |  1 1 1 1  . The ideal I X =  C ( X ) ⋊ G  p X  C ( X ) ⋊ G  is then given by the clo sed linea r hull of matrices of the form  f g f σ ( g ) σ ( f ) g σ ( f g )  , f , g ∈ C ( X ), while the cor ne r p X  C ( X ) ⋊ G  p X consists of all matrices of the form  f f f f  where σ ( f ) = f . This C*-algebra is isomor phic to C ( G \ X ). It is also useful to give a r e pr esentation theo retic description of the ideal I X in terms o f the Mack ey-Rieﬀel-Gree n machine of Theorem 3 .7. F or this recall that for any closed t wo-sided ideal I in a C*-alg ebra A the sp e ctrum b I includes as an op en s ubset o f b A via (unique) extension of irr educible r epresentations from I to A , and the resulting corr esp o ndence I ⊆ A ↔ b I ⊆ b A is one-to-one. Th us the ideal I X in C 0 ( X ) ⋊ G is uniquely determined by the set of irre ducible r epresentations of C 0 ( X ) ⋊ G whic h do not v a nish on I X . The ab ove results now c ombine to the following: Prop ositio n 3.11. The irr e ducible r epr esen t ations of C 0 ( X ) ⋊ G wh ich c orr esp ond to the de al I X ar e pr e cisely the re pr esent ations of the form π 1 G x x = Ind G G x ( x, 1 G x ) , x ∈ X , wher e 1 G x denotes the trivia l r epr esentation of the stabilizer G x . Example 3.12 . The ab ov e representation theor etic description of the Ideal I X makes it easy to identify this ideal in the case of the cro ssed pro duct C ( T 2 ) ⋊ G with G = D 4 acting on X = T 2 as describ ed in Examples 2.6 an 3 .5. If we realize C ( T 2 ) ⋊ G = { f ∈ C ( Z , K ( ℓ 2 ( G ))) : f ( s, t ) ∈ K ( ℓ 2 ( G )) G ( s,t ) } as in E x ample 3.5, then, with r e s pec t to the des cription of the ﬁbers K ( ℓ 2 ( G )) G ( s,t ) as given in that ex ample, the ideal I T 2 consists o f those functions whic h take a r- bitrary v alues in the in terior o f Z and which take v alues in the corners A 1 at the bo undaries o f Z and in d 1 and B 1 at the corner s (0 , 0), ( 1 2 , 1 2 ) and (0 , 1 2 ), resp ec- tively . Recall from § 1 tha t ev ery prop er G -space X is lo cally induced b y compact sub- groups of G , which means that eac h x ∈ X has a G -in v ariant open neighbor ho o d U of the for m U ∼ = G × K Y . F or later use we need to compare the Mackey-Rieﬀel-Green map of C 0 ( G × K Y ) ⋊ G with that of C 0 ( Y ) ⋊ K . By a version o f Green’s imprim- itivit y theorem we kno w that C 0 ( G × K Y ) ⋊ G is Morita equiv alent to C 0 ( Y ) ⋊ K . The imprimitivity bimo dule E is given b y a completion of E 0 = C c ( G × Y ) with underlying pre-Hilb ert C c ( K × Y )-structure given by h ξ , η i C c ( K × Y ) ( k , y ) = Z G ξ ( g − 1 , y ) η ( g − 1 k , k − 1 y ) dg ξ · b ( g , y ) = Z K ξ ( g k − 1 , k y ) b ( k , k y ) dk (3.8) for ξ , η ∈ E 0 and b ∈ C c ( K × Y ) ⊆ C 0 ( Y ) ⋊ K , and with left actio n of the dense subalgebra C c ( G × ( G × K Y )) of C 0 ( G × K Y ) ⋊ G on X 0 given by the cov ariant representation ( P , L ) suc h that (3.9) ( P ( F ) ξ )( g , y ) = F ([ g , y ]) ξ ( g , y ) a nd ( L ( t ) ξ )( g , y ) = ∆( t ) 1 / 2 ξ ( t − 1 g , y ) , for F ∈ C 0 ( G × K Y ). These formulas follow from [48, Cor ollary 4.17] b y identifying C 0 ( G × K Y ) with C 0 ( G × K C 0 ( Y )) (= Ind G K C 0 ( Y ) in the no tation o f [4 8]). Induction of representations fr om C ( Y ) ⋊ K to C 0 ( G × K Y ) ⋊ G via the imprimitivity bimo dule E induces a homeo morphism betw een  C 0 ( Y ) ⋊ K  b and  C 0 ( G × K Y ) ⋊ G  b . CR OSSED PRODUCT S BY PR OPER ACTIONS 21 Prop ositio n 3.13. Supp ose that K is a c omp act su b gr oup of G , Y is a K - sp ac e and X = G × K Y . Then ther e is a c ommutative diagr am of bije ctive maps K \ Stab( Y ) b ι − − − − → G \ Stab( X ) b Ind K   y   y Ind G ( C 0 ( Y ) ⋊ K ) b − − − − → Ind E ( C 0 ( X ) ⋊ G ) b wher e the vertic al maps ar e the r esp e ctive induction maps of The or em 3.7, the lower horizontal map is induction via the imprimitivity bimo dule c onstruct e d ab ove and the u pp er h orizontal map is give n by the map on orbit sp ac es induc e d by the inclusion ι : Stab ( Y ) b → St ab ( X ) b , ( y , σ ) 7→ ([ e, y ] , σ ) . Pr o of. L e t ( y , σ ) ∈ Stab( Y ) b . Let τ σ y denote the r epresentation of C 0 ( Y ) ⋊ K induced from ( y, σ ) and let π σ y denote the represen tation of C 0 ( G × K Y ) ⋊ G induced from ([ e, y ] , σ ). Let H K σ and H G σ denote the respective Hilb ert spaces on which they act. W e hav e to chec k that Ind E τ σ y ∼ = π σ y . Recall that Ind E τ σ y acts o n the Hilb ert space E ⊗ C 0 ( Y ) ⋊ K H K σ via the left action of C 0 ( G × K Y ) ⋊ G on E , as sp eciﬁed in (3.9). Recall from (3.1) that H K σ = { ϕ ∈ L 2 ( K, V σ ) : ϕ ( k l ) = σ ( l − 1 ) ϕ ( k ) ∀ l ∈ K y } (since K is unimo dular) and similarly for H G σ . W e claim that there is a uniq ue unitary op era to r Φ : E ⊗ C 0 ( Y ) ⋊ K H K σ → H G σ given o n elemen tary tens o rs ξ ⊗ ϕ , ξ ∈ E 0 , ϕ ∈ H K σ , by Φ( ξ ⊗ ϕ )( g ) = ∆( g ) − 1 / 2 Z K ξ ( g k − 1 , k y ) ϕ ( k ) dk . A quic k computation sho ws that Φ( ξ ⊗ ϕ )( g l ) = ∆ G ( l ) − 1 / 2 σ ( l − 1 )Φ( ξ ⊗ ϕ )( g ). T o see that Φ preserves the inner pro ducts we compute for all ξ , η ∈ E 0 and ϕ, ψ ∈ H K σ : h Φ( ξ ⊗ ϕ ) , Φ( η ⊗ ψ ) i H G σ = Z G h Φ( ξ ⊗ ϕ )( g ) , Φ( η ⊗ ψ )( g ) i V σ dg = Z G ∆( g − 1 ) Z K Z K h ξ ( g k − 1 , k y ) ϕ ( k ) , η ( g l − 1 , l y ) ψ ( l ) i V σ dl dk dg = Z G Z K Z K ξ ( g − 1 k − 1 , k y ) η ( g − 1 l − 1 , l y ) h ϕ ( k ) , ψ ( l ) i V σ dl dk dg while on the other side we get h ξ ⊗ ϕ, η ⊗ ψ i E ⊗ C 0 ( Y ) ⋊ K H K σ = h τ σ y ( h η , ξ i E ) ϕ, ψ i H K σ . F or a function f ∈ C c ( K × Y ) ⊆ C 0 ( Y ) ⋊ K the op erator τ σ y ( f ) acts on ϕ ∈ H K σ by  τ σ y ( f ) ϕ  ( l ) = Z K f ( k, l y ) ϕ ( k − 1 l ) dk . Applying this together with the formula for h ξ , η i E as given in (3.8), we g et h ξ ⊗ ϕ, η ⊗ ψ i E ⊗ C 0 ( Y ) ⋊ K H K σ = Z K Z K  h η , ξ i E ( l, k y ) ϕ ( l − 1 k ) , ψ ( k )  V σ dl dk = Z K Z K Z G ξ ( g − 1 k , k − 1 l y ) η ( g − 1 , l y ) h ϕ ( k − 1 l ) , ψ ( l ) i V σ dg dl dk = Z G Z K Z K ξ ( g − 1 k − 1 , k y ) η ( g − 1 l − 1 , l y ) h ϕ ( k ) , ψ ( l ) i V σ dl dk dg = h Φ( ξ ⊗ ϕ ) , Φ( η ⊗ ψ ) i H G σ , 22 ECHTERHOFF AND EMERSON where the s econd to last eq ua tion follo ws from F ubini and the transformation g 7→ l g follow ed by the tra nsformation k 7→ lk − 1 . It follows now that Φ extends to a w ell deﬁned isometry from E ⊗ C 0 ( Y ) ⋊ K H K σ int o H G σ . W e now show that it int ertwines the representations I nd E τ σ y and π σ y . Since these representations ar e irreducible, this will then also imply surjectivity of Φ. F or the left a ction of F ∈ C 0 ( G × K Y ) w e chec k Φ( P ( F ) ξ ⊗ ϕ )( g ) = ∆( g ) − 1 / 2 Z K F ([ g k − 1 , k y ]) ξ ( g k − 1 , k y ) ϕ ( k ) dk = F ([ g , y ])Φ( ξ ⊗ ϕ )( g ) =  P σ [ e,y ] ( F )Φ( ξ ⊗ ϕ )  ( g ) , where we used the equations [ g k − 1 , k y ] = [ g , y ] = g · [ e, y ]. Similarly , for the actions of G we easily chec k Φ( L ( t ) ξ ⊗ ϕ )( g ) = Φ( ξ ⊗ ϕ )( t − 1 g ) =  U σ y ( t )Φ( ξ ⊗ ϕ )  ( g ) , which now completes the proo f.  Recall that a ny A − B -imprimitivity bimodule E induces a bijection of ideals in A and B . Under the corr esp ondence betw een ideals in A (resp. B ) and open subsets of b A (resp. b B ) the corres po ndence of ideals of A and B induced by E is compatible with the co rresp ondence of op en subsets in b A a nd b B given by the homeomorphism ind E : b B → b A . This all follows fr o m the Rieﬀel- c o rresp ondence a s explained in [42, Chapter 3.3]. Using these facts together with the a bove Prop osition 3.11 a nd Prop ositio n 3.13 we get Corollary 3.14. Supp ose that K is a c omp act sub gr oup of G , Y is a K -sp ac e and X = G × K Y . Then under the ab ove describ e d Morita e quivalenc e b etwe en C 0 ( X ) ⋊ G and C 0 ( Y ) ⋊ K the ide al I X in C 0 ( X ) ⋊ G c orr esp onds to the ide al I Y in C 0 ( Y ) ⋊ K . R emark 3.1 5 . Analogues of Pr op o sition 3.13 and Cor ollary 3.14 are a lso true if the compact s ubg roup K is r eplaced by any given closed subgroup H of G which acts prop erly on the space Y . The argument s are exa c tly the same—only some formulas bec ome a bit more complicated due to the app earence of the mo dular function on H . Since we only need the compact case b elow, w e restr icted to this cas e here. 4. The spectrum of C 0 ( X ) ⋊ G As ﬁrst step to obtain any further pro gress for K-theory computations of crosse d pro ducts b y prop er actions with non- isolated free o rbits, it s hould be useful to obtain a b etter under standing of the idea l s tr ucture of the cro ssed pr o ducts. Since closed ideals in C 0 ( X ) ⋊ G corres p ond to op en subsets o f ( C 0 ( X ) ⋊ G ) b , this problem is strongly related to a computation of the top olo g y of the r e presentation spac e ( C 0 ( X ) ⋊ G ) b . So in this section we giv e a detailed descr iption of the top o logy of  C 0 ( X ) ⋊ G  b in terms of the bijection with the parameter s pace G \ Stab( X ) b as in Theorem 3.7. T o be more pr e c ise, we sha ll in tro duce a top ology o n Stab( X ) b such that the bijection of Theor em 3.7 b e comes a homeo morphism, if G \ Sta b( X ) b ca rries the corres p onding quotien t top olog y . Note that Ba ggett g ives in [6] a gener al descr ip- tion of the to po logy of th e unitary duals of semi-direct product groups N ⋊ K , with N ab elian and K compact in terms o f the F ell- topo logy on the set of subgr oup representations of K . (This is the spa ce of all pairs ( L, τ ), where L is a subgroup of K and τ is a unitary representation of L , see [28, 32] for the deﬁnition.) Since ( N ⋊ K ) b = C ∗ ( N ⋊ K ) b ∼ = ( C 0 ( b N ) ⋊ K ) b the study of such semi-direct pro ducts can b e regar ded as a special case o f the study of crossed pro ducts b y proper actions. CR OSSED PRODUCT S BY PR OPER ACTIONS 23 Since F ell’s top olo gy o n subgroup repr esentations is not very easy to understand, we aim to deﬁne a suitable topo logy o n Stab( X ) b without using this construction. T o make this p oss ible, we r estrict o ur attention to actions whic h satisfy Palais’s slice prop erty (SP). Recall that this mea ns that the pr op er G -space X is lo cally induced from actions of the stabilizers. Reca ll also that by P alais’s Theorem (see Theorem 1.6), pr op erty (SP) is alwa ys satisﬁed if G is a Lie- group. Let us introduce some further nota tion: If X is a prop er G -space with pr op erty (SP), then for ev ery x ∈ X we deﬁne S x := { y ∈ X : G y ⊆ G x } . By an almost slic e at x ∈ X we shall understand an y s et o f the form W · V x , wher e W is an op en neigh b orho o d of e in G a nd V x is an op en neighbo rho o d of x in S x , i.e., V x = S x ∩ U x for some op e n neigh b orho o d U x of x in X . W e denote by AS x the set of a ll almost slices a t x . Lemma 4.1. Le t X b e a pr op er G -sp ac e with pr op erty (SP). Then the set AS x of al l almost slic es at x forms an op en neighb orho o d b ase at x . Pr o of. L e t W V x be an y almo st slice at x . T o see that it is op en in X le t y ∈ W V x be arbitrar y . Let g ∈ W such that y ∈ g V x . L et Y y be a lo ca l slice at y , i.e., there is an op en ne ig hborho o d U y of y suc h that U y = G · Y y ∼ = G × G y Y y . T he n Y y ⊆ S y ⊆ S gx . Thu s, pa ssing to a smaller lo cal slice if necessar y , we may assume that Y y ⊆ g V x . Now c ho ose an op en neighborho o d W ′ of e in G such that W ′ g ⊆ W . Then W ′ Y y ⊆ W ′ g V x ⊆ W V x is an op en neighbor ho o d of y contained in W V x . (Note that the map G × Y y → G · Y y is op en since it coincides with the quo tien t map G × Y y → G y \ ( G × Y y ) = G × G y Y y .) Conv ersely , it follo ws f rom the con tin uity of the action that e very op en neig hbor- ho o d V of x in X contains a n a lmost slice W V x at x , which ﬁnishes the pro o f.  In what follows, if τ a nd σ are r epresentations, w e wr ite τ ≤ σ if τ is a subrep- resentation of σ . Deﬁnition 4.2. Supp ose that X is a prop er G -space whic h satisﬁes (SP). If ( x, σ ) ∈ Stab( X ) b and if W V x ∈ AS x we say that a pair ( z , τ ) ∈ Stab( X ) b lie s in the se t U ( x, σ, W V x ) if and only if there exists y ∈ V x and g ∈ W s uch that z = g y a nd τ ≤ g σ | G z . Alternatively one could describ e the s e ts U ( x, σ , W V x ) as the pr o duct W · U ( x, σ , V x ) with U ( x, σ, V x ) := { ( y , τ ) ∈ Stab( X ) b : y ∈ V x , τ ≤ σ | G y } . W e further deﬁne U ( x,σ ) := { U ( x, σ, W V x ) : W V x ∈ AS x } . Lemma 4.3. Ther e is a top olo gy on Stab( X ) b such that the elements of U ( x,σ ) form a b ase of op en neighb orho o ds for the element ( x, σ ) ∈ Stab ( X ) b in this top olo gy. Mor e over, the c anonic al action of G on Stab( X ) b is c ontinuous with r esp e ct to this top olo gy. Pr o of. W e have to show tha t if ( x, σ ) lies in the intersection of tw o sets U ( x 1 , σ 1 , W 1 V 1 ) a nd U ( x 2 , σ 2 , W 2 V 2 ), then there exists an a lmost slice W V at x s uc h that U ( x, σ, W V ) ⊆ U ( x 1 , σ 1 , W 1 V 1 ) ∩ U ( x 2 , σ 2 , W 2 V 2 ) . If this is shown, then the union S ( x,σ ) ∈ Stab( X ) b U ( x,σ ) forms a base of a top olog y with the r equired prop erties. F or this let g 1 ∈ W 1 and g 2 ∈ W 2 such that y i := g − 1 i x ∈ V i and suc h that g − 1 i σ is a sub- representation of σ i | G y i for i = 1 , 2. Since g i S x = S g i x ⊆ S x i for i = 1 , 2, we 24 ECHTERHOFF AND EMERSON may cho o se an op en neighborho o d V of x in S x such that g i V ⊆ V i for i = 1 , 2. W e may then also ﬁnd a symmetric op en neighbor ho o d W of e in G suc h that g i W ⊆ W i and W V ⊆ W i V i for i = 1 , 2. W e wan t to sho w that U ( x, σ, W V ) ⊆ U ( x i , σ i , W i V i ) for i = 1 , 2. Since U ( x,σ ) is closed under ﬁnite intersections, whic h follows easily from the deﬁnitions, it is enough to show this for i = 1. So let ( y , τ ) ∈ U ( x , σ, W V ). Let g ∈ W such that g y ∈ V and g τ is a sub-represe n tation of σ | G gy . Then g 1 g ∈ W 1 with g 1 g y ∈ V 1 and g 1 g τ is a sub-represe n tation of g 1 ( σ | G gy ), which is a subr epresentation of  σ 1 | G g 1 x  | G g 1 gy = σ 1 | G g 1 gy . Thus ( y , τ ) ∈ U ( x 1 , σ 1 , W 1 V 1 ). T o see that the action o f G on Stab( X ) b is contin uous let U ( g x, g σ, W V g x ) be a given neighborho o d o f ( gx, g σ ). Then, if W 0 is an op en neighborhho o d of e in G such that g W 2 0 g − 1 ⊆ W and if we deﬁne V x := g − 1 · V gx , we get ( g W 0 ) · U ( x, σ , W 0 V x ) = ( g W 2 0 g − 1 ) · U ( g x, g σ, V gx ) ⊆ U ( g x, g σ , W V gx ) and we are done.  R emark 4.4 . In the case where G is a discrete gr oup, the description of the top ology on Stab( X ) b b e comes easier to describ e since for ev ery po int x ∈ X the set S x ⊆ X is o pe n in X , and hence contains a neighbor ho o d base of sets V x . (Observe also that V x = W V x for W = { e } .) Ther efore a neighborho o d base of a pair ( x, σ ) ∈ Stab( X ) b is given by the sets U ( x, σ , V x ), wher e V x runs through all op en neighborho o ds o f x in S x . In this case , w e ma y e ven repla ce the V x by loca l slices Y x , since for discrete G the lo ca l slices are a lso op e n in X . R emark 4.5 . Another o bvious appro ach to de ﬁne basic neighborho o ds for t he to p ol- ogy of Stab( X ) b would be to co ns ider the sets U ( x, σ, W Y x ) := W · U ( x, σ, Y x ) where the Y x are lo cal slices at x a nd U ( x, σ, Y x ) := { ( y , τ ) : y ∈ Y x , τ ≤ σ | G y } . W e ac- tually believe that these sets do form a neig hborho o d base of the above de ﬁned top ology , but we lack a pro of. In particular, it is not clear to us whether the inter- section of tw o sets of this form will contain a thir d one o f this form. The diﬃculty comes from the fact that the intersection of tw o lo cal slices at x might not b e a lo cal slice at x – in fact the intersection will very often only contain the p oint x . How ever, it follows from the ab ov e remark that these problems disapp ear if G is discrete. In what follows, we alw ays equip ( C 0 ( X ) ⋊ G ) b with the Jaco bson top ology a nd G \ Stab( X ) b with the quotient top olo g y o f the ab ov e de ﬁned to p olo gy on Stab( X ) b . Theorem 4.6. Assume that X is a pr op er G -sp ac e which satisﬁes Palais’s slic e pr op ert y (S P). L et Stab( X ) b b e e quipp e d with the top olo gy c onst ructe d ab ove. The n the bije ction [( x, σ )] 7→ π σ x b etwe en G \ Stab( X ) b and ( C 0 ( X ) ⋊ G ) b of The or em 3.7 is a home omorphism. Before we s ta rt with the pro of, w e have to re call the deﬁnition of the F el l-t op olo gy on the s et Rep( A ) o f al l equiv alence class es of repr esentations o f a C* -algebra A with dimension dominated b y some ﬁxed cardinal κ ( κ is a lw ays chosen so big, that all re presentations w e ca re for lie in Rep( A )). A neigh b orho o d base for the F e ll top ology is g iven by the collection of all sets of the form U ( I 1 , . . . , I l ) = { π ∈ Rep( A ) : π ( I i ) 6 = { 0 } for all 1 ≤ i ≤ l } , where I 1 , . . . , I l is an y giv en ﬁnite colle c tio n of closed t wo-sided ideals in A . If we restrict this top ology to the set b A of equiv alence classes of irreducib e representations of A , we recov er the usual Jaco bson toplogy on b A . Conv ergence of nets in Rep( A ) ca n conv eniently be describ ed in terms o f we ak c ontainment : If π ∈ Rep( A ) and R is a subset of Rep( A ), then π is s aid to be we akly c ontaine d in R (denoted π ≺ R ) if ker π ⊇ ∩{ ker ρ : ρ ∈ R } . Two subsets S, R of Rep( A ) are said to b e we akly e quivalent ( S ∼ R ) if σ ≺ R for all σ ∈ S and ρ ≺ S for all ρ ∈ R . CR OSSED PRODUCT S BY PR OPER ACTIONS 25 Lemma 4.7 (F ell) . L et ( π j ) j ∈ J b e a net in Rep( A ) and let π , ρ ∈ Rep( A ) . Then (i) π j → π if and only if π is we akly c ontaine d in every subnet of ( π j ) j ∈ J . (ii) If π j → π and if ρ ≺ π , then π j → ρ . F or the pro of see [32 , Prop ositions 1.2 a nd 1 .3]. W e should also note that b y construction of the F ell-top olo g y , the top ology can only se e the kernel of a repre- sentation and not the representation itself – that means in pa rticular that if we replace a net ( π i ) b y s ome o ther net ( ˜ π i ) with ker ˜ π i = k er π i for all i ∈ I , then bo th nets have the sa me limit sets! Suppo se now that A, B are tw o C* -algebra s and let A E B be a Hilbert A − B - bimo dule. By this we under stand a Hilber t B -mo dule E B together with a ﬁxed ∗ -homomor phism Φ : A → L B ( E ). Then E induces an induction map (due to Rieﬀel) Ind E : Rep( B ) → Rep( A ) which sends a representation π ∈ Rep( B ) to the induced representation Ind π ∈ Rep( A ), which acts on the balanced tenso r pro duct E ⊗ B H π via Ind E π ( a )( ξ ⊗ v ) = Φ( a ) ξ ⊗ 1 . One can chec k that ker( Ind E π ) = { a ∈ A : Φ( a ) E ⊆ E · (ker π ) } from whic h it follows that induction v ia A E B preserves weak containmen t, and hence the map Ind E : Rep( B ) → Rep( A ) is contin uous. In particular , if A E B is an imprimitivity bimo dule w ith inverse module B E ∗ A , then Ind E ∗ gives a co nt inuous inv er se to Ind E , and therefore induction via E induces a homeomorphis m b etw een Rep( B ) and Rep( A ) (see [4 2, Chapter 3.3 ]). Basically a ll inductio n maps w e us e in this paper are co ming in one w ay o r the other from induction via bimo dules, s o the ab ove principles c a n be used. W e need the following o bserv a tio n: Lemma 4 .8. Su pp ose that ( π i ) is a net in Rep( K ) = Rep( C ∗ ( K )) for some c om- p act gr oup K . Then ( π i ) c onver ges to some σ ∈ b K if and only if ther e exists an index i 0 such that σ is a sub-r epr esentation of π i for al l i ≥ i 0 . Pr o of. Using the Peter-W eyl theorem, we can write C ∗ ( K ) = L τ ∈ b K End( V τ ). In this picture, g iven any r epresentation π of K , a n ir r educible representation τ is a sub-represe n tation o f π if a nd only if the summand End( V τ ) is not in the kernel o f π , viewed as a repr esentation of C ∗ ( K ). Thu s weak containmen t and containmen t of τ a re equiv alent. Assume now that there exists no i 0 ∈ I such that τ ≤ π i for all i ≥ i 0 . W e then construct a subnet ( π j ) o f ( π i ) such tha t τ is no t co n tained in any of the π j , which then implies that End( V τ ) lies in the kernel of all π j . But then τ is not weakly contained in the s ubnet ( π j ) which by Lemma 4.7 contradicts π i → τ . F or the construction of the subnet, we deﬁne J := { ( i, k ) ∈ I × I : k ≥ i and τ 6≤ π k } , equipp e d with the pairwise ordering. The pro jection to the second factor is clear ly order preserving, and if w e deﬁne π ( i,k ) := π k , we obtained a subnet ( π ( i,k ) ) ( i,k ) ∈ J with the desir ed prop erties.  The follo wing pr op osition will provide the main step tow ards the pro of of The- orem 4.6. 26 ECHTERHOFF AND EMERSON Prop ositio n 4.9. Su pp ose that K is a c omp act gr oup acting on a lo c al ly c omp act sp ac e Y and assume that y ∈ Y is ﬁxe d by K . L et σ ∈ b K b e identiﬁe d with the r epr esentation π σ y ∈ ( C 0 ( Y ) ⋊ K ) b in the usual way and let ( y i , σ i ) b e any net in Stab( Y ) b . Then the fol lowing ar e e quivalent: (i) The net π σ i y i = Ind K K y i ( y i , σ i ) c onver ges to π σ y in ( C 0 ( Y ) ⋊ K ) b . (ii) The n et ( y i , σ i ) c onver ges to ( y , σ ) in Stab( Y ) b . Recall that for σ ∈ b K we denote b y p σ = dim( V σ ) χ σ ∗ the central pro jection corres p onding to σ . Be fore we give the pr o of, we s hould p oint out a well-known fact on isotypes of unitary r epresentations of compact gro ups: if π : K → U ( V π ) is any such representation, then π ( p σ ) ∈ B ( V π ) is the pr o jection onto the isotyp e V σ π of σ in π , that is V σ π = ∪{ V ⊆ V π : π | V ∼ = σ } . This follows ea sily from the fa c t that for any τ ∈ b K we get τ ( p σ ) = 1 V σ if τ ∼ = σ and τ ( p σ ) = 0 else, and the well kno wn fact that every r epresentation of a compact group decomp os e s in to irr educible ones. In particular, the isotype of an irr educible representation σ ∈ b K in the left reg ular representation λ : K → U ( L 2 ( K )) is the ﬁnite dimensiona l space V σ ⊗ V ∗ σ , viewed as a subspace o f L 2 ( K ) via the unitary embedding v ⊗ w ∗ 7→ ξ v, w with ξ v, w ( k ) = √ d σ h v , σ ( k ) w i (compar e with the pro of of Lemma 3 .1). Pr o of of Pr op osition 4.9. W e may assume without loss of generality tha t Y is com- pact, since otherwise we may restrict ours elves to a K -inv a r iant compact neigh- bo rho o d o f y . W e ﬁrst show (i) ⇒ (ii). F or this consider the canonical in- clusion C ∗ ( K ) → C ( Y ) ⋊ K which is induced by the K - equiv ar ia nt embedding C → C ( Y ); λ 7→ λ 1 Y . A r epresentation π = P × U of C ( Y ) ⋊ K restricts to the unitary repre s ent ation U of K v ia this inclusion. Thus the map Rep( C ( Y ) ⋊ K ) → Rep( K ) which sends π = P × U to U can be viewed as an induction map via the C ∗ ( K ) − C ( Y ) ⋊ K -bimo dule C ( Y ) ⋊ K , and therefore is contin uous. It follows that if π σ i y i = P σ i y i × U σ i y i (w e use the notation o f § 3) conv erges to σ = π σ y as in (i), then U σ i y i conv erges to σ in Rep( K ), which by Lemma 4.8 implies that there exists i 0 ∈ I such that σ is a sub-r e presentation of U σ i y i for all i ≥ i 0 . But by the F r ob e nius recipro city theorem (e.g. see [1 3, Theor e m 7.4.1]) this implies that σ i is a sub-repr esentation of σ | K y i for all i ≥ i 0 . It remains to show that (i) implies that y i → y . F o r this we use the ca nonical inclusion o f C ( K \ Y ) in to the cen ter of M  C ( Y ) ⋊ K  . The asso ciated “induction map” from Rep( C ( Y ) ⋊ K ) to Rep( C ( K \ Y )) sends the representation π σ i y i = P σ i y i × U σ i y i to the restriction of P σ i y i to C ( K \ Y ) ⊆ C ( Y ), which is equal to ev K y i · 1 H σ i . By contin uity of “induction” we see that ev K y i · 1 H σ i conv erges to ev K y · 1 V σ in Rep( C ( K \ Y )) whic h just means that K y i → K y = { y } in K \ Y . Since K is compact and y is ﬁxed by K , this implies that y i → y in Y . W e no w prove (ii) ⇒ (i). F o r this supp ose that ( y i , σ i ) is as in (ii). Since every subnet enjoys the same pr op e rties, it is enoug h to show tha t σ = π σ y is weakly contained in { π σ i y i : i ∈ I } . F or this let a ∈ C ( Y ) ⋊ K b e a ny ele ment such that π σ i y i ( a ) = 0 fo r all i ∈ I . W e need to show tha t π σ y ( a ) = 0, too . F or this recall ﬁrst from Theo r em 2.10 that C ( Y ) ⋊ K ∼ = C ( Y × K K ( L 2 ( K ))) is a co n tinuous C* -algebra bundle o ver K \ Y with ﬁber K ( L 2 ( K )) K y i at the orbit K y i . The pro jection o f C 0 ( Y ) ⋊ K to the ﬁb er K ( L 2 ( K )) K y i at the orbit K y i is given via the representation M y i × λ with λ the regular repr esentation of K and ( M y i ( ϕ ) ξ )( k ) = ϕ ( k y i ) ξ ( k ) . CR OSSED PRODUCT S BY PR OPER ACTIONS 27 It follows fro m this that the representation of a ∈ C ( Y ) ⋊ K on these ﬁbers is given by a net ( a y i ) in K ( L 2 ( K )) K y i ⊆ K ( L 2 ( K )) whic h conv erges in the opera tor norm to the element a y ∈ K ( L 2 ( K )) K . At the p oint y w e get the decompo sition C ∗ ( K ) ∼ = K ( L 2 ( K )) K = M π ∈ b K K ( V π ) ⊗ 1 V ∗ π and at a ll o ther p oints w e get the de c o mpo sitions K ( L 2 ( K )) K y i = M τ ∈ b K y i K ( H τ ) ⊗ 1 V ∗ τ , where, for co n venience, w e write H τ for the Hilb ert space H U τ of U τ . In par ticular, if p i : L 2 ( K ) → H σ i ⊗ V ∗ σ i ⊆ L 2 ( K ) denotes the or thogonal pro jection, then the repr esentation π σ i y i ⊗ 1 V ∗ σ i = ( P σ i y i × U σ i y i ) ⊗ 1 V ∗ σ i is given by sending the element a ∈ C ( Y ) ⋊ K to the ele men t p i a y i p i ∈ K ( H σ i ) ⊗ 1 V ∗ σ i ⊆ K ( L 2 ( K )). Assume now that a ∈ C ( Y ) ⋊ K such that π σ i y i ( a ) = 0 for all i ∈ I . Then all those elemen ts p i a y i p i v anish. W e have to show that σ ( a y ) will v anish, too . F or this recall that σ ( p σ ) = 1 V σ . Thus we may replace a by p σ ap σ , where w e view p σ as an element of C ( Y ) ⋊ K via the embedding C ∗ ( K ) → C ( Y ) ⋊ K . W riting 1 i = 1 V ∗ σ i we then have π σ i y i ( a ) ⊗ 1 i = π σ i y i ( p σ ap σ ) ⊗ 1 i = ( U σ i y i ( p σ ) ⊗ 1 i )( π σ i y i ( a ) ⊗ 1 i )( U σ i y i ( p σ ) ⊗ 1 i ) , where U σ i y i ( p σ ) ⊗ 1 i is the or thogonal pro jection from H σ i ⊗ V ∗ σ i onto the isot yp e W i := ( H σ i ⊗ V ∗ σ i ) σ = ( H σ i ⊗ V ∗ σ i ) ∩ ( V σ ⊗ V ∗ σ ) ⊆ L 2 ( K ) . It thus follows that π σ i y i ⊗ 1 i represents each a i := a y i as a n op erato r on the subspace W i of the ﬁxed ﬁnite dimensionl spac e V σ ⊗ V ∗ σ of L 2 ( K ). By F rob enius recipro city , the condition that σ i is a sub-r epresentation of σ | K y i implies that σ is a sub-repre s ent ation of the repr esentation U σ i y i , which implies that W i is non-z ero for a ll i ∈ I . Now let q i : V σ ⊗ V ∗ σ → W i denote the orthogonal pro jection. Since V σ ⊗ V ∗ σ is ﬁnit e dimensional, we may pas s to a subnet if necessar y to assume that q i conv erges to some no n- zero pro jection q : V σ ⊗ V ∗ σ → W . W e then get 0 = π σ i y i ( a ) ⊗ 1 i = q i a y i q i → q a y q ∈ K ( V σ ⊗ V ∗ σ ). Moreover, s ince all W i are K-inv a riant, the same is true for W . It follo ws that the repre s ent ation λ : C ∗ ( K ) → K ( L 2 ( K )) K restricts to a multiple of σ on W . But then we have σ ( a y ) = 0 ⇔ q a y q = 0 a nd the result follo ws.  Pr o of of The or em 4.6. W e ﬁrst show that the map fro m Stab( X ) b to ( C 0 ( X ) ⋊ G ) b which assigns ( x, σ ) to the induced representation π σ x is con tinuous. By the univ ersal prop erty of the quotient top olog y on G \ Stab( X ) b , this will imply that the map Ind : G \ Stab( X ) b → ( C 0 ( X ) ⋊ G ) b of Theorem 4.6 is cont inuous, to o. So let ( x i , σ i ) b e a net in Stab( X ) b which conv erges to some ( x, σ ) ∈ Stab( X ) b . By the deﬁnition of the top ology on Stab( X ) b we may a ssume that x i = g i y i with g i → e in G , y i ∈ S x (i.e., G y i ⊆ G x ) such that y i → x and σ i = g i τ i with τ i ∈ b G y i such tha t τ i ≤ σ | G y i . Since induction is constant on G -orbits, w e may ass ume without loss of gener ality that y i = x i and τ i = σ i . It follows then from P rop osition 4.9 that the induced represe ntations ind G x G x i ( x i , σ i ) converge to ( x, σ ) in ( C 0 ( X ) ⋊ G x ) b . Using induction in steps and contin uity o f induction from a ﬁxed subgroup (due to the fact that this induction can b e p erfor med v ia an a ppropriate Hilbert mo dule), it then follo ws that π σ i x i = ind G G x i ( x i , σ i ) = ind G G x  ind G x G x i ( x i , σ i )  → ind G G x ( x, σ ) = π σ x . 28 ECHTERHOFF AND EMERSON Conv ersely , assume that we hav e a net ( π i ) in ( C 0 ( X ) ⋊ G ) b whic h conv er ges to some representation π ∈ ( C 0 ( X ) ⋊ G ) b . Choo se x i ∈ X and σ i ∈ ( C 0 ( X ) ⋊ G ) b such that π i = π σ i x i , and similar ly we realize π as π σ x for so me ( x, σ ) ∈ Stab( X ) b . By im b e dding C 0 ( G \ X ) in to Z M ( C 0 ( X ) ⋊ G )) we can see that Gx i → Gx in G \ X . Since the quo tien t map X → G \ X is op en, we can therefore pass to a subnet, if necessary , to ass ume that g i x i → x for a suitable net g i ∈ G . Cho osing a lo cal slice Y a t x , w e may even a djust the situa tion to gua rantee that g i x i ∈ Y for all i ∈ I . Thus, repla cing ( x i , σ i ) by ( g i x i , g i σ i ) we may assume fro m now on that the net ( x i , σ i ) and the pair ( x, σ ) all lie in Stab( Y ) b . Using the canonical Mo rita equiv alence C ( G × G x Y ) ⋊ G ∼ M C 0 ( Y ) ⋊ G x together with Prop osition 3.1 3 it follows that τ σ i x i = Ind G x G x i ( x i , σ i ) co nv er ges to τ σ x = σ in ( C 0 ( Y ) ⋊ G x ) b . It is then a consequence of P rop osition 4.9 that ( x i , σ i ) conv erges to ( x, σ ) in Stab( Y ) b . But this also implies conv e r gence in Sta b( X ) b a nd thus of the resp ective orbits in G \ Stab( X ) b .  Before disc ussing an example, we empha size the following consequences of the ab ov e disc us sion. • F or a prop er a ction o f a discr ete g roup G on X , a subset U ⊂ Stab( X ) b is open if and only if the following is tr ue : if ( x, π ) ∈ U , then there exists an op en slice V x around x such that U co nt ains all p oints ( y , τ ) such that y ∈ V x and τ ≤ π | G y . • In a ny case, the s e t X ∼ = { ( x, 1 G x ) : x ∈ X } ⊂ Stab( X ) b is op e n in this top ology , and by pro jection, G \ X (iden tiﬁed with G \{ ( x, 1 G x ) : x ∈ X } ) is op en in G \ Stab( X ) b . Example 4 .10 . W e co me ba ck to our e x ample C ( T 2 ) ⋊ G with G = D 4 = h R, S i as discussed in E xamples 3.5, 3.5 and 3.12. Consider the topo logical fundamen tal domain Z := { ( e 2 π is , e 2 π it ) : 0 ≤ t ≤ 1 2 , 0 ≤ s ≤ t } ⊆ T 2 for the action o f G on T 2 . Recall that this means that the ca nonical ma p from Z to G \ T 2 is a ho meomorphism. It then is easily chec ked that Stab( Z ) b = {  z , w ) , σ  ∈ Stab( T 2 ) b : ( z , w ) ∈ Z } is a to p olo gical fundament al domain fo r the action o f G in Stab( T 2 ) b . Thu s, in order to de s crib e the topolog y of C 0 (( T 2 ) ⋊ G ) b ∼ = G \ Stab( T 2 ) b it suﬃces to descr ibe the top ology of Stab( Z ) b . In what follows we iden tify Z with the triang le { ( s, t ) ∈ R 2 : 0 ≤ t ≤ 1 2 , 0 ≤ s ≤ t } . In Exa mple 3 .5 w e already computed the stabilizers and their representations. Each po in t in the interior ◦ Z has trivial stabilizer { E } . Since the trivial group has only the trivial representation and since every repres en tation of a group restricts obviously to a m ultiple of the trivial representation of the trivial group, we s ee that if ( s n , t n ) n ∈ N is a seq ue nc e in ◦ Z which co nverges to some ( s, t ) ∈ Z , then  ( s n , t n ) , 1 { E } ) conv erges to any  ( s, t ) , σ  with σ ∈ b G ( s,t ) . If ( s n , t n ) conv erges to 0, then the sequence  ( s n , t n ) , 1 { E } ) has the ﬁve limit p oints  (0 , 0) , 1 G  ,  (0 , 0) , χ 1  ,  (0 , 0) , χ 2  ,  (0 , 0) , χ 3  ,  (0 , 0) , λ  with { 1 G , χ 1 , χ 2 , χ 3 , λ } as deﬁned in Example 3.5. Let us now restrict our a tten tion to the three b order lines. F o r example, if we consider the line segment I 1 := { ( s, s ) : 0 < s < 1 2 } we hav e the constant stabilizer K 1 = h RS i and w e s e e that {  ( s, s ) , σ  ∈ Stab( Z ) b : ( s, s ) ∈ I 1 } = I 1 × { 1 K 1 , ǫ K 1 } CR OSSED PRODUCT S BY PR OPER ACTIONS 29 top ologically . Similar descr iptions hold fo r the line segments I 2 := { (0 , t ) : 0 < t < 1 2 } and I 3 := { ( s, 1 2 ) : 0 < s < 1 2 } . In orde r to des crib e the top ology o f Stab( Z ) b at the corners (0 , 0) , ( 1 2 , 1 2 ) a nd (0 , 1 2 ) of Z , we observe that 1 K 1 ≤ res G K 1 (1 G ) , res G K 1 ( χ 2 )) , res G K 1 ( λ ) , and ǫ K 1 ≤ res G K 1 ( χ 1 ) , res G K 1 ( χ 3 ) , res G K 1 ( λ ) , 1 K 2 ≤ res G K 2 (1 G ) , res G K 2 ( χ 3 )) , res G K 1 ( λ ) , and ǫ K 2 ≤ res G K 2 ( χ 1 ) , res G K 2 ( χ 2 ) , res G K 1 ( λ ) , 1 K 3 ≤ res G K 3 (1 G ) , res G K 3 ( χ 3 )) , res G K 1 ( λ ) , and ǫ K 3 ≤ res G K 3 ( χ 1 ) , res G K 3 ( χ 2 ) , res G K 1 ( λ ) , which implies, for example, that if a s equence ( s n , s n ) n ∈ N ⊆ I 1 conv erges to (0 , 0) the sequence  ( s n , s n ) , 1 K 1  has the limit points  (0 , 0) , 1 G  ,  (0 , 0) , χ 2  ,  (0 , 0) , λ  and the sequence  ( s n , s n ) , ǫ K 1  has the limit po ints  (0 , 0) , χ 1  ,  (0 , 0) , χ 3  ,  (0 , 0) , λ  . Similar descriptio ns follow from the ab ove list for s equences in I 1 , I 2 , I 3 which conv erge to any of the c orners (0 , 0) o r ( 1 2 , 1 2 ). A t the corner (0 , 1 2 ) with stabilizer H = h R 2 , S i a nd character s 1 H , µ 1 , µ 2 , µ 3 of H , as describ ed in Example 3.5, we get the r elations 1 K 2 = res H K 2 (1 H ) , res H K 2 ( µ 2 ) , and ǫ K 2 = res H K 2 ( µ 1 ) , res H K 2 ( µ 3 ) , 1 K 3 = res H K 3 (1 H ) , res H K 3 ( µ 3 ) , and ǫ K 3 = res H K 3 ( µ 1 ) , res H K 3 ( µ 2 ) , which implies, for example, that for a sequence ( s n , 1 2 ) in I 3 which conv erges to (0 , 1 2 ), the sequence  ( s n , 1 2 ) , 1 K 3  has the limit p oints  (0 , 1 2 ) , 1 H  ,  (0 , 1 2 ) , µ 3  and the sequence  ( s n , 1 2 ) , ǫ K 3  has the limit po in ts  (0 , 1 2 ) , µ 1  ,  (0 , 1 2 ) , µ 2  . F ro m this des cription we obtain an incr easing sequence of op en s ubsets of Stab( Z ) b ∅ = O 0 ⊆ O 1 ⊆ O 2 ⊆ O 3 = Stab( Z ) b (and cor resp onding o pe n subsets of ( C 0 ( T 2 ) ⋊ G ) b via induction) with O 1 = { (( x, y ) , 1) | ( x , y ) ∈ Z } ∼ = Z O 2 r O 1 =  ( s, s ) , ǫ K 1  : 0 < s < 1 / 2  ∪  (0 , t ) , ǫ K 2  : 0 < t < 1 / 2  ∪  ( s, 1 / 2) , ǫ K 3  : 0 < s < 1 / 2  ∪  (0 , 0) , χ 1  ,  (1 / 2 , 1 / 2) , χ 1  ,  (0 , 1 / 2 ) , µ 1  ∼ = ∂ Z O 3 r O 2 = { (0 , 0) , (1 / 2 , 1 / 2) } × { χ 2 , χ 3 , λ } ∪ { (0 , 1 / 2) } × { µ 2 , µ 3 } ∼ = { 1 , . . . , 8 } . Thu s we obtain a co r resp onding sequence of idea ls { 0 } = I 0 ⊆ I 1 ⊆ I 2 ⊆ I 3 = C 0 ( T 2 ) ⋊ G with b I 1 ∼ = Z, [ I 2 /I 1 ∼ = ∂ Z, [ I 3 /I 2 ∼ = { 1 , . . . , 8 } . Indeed, if w e combine the ab ove description o f the top ology of Stab( Z ) b ∼ =  C ( T 2 ) ⋊ G  b with the description o f the algebr a C ( T 2 ) ⋊ G ∼ = { f ∈ C  Z, K ( ℓ 2 ( G ))  : f ( s, t ) ∈ K ( ℓ 2 ( G )) G ( s,t ) } as giv en in Example 3.5, we see that I 1 = I T 2 as descr ibed in Example 3.12. It is Morita-equiv a lent to C ( Z ) ∼ = C ( G \ T 2 ) a nd consists of those functions f ∈ C ( Z , K ( ℓ 2 ( G ))) which only take non-zer o v alues in the matrix blocks cor resp ond- ing to the trivial r epresentations in each ﬁbe r K ( ℓ 2 ( G )) G ( s,t ) . The ideal I 2 takes arbitrar y v alues in K ( ℓ 2 ( G )) G ( s,t ) at each ( s, t ) ∈ Z r { (0 , 0 ) , ( 1 2 , 1 2 ) , (0 , 1 2 ) } and it takes non-zer o v alues only in the dia gonal en tries d 1 , d χ 1 at the corners (0 , 0) a nd ( 1 2 , 1 2 ), with resp e c t to the block decomp ositio n of K ( ℓ 2 ( G )) G as given in E x ample 3.5, and it tak es only non-zero v a lues in the 2 × 2 blo ck ent ries B 1 , B µ 1 at the corner (0 , 1 2 ). It is then clear that I 2 /I 1 is iso mo rphic to the a lg ebra of contin uous functions f : ∂ Z → M 8 ( C ) such that o n the line segments I i , i = 1 , 2 , 3, f ( s, t ) 30 ECHTERHOFF AND EMERSON has no n-zero entries only in the 4 × 4 matrix blo cks A ǫ K i , at the corners (0 , 0) and ( 1 2 , 1 2 ) it has non-zer o en try only in the diag onal entry d χ 1 and at the corner (0 , 1 2 ) it ha s non-zer o en tries only in the 2 × 2 blo ck C µ 1 . It follows that I 2 /I 1 is Morita equiv alent to C ( ∂ Z ). Finally , the quotient ( C ( T 2 ) ⋊ G ) /I 2 is isomorphic to M 2 ( C ) 4 L C 4 . R emark 4.11 . If G acts freely and pr op erly on X , then Green’s theorem ([29]) implies that C 0 ( X ) ⋊ G ∼ = C 0 ( X × G K ) is Mo r ita equiv a lent to C 0 ( G \ X ). If everything in sight is second countable, this implies that the bundle C 0 ( X × G K ) is (stably ) isomor phic to the trivial bundle C 0 ( G \ X , K ) (after s tabilization, if necessary , w e may assume K ∼ = K ( ℓ 2 ( N )) everywhere). So one may wonder, whether a similar result can be true in general, i.e., is there a chance to show that for any prop er actio n of G o n a lo ca lly compact space X the bundle C 0 ( X × G K ) is stably isomor phic to a subbundle of the trivial bundle C 0 ( G \ X , K ), s o that the ﬁbre over the orbit Gx would b e a suitable s ubalgebra of K = K ( ℓ 2 ( N )) stably isomorphic to K ( L 2 ( G )) Ad ρ ( G x ) . Actually it follows from Prop ositio n 2.7 that this is alw ays the case if there exists a top olo g ical fundamen tal domain Z ∼ = G \ X for the action of G on X as in Deﬁnition 2.4. Unfortunately , s uc h a trivia lization is not possible in general. Indeed, the prob- lem already app ears in the case of linear actions of ﬁnite groups o n the clo sed unit ball in R n . W e shall pres e n t a co ncrete counter-example in the following section. 5. K -theor y of proper actions In this chapter we will c o nsider the pr o blem of calcula ting equiv ariant K-theory for prop er a ctions. Deﬁnition 5. 1. L e t X be a prop er G -space where G is a lo cally compact group. The G -e quivariant K -the ory of X , denoted K ∗ G ( X ), is the K-theo ry o f the cro ssed pro duct C*-alg ebra C 0 ( X ) ⋊ G . There are v ery few gener al results abo ut equiv ariant K-theory . The ones we discuss b elow all treat simplica tions of the problem. There are s everal kinds of simplications p ossible. One inv olves ignoring torsion in K ∗ G ( X ). Thus, one can a im for a co mputation of equiv ariant K-theory with rational co eﬃcients K ∗ G ( X ) ⊗ Z Q . F or tec hnical pur po ses it is often co nv enient to tensor with C instead, this gives equiv alent results. Another simplication is to consider only compact gro ups. A further simpliﬁcation is to restrict to ﬁnite groups. W e start with the question of whether or not equiv ar iant K-theory ca n b e de- scrib ed, as with non-equiv ariant K - theory , in terms of vector bundles. Let G b e pos sibly non- c o mpact, lo cally compact gr oup, acting prop erly o n X with c omp act sp ac e G \ X of orbits. Let E b e a G -equiv a riant vector bundle on X . Because the action is pro per , ther e is a G -inv ar iant Hermitian structure o n E . W e can equip the space Γ c ( E ) o f contin uous sections of E with compact supp or ts with the right C c ( G × X )-mo dule structure via ( s · a )( x ) = Z G g · s ( g − 1 x ) a ( g − 1 , g − 1 x ) p ∆( g − 1 ) dg , s ∈ Γ c ( E ) , a ∈ C c ( G × X ) where we regar d C c ( G × X ) as a dense subalge br a of C 0 ( X ) ⋊ G . One can als o deﬁne a r ig ht C 0 ( X ) ⋊ G -v alue d inner pro duct on Γ 0 ( E ), by the form ula (5.1) h s 1 , s 2 i ( x, g ) := p ∆( g − 1 ) h s 1 ( x ) , g · s 2 ( g − 1 x ) i E . So the completion e Γ( E ) := Γ c ( E ) with resp ect to this inner product b ecomes a right C 0 ( X ) ⋊ G -Hilbert mo dule. If G \ X is co mpact, this Hilb e rt module can be chec ked to b e ﬁnitely genera ted. The co-compactness ass umption implies tha t the CR OSSED PRODUCT S BY PR OPER ACTIONS 31 ident ity o per ator is a compa ct Hilbert mo dule op erato r . See e.g. [24] for a pro of, and the following example for the case where E = 1 X is the trivia l line bundle. Example 5.2 . Let G b e locally compac t, and a ct pr op erly and c o -compactly on X . If E is the trivial line bundle 1 X ov er X , w ith the trivia l action of G on the ﬁbre s then the ﬁnitely genera ted pro jective Hilb ert C 0 ( X ) ⋊ G -mo dule e Γ(1 X ) as ab ove is isomorphic to the range of an y of the idempotents p X ∈ C 0 ( X ) ⋊ G , constructed by a cut-oﬀ function c . Recall that, since G \ X is compact, c : X → [0 , 1] is a co mpactly suppo rted contin uous function such that R G c ( g − 1 x ) 2 dg = 1 for all x ∈ X . Then p X := h c, c i is a pro jection in C 0 ( X ) ⋊ G (co mpare with the discussion around (3.6)) and we get e Γ(1 X ) ∼ = p X ·  C 0 ( X ) ⋊ G  as r ight C 0 ( X ) ⋊ G -mo dules. F or the pro o f, regard c as an elemen t o f e Γ(1 X ). Deﬁne Q : e Γ(1 X ) → C 0 ( X ) ⋊ G, Q ( s ) := h c, s i , R : C 0 ( X ) ⋊ G → e Γ(1 X ) , R ( a ) := ca, using the inner pro duct in (5.1). Then fo r all s ∈ e Γ(1 X ) , a ∈ C 0 ( X ) ⋊ G , RQ ( s ) = c · h c, s i = s, and QR ( a ) = h c, c i a = p X a. In particula r this shows that e Γ(1 X ) is a ra nk-one mo dule. It a lso gives an- other pro of that the K-theory classes [ p X ] ∈ K 0 G ( X ) are indep endent of the choice of cut-oﬀ function. Note a lso that the natural repres e ntation of C ( G \ X ) → L C 0 ( X ) ⋊ G  e Γ(1 X )  by letting G -inv ariant functions a ct as multiplica- tion op era tors o n e Γ(1 X ), gives an isomorphis m C ( G \ X ) ∼ = K  e Γ(1 X )  . Consequently , we g e t a n isomor phism C ( G \ X ) ∼ = K ( p X · C 0 ( X ) ⋊ G ) ∼ = p X · C 0 ( X ) ⋊ G · p X (see the discuss io n around (3.7).) Let VK 0 G ( X ) b e the Grothendiec k gr oup of th e monoid of G -equiv aria n t complex vector bundles on X . Let VK 1 G ( X ) be deﬁned to b e the k ernel of the map VK 0 G ( X × S 1 ) → VK 0 G ( X ) b y restriction of G -equiv ariant vector bundles to X × { 1 } , where we let G act trivially on S 1 . Then applica tion of the above pro cedure to cycles yields a map (5.2) VK ∗ G ( X ) → K ∗ ( C 0 ( X ) ⋊ G ) . Is (5.2) a n isomo rphism in gene r al? Co rollar y 5.3 of [24] s hows that fo r any prop er action of a lo cally compact gr oup G , the mono id of isomorphism clas s es of G -equiv ar iant vector bundles on X is isomorphic to the mo no id of pro jections in  C 0 ( X ) ⋊ G  ⊗ K . Howev e r , the K-theo ry of C 0 ( X ) ⋊ G is deﬁned in terms of the monoid of pro jections in  C 0 ( X ) ⋊ G  + ⊗ K , wher e the + denotes unitization, so that a priori there c ould be classes whic h do not come from equiv ar iant v ector bundles. In fact, the q uestion is so mewhat delicate. See the analys is [24], following work of L ¨ uc k a nd O liver in [37], who are partially re spo nsible for the following result. Theorem 5.3. If G is discr ete, c omp act or an almost-c onne cte d gr oup, and if X is G -c omp act, t hen the map (5.2) is an isomorphism. F or gr oups not satisfying one of the conditions of Theo rem 5.3, equiv ariant vector bundles need not g enerate all o f equiv aria nt K-theory . A counter-example to iso morphism of (5.2) can be built using the following ideas, es sentially due to Juliane Sauer . Let G be the semi- direct pro duct T 2 ⋊ A Z 32 ECHTERHOFF AND EMERSON where Z acts by the automo rphism of the compact group T 2 induced b y a ma trix A ∈ GL(2 , Z ). Then G acts pro pe rly on R with Z acting b y transla tions and T 2 acting trivia lly . The res triction of any G -equiv aria nt vector bundle on R to a po in t of R deter mines a ﬁnite-dimensional repres ent ation of the compact subgroup T 2 ⊂ G . It can b e shown that if A is a hyperb olic matrix, then only multiples of the trivia l r epresentation can b e obtained in this w ay . This shows that there are rather few G -equiv ariant v ector bundles on R . Of course G is neither compact, nor a Lie g roup. Equiv ar iant Kaspa rov theor y o ﬀer s another po in t o f view on equiv ariant K-theory for G -co mpact spaces. The re pr esent able G -equiv a r iant K-theo ry of a prop er G - space X – denoted RK ∗ G ( X ) – is deﬁned in terms o f G -equiv ar iant maps from X to suitable spaces o f F redholm o p er ators on G -Hilb ert spaces; s uc h maps may alwa ys be norma lized so that the Hilb ert space ma y b e taken ﬁxed, so the r epresentable K-theory has cycles maps X → F red( H G ). An equiv ariant v ersion of the Kuip er theorem implies s uc h maps ma y b e tak en norm co ntin uous. F or G -compact spa c e s, it can b e sho wn that RK ∗ G ( X ) ∼ = K ∗ G ( X ) (see [24, Theorem 3 .8]). This giv es a reasona bly satisfac tory ho mo topy-theoretic description of equiv ar ia nt K-theory for some purp oses, but it is not very concrete. F or gener al proper actions of lo cally compact groups G on spaces X , K ∗ G ( X ) is a G -compactly s uppo rted theory and RK ∗ G ( X ) is no t, so the theories deﬁnitely diﬀer in this c a se. R emark 5.4 . Unlike non-eq uiv ariant K-theor y , whic h is r ationally isomorphic to co- homology , equiv aria nt K-theory is de ﬁnitely diﬀerent fro m equiv a riant c ohomolo gy – even for ﬁnite groups, a nd even after tenso ring b oth with the c o mplex num ber s. Equiv ar iant cohomolo g y for a compact gro up a ction is deﬁned to b e the ordinar y cohomolog y of X × G E G , where E G is the classifying space for fr e e ac tions of G . It is not hard to chec k that equiv ariant cohomolo gy with c omplex (or rational) co eﬃcients for ﬁnite groups G is the same as the G -inv ariant part of ordinary co - homology , and hence (b y Pr op osition 5.5 below) it agrees with just the ordinary cohomolog y of the q uotient space, whereas equiv a riant K -theory do es not behave like this, even for ﬁnite gr oups, for a lready in tegrally K ∗ G (pt) ∼ = Rep( G ) is free ab elian wiht o ne generator for each ir reducible represe ntation of G . What is true is that equiv aria nt co homology with complex coeﬃcients and ﬁ nite groups, is iso- morphic to the lo c alization of equiv ariant K-theo ry with complex coeﬃcients at the ident ity conjugacy class of the group, c.f. [7]). Equiv aria nt K-theory with complex co eﬃcients ma y thus b e view ed as ‘de-lo calized’ equiv ariant cohomolog y – the point of view taken in the a rticle [7] of Baum and Connes, and explained below in § 5.1. W e start b y justifying o ne of the statemen ts made in the above Remar k . Prop ositio n 5.5. F or any G -sp ac e X , G ﬁnite, (5.3) K ∗ ( X ) G ⊗ Z Z [ 1 | G | ] ∼ = K ∗ ( G \ X ) ⊗ Z Z [ 1 | G | ] as Z [ 1 | G | ] -mo dules. The i somorphism is induc e d by the pul l-b ack map π ∗ : K ∗ ( G \ X ) → K ∗ ( X ) . In p articular, X 7→ K ∗ ( X ) G and X 7→ K ∗ ( G \ X ) agr e e for ﬁ nite gr oups, after tensoring by C . Prop ositio n 5.5 do es not ho ld for compact gr oups in general. It is easy to ﬁnd a counter-example: take the T -actio n on the 2-sphere which rotates ar ound the z - axis. The quotien t space is the c lo sed in terv al [0 , 1], so K ∗ ( T \ S 2 ) ∼ = Z . But since T is connected, it acts trivially on K-theor y o f S 2 , so K ∗ ( S 2 ) T ∼ = K ∗ ( S 2 ) ∼ = Z L Z . These are o b viously diﬀerent even after tens o ring with C . CR OSSED PRODUCT S BY PR OPER ACTIONS 33 Nor do es the P rop osition ho ld for inﬁnite dis crete g roups, for a diﬀerent reaso n. If G is inﬁnite and X = G with G acting by translation, then K ∗ ( G \ X ) ∼ = K ∗ ( C ) ∼ = Z but since there a re no nonzer o ﬁnitely suppor ted G -inv ariant maps G → Z , a nd since K-theor y is compactly supp orted, K ∗ ( X ) G = 0. Pr o of of Pr op osition 5.5. Closed G -inv a riant subspaces o f X ar e in 1-1 corresp on- dence with closed subspaces of G \ X . If Z ⊂ X is a closed inv a riant subspace, then the exact se q uence 0 → C 0 ( X r Z ) → C 0 ( X ) → C 0 ( Z ) → 0 of G -C*-alg ebras induce s a long exa ct sequence of G -e q uiv aria nt K - theory groups (5.4) · · · → K ∗ ( G \ Z ) → K ∗ ( G \ X ) → K ∗ ( G \ Z ) ∂ − → K ∗ +1 ( G \ ( X r Z )) → · · · and also a n e x act sequence · · · → K ∗ ( X r Z ) → K ∗ ( X ) → K ∗ ( Z ) ∂ − → K ∗ +1 ( X r Z ) → · · · of non-equiv aria nt groups , eac h of which carr ies an action of G , which therefore induces a lo ng exact sequence (5.5) · · · K ∗ ( X ) G ⊗ Z Z [ 1 | G | ] → K ∗ ( Z ) G ⊗ Z Z [ 1 | G | ] ∂ − → K ∗ +1 ( X r Z ) G ⊗ Z Z [ 1 | G | ] → · · · of the G -in v ar iant elements. Note that we hav e to tenso r with Z [ 1 | G | ] to guarantee that this sequence is exact. The map π ∗ of the theore m induces a map b etw een the sequence (5.4), tensored b y Z [ 1 | G | ], and seq ue nce (5.5), so it follows from the Five-lemma that π ∗ is an iso morphism for X , if this is true for Z and X r Z . W e no w pro ceed by induction on | G | . If Z ⊆ X is the closed set of G -ﬁxed po in ts, we ha ve K ∗ ( G \ X ) = K ∗ ( X ) = K ∗ ( X ) G , s o by the a bove discussion we may assume that all stabilizers for the actio n of G on X a re prop er subg r oups of G . W e then ﬁnd a cov er U o f X by in v ariant open sets U ea ch is omorphic to some induced G -space G × H Y with | H | < | G | . F or each suc h set U ∼ = G × H Y we have K ∗ ( G \ U ) ∼ = K ∗ ( H \ Y ). On the other hand G × H Y ﬁbres by the ﬁrst co ordinate pro jection o ver the ﬁnite G -set G/H ; the ﬁbre ov er g H is Y . Hence the or dinary K-theo ry o f U decompo s es as a direct sum K ∗ ( G × H Y ) ∼ = M gH ∈ G/H K ∗ ( Y ) . The group G p ermutes the summands b y left trans la tion and the in v ar iant pa rt K ∗ ( G × H Y ) G is isomorphic to K ∗ ( Y ) H . By induction, the formula o f the lemma is true for H , and and since | H | devides | G | we ge t K ∗ ( U ) G ⊗ Z Z [ 1 | G | ] ∼ = K ∗ ( G \ U ) ⊗ Z Z [ 1 | G | ] . Now suppo se a G -space X is cov ered by n induced op en sets U 1 , . . . , U n . Put Z = X r U n . Then Z is covered by the n − 1 op en sets U 1 , . . . , U n − 1 . By Meyer- Vietoris a nd the a bove dis c ussion for induced sets, the lemma holds for X if it holds for Z . Thus by induction o n n we ma y therefor e a ssume the lemma to b e prov ed for all G -sets whic h a re covered b y ﬁnit ely man y properly induced op en sets. The result then follows from the fact that the theor ie s K ∗ ( X ) G and K ∗ ( G \ X ) bo th resp ect inductive limits.  W e are going to start o ur inv estigation o f equiv ariant K-theory with a summary of wha t is known ab out ﬁnite group a ctions. W e will als o neglect torsion for the moment – in fact, it will be helpful to tenso r all equiv a riant K-theory groups in the following by C , since this has th e conseq ue nce that if G is ﬁnite, then Rep( G ) ⊗ Z C , 34 ECHTERHOFF AND EMERSON as a ring, is simply the r ing of co mplex v alued functions on the set of co njugacy classes in G (this would not b e true ev en if w e tensored b y Q .) Now K ∗ G ( X ) is alwa ys a mo dule ov er Rep( G ). T e nsoring everything b y C then gives K ∗ G ( X ) ⊗ Z C the structure of a mo dule over Rep( G ) ⊗ Z C , and a n y module ov er the ring of complex-v a lued functions o n a ﬁnite set of p oints deco mpo ses as a direct sum ov er the sp ectrum of the ring (which in this case is the set of conjugacy cla s ses in G .) This provides some additional a lgebraic structure which prov es to be v ery useful. In what follws, w e write K ∗ G ( X ) C := K ∗ G ( X ) ⊗ Z C for the usual in tegral G - equiv ar ia nt K-theory of X , tenso red by the complex num b ers a nd we wr ite Rep( G ) C for Rep( G ) ⊗ Z C . W e star t b y simplifying even further, and discus s the diﬀer enc e of the ra nks of K 0 G ( X ) C and K 1 G ( X ) C . This integer is ca lled the e quivariant Euler char acteristic of X . Denote the equiv ariant Euler characteristic by Eul( G ⋉ X ). Although a crude inv a riant, it is at leas t easily geometrica lly c o mputable, by a version of the Lefschetz ﬁxed-p oint theorem (proved by Atiy ah, Theo rem 5.9 b elow.) Clearly the equiv ariant Euler c haracteris tic of a free action is the ordinar y Euler characteristic of the q uo tien t spa ce. On the o ther hand, the Euler characteristic is m ultiplicative under cov erings, so for a fr ee action o f a ﬁnite g roup Eul( G ⋉ X ) = Eul( G \ X ) = 1 | G | Eul( X ) . The following lemma desc r ibe s χ ( G \ X ) geometr ically , even in the presence of isotropy , with the additional hypo thesis of a smooth ac tio n on a smo oth manifold. Lemma 5 . 6. L et X b e a s mo oth c omp act manifold and G a ﬁnite gr oup acting smo othly on X . Then (5.6) Eul( G \ X ) = 1 | G | X g ∈ G Eul( X g ) , wher e X g is the ﬁxe d-p oint submanifold of g . Pr o of. Using the standar d formula for the dimension of the space of G -in v ar iants in a representation and the fact that K ∗ ( G \ X ) ⊗ Z C is the same a s the G -inv aria nt part of K ∗ ( X ) ⊗ Z C , Eul( G \ X ) = 1 | G | X g ∈ G χ Eul ( g ) where χ Eul is the (virtual) character of the Z / 2 -graded representation of G on K ∗ ( X ) C ( ∼ = H ∗ ( X, C )), that is, the diﬀerence o f the characters of G ac ting on even and o dd K- theo ry tensored with C , or cohomology with c o e ﬃcien ts in C . Now b y the Lefschetz ﬁxed-p oint formula, χ Eul ( g ) := trace s ( g ) = Eul( X g ), which prov es the result.  In the la st pro of, we used the very str ong version of the Lefschetz ﬁxed-p oint formula applicable to an isometry of a co mpact Riemannian ma nifold. This classica l fact seems quite well-known to top olo gists, but for lack of a reference, w e cite the second author’s article [22] for this (see Theore m 10 . F or the connection b etw een the Lefschetz map a nd traces, see [21].) It also follows immediately from the Atiy ah- Segal-Singer index theor em (see Theo r em 2.12 of [4].) The follo wing g ives a n um b er of exa mples o f interesting ﬁnite gr oup actions on surfaces. Example 5.7 . This exa mple is due P .E. Conner and E.E. Floyd in [12]. Let p and q be t wo odd primes , let n = pq and λ b e a primitive n th ro ot of unity . Let S ⊂ C P 2 be the zer o locus of the homogeneous polyno mial f ( z 1 , z 2 , z 3 ) = z n 1 + z n 2 + z n 3 . Then CR OSSED PRODUCT S BY PR OPER ACTIONS 35 S is a smo oth complex submanifold o f C P 2 of complex dimensio n 1, i.e. is a curve. Let T ([ z 1 , z 2 , z 3 ]) := [ z 1 , λ p z 2 , λ q z 3 ]. Then T has order n . Let Γ = h T i . Let τ : S → S, τ ([ z 1 , z 2 , z 3 ]) := [ λ z 1 , z 2 , z 3 ] . It co mm utes with Γ, has o rder n . Thus we get an a ction of Z /n × Z /n on S . This is the r estriction of a n actio n on CP 2 of course. The ﬁxed-p oint set o f τ in CP 2 is points with ﬁrst homogeneous co ordinate zero, so it is a copy of CP 1 . The quotient X := Γ \ S is a closed Riemann surface. Its gen us is deduced from its Euler characteristic, which b y the formula ab ov e is given b y 1 n P 0 ≤ j → T denote the corr esp onding char- acter o f the cyc lic group h g i gener a ted b y g , i.e., χ ζ ( g l ) = ζ l . Let V ζ τ denote the eigenspace in V τ for the eigenv alue ζ of τ ( g ). Its dimension d τ ζ gives the m ultiplicit y of the c haracter χ ζ in the r estriction of τ to h g i . Computing χ τ ( g ) = trac e τ ( g ) with resp ect to a co rresp onding basis, w e get χ τ ( g ) = P ζ ∈ C d d τ ζ ζ . W e therefore get (5.7) Φ 0 [ g ] ([ V ]) = X τ ∈ b G χ τ ( g )[Hom G ( V τ , V )] = X ζ ∈ C d X τ ∈ b G d τ ζ ζ [Hom G ( V τ , V )] . CR OSSED PRODUCT S BY PR OPER ACTIONS 37 On the other hand, using the isomo r phism V ∼ = L τ ∈ b G  V τ ⊗ Hom G ( V , V τ )  we get V ζ ∼ = M τ ∈ b G  V ζ τ ⊗ Ho m G ( V τ , V )  ∼ = M τ ∈ b G Hom G ( V τ , V ) d τ ζ , so that we g et the equa lit y [ V ζ ] = P τ ∈ b G d τ ζ [Hom G ( V τ , V )] in K 0 ( X ) C . T og ether with (5.7) this giv es X ζ ∈ C d ζ [ V ζ ] = X ζ ∈ C d X τ ∈ b G ζ d τ ζ [Hom G ( V τ , V )] = Φ 0 [ g ] ([ V ]) .  If a ﬁnite g roup acts on a c o mpact space X , the mo dule K ∗ G ( X ) C decomp oses as K ∗ G ( X ) C ∼ = M [ g ] ∈ [ G ] K ∗ G ( X ) C , [ g ] and to cla ssify the mo dule it suﬃces to analyse the contributions K ∗ G ( X ) C , [ g ] from each conjugacy clas s [ g ]. The above lemma does this job in ca se where G acts trivially on X . In what fo llows next, we w ant t o extend this description to the case of arbitrar y compact G -spaces X . R emark 5.12 . Suppo se that a ﬁnite gr oup G a cts on the co mpact space X . Then we alwa ys ge t a pro jection from K ∗ G ( X ) to the G -inv ariant part K ∗ ( X ) G of K ∗ ( X ) which is given b y sending the class [ V ] of a G -equiv aria nt vector bundle to the class [ V ] with forgotten G -action. T o see that V gives a G -inv ariant class in K 0 ( X ) observe that the actio n of an ele men t g ∈ G on V provides a con tinuous family of isomorphisms α g x : V x → V gx = g ∗ ( V ) x , hence an iso morphism betw e e n the bundles V and g ∗ V . Note that the pro jectio n K ∗ G ( X ) → K ∗ ( X ) G bec omes surjectiv e a fter complexiﬁcation, since for any G - inv aria n t cla ss [ V ] the cla s s 1 | G |  L g ∈ G g ∗ V  ∈ K 0 G ( X ) C maps to [ V ] in the complexiﬁcation of K 0 ( X ) G . Mo reov er, by Pr op osition 5.5 we see that the co mplexiﬁcation of K ∗ ( X ) G is isomorphic to K ∗ ( G \ X ) C , so w e obtain a cano nical surjective pro jection (5.8) P ∗ G : K ∗ G ( X ) C → K ∗ ( G \ X ) C which se nds a class [ V ] ∈ K 0 G ( X ) to the class 1 | G | P g ∈ G [ g ∗ V ] ∈ K ∗ ( G \ X ) C . By passing to one-p oint c ompactiﬁcations, we obtain a similar pro jection for loca lly compact G -spaces . Suppo se now that X is a compact G -spa ce for the ﬁnite g roup G . Le t g ∈ G . An y G -equiv a r iant vector bundle on X restricts to the s et of g -ﬁxed p oints X g = { x ∈ X : g x = x } . It then decomp oses into g - eigenbundles V ζ as in Lemma 5.11. Let Z g denote the cen tralizer o f g in G . Each of the bundles V ζ are Z g -equiv ar iant. Deﬁne a map (5.9) φ 0 [ g ] : K 0 G ( X ) C − → K 0 Z g ( X g ) C , φ [ g ] ([ V ]) := X ζ ζ [ V ζ ] . Comp osing this with the pro jectio ns P 0 Z g : K 0 Z g ( X g ) C → K 0 ( Z g \ X g ) C and summing ov er [ g ] yields a map φ 0 : K 0 G ( X ) C → L [ g ] ∈ [ G ] K 0 ( Z g \ X g ) C . Replacing X b y X × S 1 (or by the one-p oint compactiﬁcation X + ) and rep eating the construc tio n gives a map (5.10) φ ∗ : K ∗ G ( X ) C → M [ g ] ∈ [ G ] K ∗ ( Z g \ X g ) C . for any loca lly compact G -spac e X . 38 ECHTERHOFF AND EMERSON Consider no w the gr oup stabilizer bundle Stab( X ) = { ( x, g ) : x ∈ X , g ∈ G x } . G acts on Sta b( X ) via h ˙ ( x, g ) = ( h · x, hg h − 1 ) and Sta b( X ) decompo ses as a dis joint union Stab( X ) ∼ = G [ g ] ∈ [ G ] G [ h ] ∈ G/ Z g h · X g ∼ = G [ g ] ∈ [ G ] ( G × Z g X g ) , which implies a decompo sition (5.11) G \ Stab( X ) ∼ = G [ g ] ∈ [ G ] Z g \ X g . F ro m this we o btain a deco mpos ition K ∗ ( G \ Stab( X )) C ∼ = M [ g ] ∈ [ G ] K ∗ ( Z g \ X g ) C . Therefore we may reg ard the map (5.10) as having target K ∗ ( G \ Stab( X )) C and we obtain a well deﬁned ma p (5.12) Φ X : K ∗ G ( X ) C − → K ∗ ( G \ Stab( X )) C ∼ = M [ g ] ∈ [ G ] K ∗ ( Z g \ X g ) C . Baum and Connes [7 ] prove the following theo rem. Theorem 5.13 (Baum-Co nnes ) . If G is a ﬁnite gr oup, X a G -sp ac e, then (5 .12) is an isomorp hism K ∗ G ( X ) C ∼ = − → K ∗ ( G \ Stab( X )) C which sends the c omp onent K ∗ G ( X ) C , [ g ] of K ∗ G ( X ) C to t he c omp onent K ∗ ( Z g \ X g ) C of K ∗ ( G \ Stab( X )) C . The idea of the pro of is given b y observing that for any induced G -space X = G × H Y the assertion of t he theorem is true if (and only if ) it is true for the H -spa ce Y . T ogether with the fact tha t for trivial G -spaces the theorem has been chec ked already in Lemma 5 .11, this allows to combine induction over the order o f G with a Meyer-Vietoris argument to obtain the pro of. W e ﬁrst wan t to v erify that the map (5.12) is a natura l tr a nsformation betw een t wo (compactly suppor ted) coho mology theo ries on G -spaces, namely X 7→ K ∗ G ( X ) C and X 7→ K ∗ ( G \ Stab( X )) C . Indeed, let Z ⊆ X b e a close d, G -in v aria n t s ubs pace of a G -spa ce X . Then Stab( Z ) ⊂ Stab( X ) equiv ariantly so G \ Stab( Z ) ⊂ G \ Stab( X ) naturally . Hence a G -inv a riant clo sed subspace of X yields a 6-term exact sequence · · · ∂ − → K ∗  G \ Stab( X r Z )  − → K ∗  G \ Stab( X )  − → K ∗  G \ Stab( Z )  ∂ − → · · · W e lea ve the pro o f of the following lemma to the reader. Lemma 5.14 . If Z ⊆ X is a close d, G -invariant subsp ac e, then the diag r am (5.13) · · · / / K ∗ G ( X r Z ) C / / Φ X r Z   K ∗ G ( X ) C / / Φ X   K ∗ G ( Z ) C / / Φ Z   · · · · · · / / K ∗  G \ Stab( X r Z )  C / / K ∗  G \ Stab( X )  C / / K ∗  G \ Stab( Z )  C / / · · · c ommut es. R emark 5.15 . If X = G × H Y for so me H -space Y , we obtain a n isomor phism R : K ∗ G ( X ) → K ∗ H ( Y ) given by comp osing the obvious restriction map K ∗ G ( X ) → K ∗ H ( X ) with the map K ∗ H ( X ) → K ∗ H ( Y ) coming from the inclusion of the op en subs et Y in X (see [47, CR OSSED PRODUCT S BY PR OPER ACTIONS 39 p. 132 ]). Its inv erse Ind : K ∗ H ( Y ) → K ∗ H ( X ) is given on the level of vector bundles by [ V ] 7→ [ G × H V ]. It is then easy to chec k tha t the diag ram (5.14) K ∗ G ( X ) ⊗ Rep( G ) m G − − − − → K ∗ G ( X ) R ⊗ res   y   y R K ∗ H ( Y ) ⊗ Rep( H ) m H − − − − → K ∗ H ( Y ) commutes, where m G and m H denote the respe c tiv e mo dule a ctions of Rep( G ) and Rep( H ) and res : Rep( G ) → Rep( H ) is the res tr iction map. Lemma 5. 16. L et G b e a ﬁnite gr oup, H ⊆ G a sub gr oup and X := G × H Y b e an induc e d G -sp ac e. Then the r esu lt of Baum and Connes for H acting on Y implies the r esult of Baum and Connes for G acting on X . Pr o of. Rec all that G × H Y is the or bit space of the spa ce G × Y for the (righ t) action ( γ , y ) h = ( γ h, h − 1 y ), of the subgroup H with G -action given by g [ γ , y ] = [ g γ , y ]. Fix g ∈ H and observe that X g = { [ γ , y ] ∈ G × H Y | γ − 1 g γ ∈ H y } . W e therefore obtain a map Φ ′ : X g → H \ Stab( Y ); [ γ , y ] 7→ [ γ − 1 g γ , y ] . which is well deﬁned, b ecause if we replace ( γ , y ) by the pair ( γ h , h − 1 y ) for some h ∈ H , then ( γ − 1 g γ , y ) is replaced b y ( hγ g γ − 1 h, h − 1 y ) = ( γ g γ − 1 , y ) h . Moreover, if z ∈ Z g , the centralizer o f g in G , then o ne chec k s that Φ ′ ( z [ γ , y ]) = Φ ′ ([ γ , y ]), so that Φ ′ descends to a map Φ : Z g \ X g → H \ Stab( Y ) . As discussed ab ov e (see (5 .11)), we can ﬁbr e the quotient of the group stabilizer bundle H \ Stab( Y ) ov er the set of co njugacy classes in H , H \ Stab( Y ) = G [ h ] ∈ [ H ] ( Z h ∩ H ) \ Y h where Z h is the cen tralizer in G of a chosen repres e n tative h of [ h ]. In this pictur e, the range o f Φ is exactly the union of the comp onents of H \ Stab( Y ) which ar e lab el le d by [ h ] with h c onjugate in G to g , and Φ induces a homeomo rphism be tw een Z g \ X g and F [ h ] ∈ [ H ] , [ h ] ⊆ [ g ] ( Z h ∩ H ) \ Y h . This shows that Φ induces an isomorphism (5.15) Φ ∗ : M [ h ] ∈ [ H ] , [ h ] ⊂ [ g ] K ∗ (( Z h ∩ H ) \ Y h ) ∼ = − → K ∗ ( Z g \ X g ) . This can b e viewed, in a rather trivial wa y , as a Rep( G )-mo dule is o morphism, with Rep( G )-module structure g iven (on each side) by ev aluation of characters at [ g ]. Now, if g ∈ G such that the conjugacy class [ g ] G of g in G do es not intersect with H , then it follows fro m the co mm utativity of diagram (5.14) that K ∗ G ( X ) C , [ g ] = { 0 } , since the re s triction of the characteris tic function of [ g ] G to H v anis hes. On the other hand, if g is not conjugate to any element in H , it follows that X g = ∅ , so that we also hav e K ∗ ( Z g \ X g ) C = { 0 } . Using the above res ults together with the fact that the action of Rep( G ) C on K ∗ H ( Y ) C is given via the r estriction res : Rep( G ) C → Rep( H ) C , we now obtain a 40 ECHTERHOFF AND EMERSON diagram of Rep( G ) C -mo dule maps K ∗ H ( Y ) C Φ Y − − − − → L [ g ] ∈ [ G ]  L [ h ] ∈ [ H ] , [ h ] ⊆ [ g ] K ∗ (( Z h ∩ H ) \ Y h ) C  Ind   y   y Φ K ∗ G ( X ) C − − − − → Φ X L [ g ] ∈ [ G ] K ∗ ( Z g \ X g ) C W e leav e it as a n exe r cise to the rea der to c heck that this diagram comm utes (do this ﬁr st in case wher e Y is compact, and then obtain the general case by passing from Y to Y + ). Assume tha t the theorem of Baum and Connes holds for H a cting on Y , i.e., Φ Y is an isomor phism. Since Φ a nd Ind are also isomorphisms, the sa me m ust then be true for Φ X .  W e are now ready for the Pr o of of The or em 5.13. W e induct on the or der of the group G . F or trivial groups there is nothing to prov e. Supp os e the result is true for all gro ups G of cardina lit y ≤ n . Let X b e a G -spa ce, G ﬁnite, o f ca rdinality ≤ n + 1. Let F ⊂ X b e the stationary set. F rom Lemma 5.1 4 the diagr am (5.16) · · · / / K ∗ G ( X r F ) C / / Φ X r Y   K ∗ G ( X ) C / / Φ X   K ∗ G ( F ) C / / Φ F   · · · · · · / / K ∗  Stab( X r F )  C / / K ∗  Stab( X )  C / / K ∗  Stab( F )  C / / · · · commutes. The vertical map Φ F is an isomor phism by Lemma 5.11. Th us, by the Five Lemma, it suﬃces to show that the ﬁrst vertical map Φ X r F : K ∗ G ( X r F ) − → K ∗  Stab( X r F )  is an iso morphism, to o. So fro m now on we may ass ume without loss of gener ality , that X do es not contain a ny G -ﬁxed p oints. In this case we ﬁnd a cov er { U i : i ∈ I } o f G -inv ariant op en subsets U i of X suc h that ea ch o f these sets is G -homeomor phic to G × H i Y i for some prop er subgro up H i of G . It follows then from the induction hypothesis together with Lemma 5.16 that the theorem holds for e ach U i . By Meyer-Vietoris, using L emma 5.1 4, this implies the theorem for all unions U F := ∪ i ∈ F U i , with F ⊆ I ﬁnite. The result now follows fr o m con tinuit y of K-theor y with r esp ect to inductiv e limits a nd the fact that C 0 ( X ) ⋊ G = lim F C 0 ( U F ) ⋊ G and C 0 ( G \ Stab( X )) = lim F C 0 ( G \ Stab( U F )).  Example 5.17 . Consider the ac tion of the dihedral gr oup G = D 4 on T 2 as intro- duced in E xample 2.6. Reca ll that it is generated by the matrices R , S ∈ GL(2 , Z ) with R =  0 − 1 1 0  and S =  1 0 0 − 1  and the canonical action of GL(2 , Z ) on T 2 = R 2 / Z 2 . This group of order eight has ﬁve conjugacy classes given b y [ E ] , [ R 2 ] , [ R ] , [ S ] , [ S R ] . The cor resp onding c e n tralizer s are Z E = Z R 2 = G, Z R = h R i , Z S = { E , R 2 , S, S R 2 } , Z RS = { E , R 2 , RS, S R } , while the c o rresp onding ﬁxed-p oint sets are X E = T 2 , X R 2 = { ( 1 1 ) ,  − 1 − 1  ,  − 1 1  ,  1 − 1  } , X R = { ( 1 1 ) ,  − 1 − 1  } , X S = { ( z , 1 ) : z ∈ T } ∪ { ( z , − 1) : z ∈ T } , X RS = { ( z , z ) : z ∈ T } . CR OSSED PRODUCT S BY PR OPER ACTIONS 41 With Z = { ( s, t ) : 0 ≤ s, t ≤ 1 2 , 0 ≤ t ≤ s } as in Example 2.6 a nd I = [0 , 1] w e get Z E \ X E = G \ T 2 ∼ = Z, Z S \ X S ∼ = I ⊔ I , Z RS \ X RS ∼ = I , the set Z R 2 \ X R 2 has three elemen ts and Z R \ X R = X R has t wo elements. Th us, as a cons e quence of the theor em of Baum and Connes we see tha t K 0 G ( T 2 ) C = C 9 and K 1 G ( T 2 ) C = { 0 } . The formula of A tiyah a nd Segal g iven in Theo rem 5.9 follows from a little manipulation similar to tha t of the pro of of Lemma 5.6, and Theorem 5.13. This is not hard to c heck, and the orig inal reference explains this quite clearly , so w e will omit the pr o of. The result of Baum and Co nnes is of co urse stronger ; it gives the dimensions of K 0 G ( X ) c and K 1 G ( X ) c separately . 5.2. Computation of in tegral K -theory. W e now r e turn to in tegr al equiv ariant K-theory . In the previous section, w e e x plained th e result of Baum and Connes , whic h ga ve a formula for the r anks of K i G ( X ), i = 0 , 1. A ﬁnitely generated ab elian group is determined b y its fre e par t and torsion part. In this section w e will dis cuss the torsion part o f K ∗ G ( X ). Computing this – ev en for ﬁnite gr oup actions – seems to be a muc h mor e diﬃcult pr oblem than computing the free part. Since part of what we are going to des c rib e is quite general and works for locally compact g roups, w e now let G b e an arbitrar y lo cally compa ct gro up and X a prop er and G -co mpact G -space. Let I X be the ideal in C 0 ( X ) ⋊ G as in (3.7). Let Q X :=  C 0 ( X ) ⋊ G  /I X . Then the Mor ita e q uiv alence C 0 ( G \ X ) ∼ = I X and excisio n yie lds a six-term exa ct sequence (5.17) K 0 ( G \ X ) − − − − → K 0 G ( X ) − − − − → K 0 ( Q X ) ∂ x     y ∂ K 1 ( Q X ) ← − − − − K 1 G ( X ) ← − − − − K 1 ( G \ X ) . In the ca se of gro up actions in which ﬁxed-p oints are isola ted (more pre c isely below), this exa ct sequence beco mes quite tractable and w e will study this situation in more detail b elow. In eﬀect, this is a situation in which the F ell top olog y on G \ Stab( X ) b is very easy to understand. When ﬁxed- po in t sets a re not just zero-dimensiona l, the problem obviously b e- comes mor e complicated. W e are no t go ing to consider the g eneral case her e. How ever, we will analyse Example 5.24 in s ome detail. If the set of p oints in G \ X with non-trivial isotr o py is a disc rete subset, then b y Lemma 3.9, the C * -algebra Q X is isomorphic to a direct sum of c o mpact oper ators, with o ne summand contributed b y each pair ( Gx, σ ), wher e Gx is a n or bit with nontrivial stabilizer subgroup G x , a nd σ 6 = 1 G x in b G x is an ir r educible represention diﬀerent fr om the trivial r epresentation. More formally , if I denotes the set of orbits with nontrivial s tabilizers, (5.18) Q X ∼ = M Gx ∈I   M σ ∈ b G x r { 1 G x } K ( H U σ )   , where for e ach orbit Gx ∈ I we cho o se o ne repres e n tative x of that o rbit and where H U σ is as in (3 .1). If we write Rep ∗ ( G x ) for the subgr oup of Rep( G ) gener - ated b y the non-trivial r reducible representations of G x it follows that K 0 ( Q X ) ∼ = 42 ECHTERHOFF AND EMERSON L Gx ∈I Rep ∗ ( G x ) is a free abelian group a nd K 1 ( Q X ) = { 0 } , so that the six-term sequence (5.17) b eco mes (5.19) 0 − → K 0 ( G \ X ) − → K 0 G ( X ) − → M Gx ∈I Rep ∗ ( G x ) ∂ − → K 1 ( G \ X ) − → K 1 G ( X ) − → 0 . In particular we see that K 0 ( G \ X ) injects in to K ∗ G ( X ) and K 1 ( G \ X ) surjects to K 1 G ( X ), in the case of isola ted ﬁxed-po in ts. Note that, in case of ﬁnite G , according to Theorem 5 .13 of Baum a nd Connes; K i ( G \ X ) injects in b oth after complexiﬁcation: it is the contribution of the trivia l conjugacy class [1]. Ther efore, our statement adds to theirs that this is a n injection in dimens ion zero inte gr al ly , and that K 1 ( G \ X ) → K 1 G ( X ) is a sur jection, likewise, even integrally . As we will see, non-v anishing of ∂ ([ σ ]) for a g enerator in K 0 ( Q X ) co rresp onding to some non-trivial representation σ of one of the isotro py groups G x , obstructs ex- tending the corr esp onding induced G -equiv aria n t v ector bundle [ V σ ] on the orbit o f x to a G -equiv ariant v ector bundle o n X . The statemen t is that these obstructions are tor sion: they v anish after m ultiplication by a suitable integer. Thus nV σ can alwa ys be extended, for appro priate n . W e also note that the p oint of view suggested b y (5.19) y ields a somewhat diﬀerent fo rmula for the Euler characteristic. F or a ﬁnite gro up G , let b G ∗ denote b G − { 1 } where 1 is the trivia l representation. Prop ositio n 5. 18. L et G b e a discr ete gr oup acting pr op erly and c o-c omp actly on X such that t he set D of p oints x ∈ X with non- trivial isotr opy is discr ete. Then Eul( G ⋉ X ) = Eul( G \ X ) + X Gx ∈ G \ D | b G ∗ x | . W e p oint this o ut ma inly to emphasise that we are counting in a ‘tra ns verse’ direction to Atiy ah, Baum and Connes, wher e our theorems intersect, when G itself is a ﬁnite group. They ﬁbre G \ Stab( X ) over the conjugacy classes, while we are ﬁbring it o ver G \ X . T o chec k that the t wo form ulas for the Euler c haracteris tic are the same, it suﬃces to obser ve that b oth – clearly – agr ee with the Euler characteristic in ordinary K-theor y o f the sa me Hausdorﬀ space G \ Stab( X ). In a ny case, since both the Ba um-Connes formula and ours contain the term Eul( G \ X ), we ca n remov e it from each side. The identit y (5.20) X [ g ] ∈ [ G ] r { [1] } card( Z g \ X g ) = X Gx ∈ I | b G ∗ x | . is a go o d exercise to prov e directly (the t wo sides of it are, ro ughly sp ea k ing, in a relation of duality , s ince if X is a po in t, the left-hand-side computes the num b e r o f nontrivial conjugacy class e s in G and the right-hand-side computes the n umber o f nontrivial irr educible representations.) T o compute equiv ariant K-theor y in the ca se of isolated ﬁxed po in ts, it r e mains to solve the following Problem: Descr ibe the bounda ry map ∂ : M Gx ∈I M σ ∈ b G x r { 1 G x } Z − → K 1 ( G \ X ) in (5.19). R emark 5 .19 . In the case of a fr e e action there is of course nothing to do. In this case the q uotient term v anishes b ecause Q X is itself the zero C*-algebr a. CR OSSED PRODUCT S BY PR OPER ACTIONS 43 If ( c.f. Exa mple 5.7) there is a single p oint x 0 in G \ X with non-tr iv ial isotropy , and if this p oint is ﬁxed by the entire group G (so that G must b e compact) then (5.19) b ecomes (5.21) 0 − → K 0 ( G \ X ) − → K 0 G ( X ) − → Rep ∗ ( G ) ∂ − → K 1 ( G \ X ) − → K 1 G ( X ) − → 0 . The quotient map K 0 G ( X ) → Rep ∗ ( G ) in this sequence is induced b y the G -map pt → X cor resp onding to the statio nary p oint. If X is compact, this map is split by the ma p X → pt and hence the b ounda r y map ∂ v anis hes in this case. Hence K 0 G ( X ) ∼ = K 0 ( G \ X ) M Rep ∗ ( G ) , K 1 G ( X ) ∼ = K 1 ( G \ X ) . The general case is as follows. Theorem 5.20. Supp ose that t he lo c al ly c omp act gr oup G acts pr op erly on X such that the fol lowing ar e satisﬁe d: (i) ( G, X ) satisﬁes Palais’s slic e pr op erty (SP); (ii) the orbits Gx with nontrivial stabilizers G x ar e isolate d in G \ X ; and (iii) al l stabilizers ar e ﬁnite. Then the b oundary map ∂ : K 0 ( Q X ) → K 1 G ( X ) of (5.19) is r ational ly trivial (i.e., the image of ∂ is a torsion su b gr oup of K 1 ( G \ X ) ). Mor e over, if an element ν ∈ K 0 ( Q X ) is an element of t he summand L σ ∈ b G x \{ 1 G x } K 0 ( K ( H U σ )) in the dir e ct-sum de c omp osition (5.18) of K 0 ( Q X ) , then the or der of ∂ ( ν ) is a di visor of the or der | G x | . The following is a well-kno wn fact ab out ﬂat v ector bundles. Lemma 5.21. L et H b e a ﬁnite gr oup acting fr e ely on the c omp act sp ac e Y . Then for e ach σ ∈ b H , the diﬀer enc e of classes [ Y × V σ ] − dim( V σ )[1 Y ] ∈ K 0 H ( Y ) is torsion of or der a divisor of | H | . Thus, 0 = | H | ·  dim( V σ ) · [ Y × C ] − [ Y × V σ ]  ∈ K 0 H ( Y ) . Pr o of. If H acts freely on Y we have K 0 H ( Y ) ∼ = K 0 ( H \ Y ) via sending the class of an equiv ariant vector bundle V ov er Y to the class of the bundle H \ V ov er H \ Y (e.g. see [47, Prop osition 2.1]). Co nsider the seq ue nce K 0 ( H \ Y ) q ∗ → K 0 ( Y ) q ∗ → K 0 ( H \ Y ) in which the ﬁr st map is given b y the pull- back of vector bundles wit h r esp ect to t he quotient map q : Y → H \ Y and the second map is the transfer map which sends a vector bundle V over Y to the bundle with ﬁbr e L g ∈ H V gy ov er H y ∈ H \ Y . It is clear that the compositio n q ∗ ◦ q ∗ acts on K 0 ( H \ Y ) as m ultiplication by | H | . If σ ∈ b H , then the H - bundle [ Y × V σ ] ∈ K 0 H ( Y ) (with resp ect to the diagona l action) co rresp onds to the (ﬂat) bundle [ Y × H V σ ] ∈ K 0 ( H \ Y ). The pull-bac k q ∗ ([ Y × H V σ ]) ∈ K 0 ( Y ) is the cla ss of the trivial bundle [ Y × V σ ] = dim( V σ )[ Y × C ] which is mapp ed to | H | dim( V σ )[( H \ Y ) × C ] b y the transfer map q ∗ : K 0 ( Y ) → K 0 ( H \ Y ). Th us we s ee that | H | [ Y × H V σ ] = q ∗ ◦ q ∗ ([ Y × H V σ ]) = | H | dim( V σ )[( H \ Y ) × C ] and the res ult follows.  44 ECHTERHOFF AND EMERSON In wha t follows next w e w ant to cons ider the case of an induced space X = G × H Y , where H is a ﬁnite subgroup of the locally compact group G and Y is a compact H -space with isolated H -ﬁxed p oint y ∈ Y such that H acts freely o n Y r { y } . Regarding Y as a clos e d subspace of X , it is clear that Gy is the only non-free orbit in X , and its stabilizer is G y = H , s o that the K- theory o f Q X = ( C 0 ( X ) ⋊ G ) /I X is the free ab elian g roup Rep ∗ ( H ) ge ner ated by the represe ntations σ ∈ b H r { 1 H } . Note also that if X is a G -space and Z is a closed G -in v ar iant subspace of X , then the r estriction map res Z : C 0 ( X ) → C 0 ( Z ) induces a quotient map res Z ⋊ G : C 0 ( X ) ⋊ G → C 0 ( Z ) ⋊ G. Prop ositio n 5.22. L et X = G × H Y and Q X = ( C 0 ( X ) ⋊ G ) /I X b e as ab ove. L et ν σ ∈ K 0 ( Q X ) b e the class c orr esp onding to the given r epr esentation σ ∈ b H r { 1 H } . Then ther e exists a class µ σ ∈ K 0 G ( X ) su ch that t he fol lowing ar e true (i) If q X : C 0 ( X ) ⋊ G → Q X denotes the quotient map, then q X, ∗ ( µ σ ) = | H | ν σ . (ii) if Z is any close d G -invariant subsp ac e of X which do es not c ontain the orbit Gy , then (res Z ⋊ G ) ∗ ( µ σ ) = 0 in K 0 ( C 0 ( Z ) ⋊ G ) . Pr o of. If we regard Y as a closed subspace of X = G × H Y via y 7→ [ e, y ], we see that Z ∼ = G × H Y Z with Y Z := Y ∩ Z . It is not diﬃcult to chec k that the ca nonical Morita equiv a lence C 0 ( G × H Y ) ⋊ G ∼ M C ( Y ) ⋊ H , as explained b efore Prop osition 3.13, restricts to the ca nonical Mo rita equiv alence C 0 ( G × H Y Z ) ⋊ G ∼ M C ( Y Z ) ⋊ H and it fo llows from Corolla ry 3 .14 and the Rieﬀel corr esp ondence (see the discussion befo re Corolla ry 3 .14) that this factors through a Morita equiv alence Q X ∼ M Q Y . Thu s these Morita equiv alenc e s provide us with a commut ative diagram K 0 ( C ( Y Z ) ⋊ H ) (res Y Z ⋊ H ) ∗ ← − − − − − − − − K 0 ( C ( Y ) ⋊ H ) q Y , ∗ − − − − → K 0 ( Q Y ) ∼ =   y ∼ =   y   y ∼ = K 0 ( C 0 ( Z ) ⋊ G ) ← − − − − − − − (res Z ⋊ G ) ∗ K 0 ( C ( X ) ⋊ G ) − − − − → q X, ∗ K 0 ( Q X ) . Thu s w e may as sume that G = H a nd Z = Y Z . This also a llows the use o f vector bundles for the description of K- theory classes. In this picture, the quo tient map q Y , ∗ : K 0 ( C ( Y ) ⋊ H ) → K 0 ( Q Y ) transla tes to the map ˜ q : K 0 H ( Y ) → K 0 ( Q Y ) whic h maps an H -bundle V ov er Y to the cla ss P τ 6 =1 H n τ · ν τ ∈ K 0 ( Q Y ), where the sum is ov er the non-trivial irreducible r epresentations of H and n τ denotes the m ultiplicit y of τ in the represe ntation of H o n the ﬁbre V y . Deﬁne µ σ := | H | ·  [ Y × V σ ] − dim( V σ ) · [ Y × C ]  ∈ K 0 H ( Y ) ∼ = K 0 G ( X ) . Then ˜ q ( µ σ ) = | H | · ν σ and (res Y Z ⋊ F ) ∗ ( µ σ ) = | H | ·  [ Y Z × V σ ] − dim( V σ ) · [ Y Z × C ]  = 0 by Lemma 5.21 (beca us e H a cts freely on Y r { y } .)  W e are now ready for the Pr o of of The or em 5.20. F o r the pro o f it suﬃces to show that the canonical g enera- tors of K 0 ( Q X ) are ma pped to elemen ts o f ﬁnite order in K 1 ( C 0 ( G \ X )) under the bo undary map ∂ : K 0 ( Q X ) → K 1 ( C 0 ( G \ X )). If the a ction of G on X is free, then Q X = { 0 } and the res ult is trivial. So assume now that Gy is any ﬁxed orbit with non-trivial stabilizer G y and let ν σ denote the gener ator of K 0 ( Q X ) corr esp onding to the representation σ ∈ c G y r { 1 G y } . W e claim that | G y | ∂ ( ν σ ) = 0. This will a lso prov e the last statement of the theorem. Since, by assumption, there exists a neighborho o d of Gy in G \ X suc h that a ll orbits in this neighborho o d are fr ee, and since the actions of G o n X s atisﬁes the slice prop erty (SP), w e may choose an op en G -in v ar iant neighborho o d U of y with G -compact closur e W = ¯ U with the following prop erties : CR OSSED PRODUCT S BY PR OPER ACTIONS 45 (i) W ∼ = G × G y Y for so me compact G y -space Y . (ii) G acts freely on W r { Gy } . Let Z = W r U . Then Prop osition 5.22 implies that we can ﬁnd a class µ σ ∈ K 0 ( C 0 ( W ) ⋊ G ) suc h that q W , ∗ ( µ σ ) = | G y | · ν σ and (res Z ⋊ G ) ∗ ( µ σ ) = 0. Applying the Meyer-Vietoris sequence in K-theory (e.g. see [9, Theorem 21.5 .1]) to the pull- back dia g ram C 0 ( X ) ⋊ G res X r U ⋊ G − − − − − − − → C 0 ( X r U ) ⋊ G res W ⋊ G   y   y res Z ⋊ G C 0 ( W ) ⋊ G − − − − − → res Z ⋊ G C 0 ( Z ) ⋊ G we see that we may glue the class µ σ ∈ K 0 ( C 0 ( W ) ⋊ G ) with the zero-class in K 0 ( C 0 ( X r U ) ⋊ G ) to obtain a cla ss µ ∈ K 0 ( C 0 ( X ) ⋊ G ) such tha t q X, ∗ ( µ ) = | G y | · ν σ ∈ K 0 ( Q X ). This implies that ∂ ( | G y | · ν σ ) = 0 in K 1 ( G \ X ).  In the a bove a rgument, we sho wed that Rep ∗ ( H ) is a direct summand o f K 0 ( Q X ) in the cas e of isolated ﬁx ed points with ﬁnite stabilizer H , a nd w e computed that the comp osition Rep ∗ ( H ) ⊆ Rep( H ) ∼ = K 0 G ( Gy ) → K 0 ( Q X ) ∂ − → K 1 ( G \ X ) maps any generator [ σ ] ∈ Rep ∗ ( H ) to a tor sion class in K 1 ( G \ X ). If X is a s mo o th manifold and the Lie g r oup G acts smo othly , this comp osition may b e made slightly mo re explicit using the language of diﬀer en tial topo logy . In the notation of the pro of of Theor e m 5 .20, take a p oint y ∈ X with iso tr opy H . Let ν be the normal bundle to Gy ; we ma y equip it with a G -in v ar iant Riemannian metric, then a re-scaling c o mpo sed with the ex p one ntial map determines a tubular neighbourho o d embedding ν ∼ = U ⊂ X for a n op en G -inv ariant neighbo urho o d U of Gy . Thus, U ∼ = G × H Y for the co r resp onding or thogonal linear action o f H on a Euclidean space Y := ν y ∼ = R k . Let S ν be the unit spher e bundle of ν , let i : S ν → U r Gy b e the co rresp onding smo oth equiv ar iant embedding; its normal bundle is equiv ar iantly trivializable (it is isomorphic to a trivial G -vector bundle with trivial G -action). Thus, there is a G -equiv ar iant s mo oth open embedding S ν × R → U r Gy . If ﬁxed-po int s are isolated, then G ac ts freely on S ν and U r Gy so that we obtain an open embedding ϕ : G \ S ν × R − → G \ X . Now the b o undary ma p ∂ : Rep ∗ ( H ) → K 1 ( G \ X ) may be describ ed simply as follows: it is the comp osition (5.22) Rep ∗ ( H ) ⊂ Rep( H ) ∼ = K 0 G ( Gy ) p ∗ − → K 0 G ( S ν ) ∼ = K 0 G ( G \ S ν ) ⊗ β − − → K 1 ( G \ S ν × R ) ϕ ! − → K 1 ( G \ X ) , where p : S ν → Gy is the pro jection map, β ∈ K 1 ( R ) the Bo tt class. This cons tr uction, as mentioned ab ove, pro duces torsion class es in K 1 ( G \ X ) of order a diviso r of | H | . W e will s ee in Exa mple 5.26 b elow that these classe s ar e not a lwa y s trivial, so that the b oundar y map in (5 .1 9) do es no t always v anish in general. But w e ﬁrst presen t a n in teresting example of a n action with isola ted orbits with non-trivial stabilizers, in whic h K 1 ( G \ X ) = { 0 } , so that the exact sequence (5.19) computes everything. Example 5.2 3 . In this example w e consider the cyclic group H = h R i of or der four with R =  0 − 1 1 0  ∈ GL (2 , Z ) acting on T 2 = R 2 / Z 2 . This is the action 46 ECHTERHOFF AND EMERSON of the dihedral gr oup G = h R, S i on T 2 as considered in Ex amples 2.6, 3.5 a nd 4.10 r estricted to the subgr oup H ⊆ G . Deformations of the crossed pro duct C ( T 2 ) ⋊ H , known as non-commut ative 2-spheres, hav e been s tudied extensively in the literature , and it is shown in [18] that the K-theory groups of these deformations are isomo rphic to the K-theor y g roups of C ( T 2 ) ⋊ H . If we study this a ction on the fundamental doma in { ( s, t ) : − 1 2 ≤ s, t ≤ 1 2 } ⊆ R 2 for the action of Z 2 on R 2 , w e see tha t (the ima ge in T 2 of ) { ( s, t ) : 0 ≤ s, t ≤ 1 2 } in T 2 is a fundamental doma in (but not a topolo gical fundamental domain a s in Deﬁnition 2.4) for the action of H on T 2 such that in the quotient H \ T 2 the line { ( s, 0) : 0 ≤ s ≤ 1 2 } in the b oundary is glued to (0 , t ) : 0 ≤ t ≤ 1 2 } and the line { ( s, 1 2 ) : 0 ≤ s ≤ 1 2 } is glued to { ( 1 2 , t ) : 0 ≤ t ≤ 1 2 } . Thu s w e see that the q uotient H \ T 2 is homeo morphic to the 2 -sphere S 2 . There are o nly three orbits in H \ T 2 with non trivial stabilizers: the points cor - resp onding to (0 , 0) and ( 1 2 , 1 2 ) in the fundamental domain have full stabilizer H and the or bit of the point corres p onding to (0 , 1 2 ) ha s stabilizer h R 2 i . Since K 0 ( S 2 ) = Z , K 1 ( S 2 ) = { 0 } a nd sinc e H has three non-trivia l c hara cters and h R 2 i has o ne non-trivial character, we see instant ly fr o m the exact seq uence (5.19), that K 0 (( C ( T 2 ) ⋊ H ) ∼ = Z 8 and K 1 ( C ( T 2 ) ⋊ H ) = { 0 } . In [1 8] we used a quite diﬀ erent and more co mplicated metho d for computing the K-theory groups of C ( T 2 ) ⋊ H . See [18] for an explicit description of the K-theory generator s. W e pro cee d w ith a n example of the equiv a riant K-theory computatio n for the action o f G = D 4 on T 2 as studied earlier in Examples 2.6, 3.5 and 4 .10. In this case the o rbits with no n-trivial stabilizer s ar e not isola ted and we use the descr iption of the ideal struc tur e as given in Example 4 .10 for the computatio n. Example 5.24 . Consider the c r ossed pro duct C ( T 2 ) ⋊ G . It is shown in Example 4.10 that we get a se q uence of ideals { 0 } = I 0 ⊆ I 1 ⊆ I 2 ⊆ I 3 = C ( T 2 ) ⋊ G with I 1 Morita equiv alent to C ( Z ), I 2 /I 1 Morita equiv alent to C ( ∂ Z ) a nd I 3 /I 2 Morita equiv alent to C 8 . W e ﬁr s t compute the K-theory of I 2 . Since K 0 ( I 1 ) = Z and K 1 ( I 1 ) = { 0 } and K 0 ( I 2 /I 1 ) = K 1 ( I 2 /I 1 ) = Z , the six-term s equence w ith resp ect to the ideal I 1 ∼ M C ( Z ) rea ds Z − − − − → K 0 ( I 2 ) − − − − → Z x     y Z ← − − − − K 1 ( I 2 ) ← − − − − 0 On the other hand, the ideal J := C 0 ( ◦ Z , M 8 ( C )) ⊆ I 2 with quotient I 2 /J Morita equiv alent to C ( T × { 0 , 1 } ) gives a six-term sequence Z − − − − → K 0 ( I 2 ) − − − − → Z 2 x     y Z 2 ← − − − − K 1 ( I 2 ) ← − − − − 0 It follows that K 0 ( I 2 ) ∼ = Z 2 and K 1 ( I 2 ) = Z . W e then get the six-term sequence Z 2 − − − − → K 0 ( C ( T 2 ) ⋊ G ) − − − − → Z 8 x     y 0 ← − − − − K 1 ( C ( T 2 ) ⋊ G ) ← − − − − Z CR OSSED PRODUCT S BY PR OPER ACTIONS 47 W e lea ve it a s an interesting exercis e for the rea de r (using the structure of I 3 = C ( T 2 ) ⋊ G a s indicated in Ex ample 3.5) to chec k dire ctly that the cla ss of K 0 ( I 3 /I 2 ) corres p onding to the character χ 2 ∈ b G at the p oint (0 , 0) ∈ Z ca nno t b e lifted to a class in K 0 ( I 3 ), a nd thus maps to a non-zer o element in K 1 ( I 2 ) ∼ = Z . Indeed, with a bit o f work o ne can show tha t it maps to a generator o f K 1 ( I 2 ) ∼ = Z whic h then implies that K 1 ( C ( T 2 ) ⋊ G ) = { 0 } a nd K 0 ( C ( T 2 ) ⋊ G ) = Z 9 . W e should mention that the alg ebra C ( T 2 ) ⋊ D 4 of the ab ov e exa mple is j ust the group alg ebra C ∗ ( Z 2 ⋊ D 4 ) o f the cristallogra phic group Z 2 ⋊ D 4 . The K-theory of this g roup a lgebra (toge ther with the K-theor ies of all other crystallogra phic groups of rank 2) has be e n computed b y co mpletely diﬀeren t methods by L ¨ uck and Stamm in [38]. Even befo r e tha t, the K -theory of the group algebr as of crystallog raphic groups of rank 2 were computed by Y ang in [5 0] b y methods muc h clo ser to o urs. In fact, many of the general results obtained in this article ha ve b een obtained in [50] in ca se of ﬁnite gr oup actions. W e now provide the pr omised count er-ex a mple for the tr ivialization pro blem a s po sed in Remar k 4.11. Example 5 .25 . Let G = Z /n be a ﬁnite cyclic group acting linearly on some R m such that the action o n R m \ { 0 } is free (it follows that n = 2 or m = 2 k is ev en). Consider the cr ossed-pr o duct C ( B ) ⋊ G , wher e B deno tes the closed unit ball in R n . Let res : C ( B ) → C ( S m − 1 ) denote the restr iction map. W e then obtain a map (5.23) φ : R ( G ) ∼ = K 0 ( C ( B ) ⋊ G ) res ∗ − → K 0 ( C ( S m − 1 ) ⋊ G ) ∼ = K 0 ( G \ S m − 1 ) . It is shown in [2] (see also [26, Theorem 0 .1]) that this map is surjective. Using the bundle s tr ucture o f C ( B ) ⋊ G ∼ = C ( B ⋊ G M n ( C )), this map can b e describ ed as follows: Firs t of a ll consider C ( B ⋊ G M n ( C )) a s a bundle over [0 , 1] with ﬁbre C ∗ ( G ) ∼ = M n ( C ) G at 0 and ﬁbre C ( S m − 1 ) ⋊ G ∼ = C ( S m − 1 × G M n ( C )) at every t 6 = 0. This bundle is trivial outside zero , and w e obtain the ab ov e sequence of K-theory maps b y ﬁrst extending a pro jetion o f the zero ﬁb er to a pro jectio n on a small neighborho o d, then use triviality outside zero to extend the pro jection to the whole bundle, and ﬁnally ev a luate this extended pro jection at the ﬁb er at 1. Assume no w that C ( B × G M n ( C )) would b e trivializa ble in the sense that it would b e stably isomorphic to so me s ubbundle of the trivia l bundle C ( G \ B , K ) with full ﬁbres outside (the o rbit of ) the origin. Then a n y pro jection in the ﬁber a t the orig in has a trivial ex tension (by the constant section) to all of C ( G \ B , K ) and the restric tion of s uch pr o jection to G \ S m − 1 would lie in the subgroup Z · 1 G \ S m − 1 of K 0 ( G \ S m − 1 ). Thus the ex istence of a trivialization together with surjectivit y of the map res ∗ in (5.23) would imply that K 0 ( G \ S m − 1 ) ∼ = Z · 1 G \ S m − 1 . But one can see in the app endix of [26] that e K 0 ( G \ S m − 1 ) := K 0 ( G \ S m − 1 ) / Z · 1 G \ S m − 1 is non-trivial (of ﬁnite or der) for many choices of groups Z n acting o n a suitable R m . F or instance: if m = 4 and n = 2, w e get e K 0 ( G \ S 3 ) ∼ = Z / 2. The above example also pro vides the basis for the following example of a n a ction with isolated ﬁxed points, such that the b oundary map in (5 .19) do es not v anish. W e are grateful to W olfgang L ¨ uck for p ointing out this example to us. Example 5 .26 . Cho o s e n and m and an action of G = Z /n on the unit ba ll B ⊆ R m as in the previous example such that e K 0 ( G \ S m − 1 ) := K 0 ( G \ S m − 1 ) / Z · 1 G \ S m − 1 is non-trivial. Notice that the map φ of (5.23) factors thr o ugh a surjective map e φ : Rep ∗ ( G ) → e K 0 ( G \ S m − 1 ) , 48 ECHTERHOFF AND EMERSON since Z · 1 G \ S m − 1 is the image of Z · 1 G ⊆ Rep( G ). Glueing tw o such ba lls at the bo undary ∂ B = S m − 1 we obtain an action of G on S m which ﬁxes the tw o points (0 , . . . , 0 , ± 1) and which is free on S m r { (0 , . . . , 0 , ± 1) } . Let us consider C ( S m ) ⋊ G as a bundle ov er [ − 1 , 1 ] with ﬁber C ( S m − 1 ) ⋊ G over each t 6 = ± 1 and ﬁb er C ∗ ( G ) at t = ± 1. W e wr ite Rep( G ) + and Rep( G ) − for the representation r ing of G when identiﬁed with the K-theory of the ﬁb er at t = 1 and t = − 1 , resp ectively . Then it follows from the descr iption of the map φ in (5.2 3) of the previous example that for a given pair ([ p + ] , [ p − ]) ∈ Rep( G ) + L Rep( G ) − there exists a class [ p ] ∈ K 0 ( C ( S m ) ⋊ G ) restricting to the pair if a nd only if φ ([ p + ]) = φ ([ p − ]). Now let I S m and Q S m as in Theo rem 5.20. Then K 0 ( Q S m ) ∼ = Rep ∗ ( G ) + M Rep ∗ ( G ) − . and the sequence (5.19) beco mes (5.24) 0 − → K 0 ( G \ S m ) − → K 0 ( C ( S m ) ⋊ G ) − → Rep ∗ ( G ) + M Rep ∗ ( G ) − ∂ − → K 1 ( G \ S m ) − → K 1 ( C ( S m ) ⋊ G ) − → 0 . Since a pair ([ p + ] , [ p − ]) ∈ Rep ∗ ( G ) + L Rep ∗ ( G ) − lies in the k ernel of the bo undary map ∂ if a nd only if it ex tends t o a class in K 0 ( C ( S m ) ⋊ G ), it follows from the abov e consideratio ns that this is possible if and only if the v alues φ ([ p + ]) and φ ([ p − ]) diﬀer by some cla ss in Z · 1 G \ S m − 1 . Thus w e see that ker ∂ = { ([ p + ] , [ p − ]) ∈ Rep ∗ ( G ) + M Rep ∗ ( G ) − : e φ ([ p + ]) = e φ ([ p − ]) } which is a prop er subgroup o f Rep ∗ ( G ) + L Rep ∗ ( G ) − . Thus it follows that the bo undary map in (5.24) is not trivia l. Note that a mo r e elab or ate study of this action (which we omit) reveals that K 1 ( G \ S m ) ∼ = e K 0 ( G \ S m − 1 ) and that the bo undary map in (5.24) is given by sending a pair ([ p + ] , [ p − ]) to the diﬀerence e φ ([ p + ]) − e φ ([ p − ]) a nd hence is surjective. Thus it follows that K 0 G ( S m ) = K 0 ( C ( S m ) ⋊ G ) ∼ = K 0 ( G \ S m ) M ker ∂ a nd K 1 G ( S m ) = { 0 } . References [1] H. Abels. A unive rsal pr op er G -sp ac e . Math. Z. 15 9 (1978), no. 2, 143–158 . [2] M.F. A tiy ah. K -theory . Amsterdam: Benjamin 1967. [3] M.F. Atiy ah, R. Bott. The moment map in e quivariant c ohomolo gy . T op ology 23, no.1 (1984), 1–28. [4] M.F. Atiy ah, G. Segal. The index of el liptic op er ators II , Ann. Math., 2nd series, 87, no. 3 (1968), 531–54 5. [5] M.F. Atiy ah, G. Segal. On equivaria nt Euler char acteristics , J.G.P . , 6 , no. 4, 1989 , 671–677. [6] L. Baggett. A description of the top olo gy on the dual sp ac es of c ert ain lo c al ly c omp act gr oups . T rans. Amer. Math. So c. 132 (1968), 175–215. [7] P . Baum, A . Connes. Chern cha r acter for discr ete gr oups . A f´ ete of topology . Acade mic Press, Boston, MA (198 8), pp. 163–232. [8] P . Baum, A. Connes, N. Higson. Classifying sp ac e for pr op er actions and K -the ory of gr oup C*-algebr as , Contemporary M athematics, 167 , 241-291 (1994). [9] B. Blac k adar. K -the ory for op er ator algebr as . Second edition. Mathematical Sciences Research Institute Publications, 5. Cambridge Universit y Press, Cambridge, 1998. xx+300 pp. ISBN: 0-521-63532-2 [10] R.J. Blattne r. On a the or em of G. W. Mackey . Bull. Amer. Math. Soc. 68 (1962), 585–58 7. [11] F. Combes. Cr osse d p r o ducts and Morita e quivalenc e . Proc. London Math. Soc. (3) 49 (1984), no. 2, 289– 306. [12] P . E . Conn er, E.E. Floyd . Ma ps of Odd Perio d . Ann. Math., Ser. 2, 84 , no. 2 (1966 ) 132–1 56. [13] A. Deitmar, S. Ec hterhoﬀ. Principles of harmonic analysis . Universitext. Springer, New Y or k, 2009. xvi+333 pp. ISBN: 978 -0-387-85468-7 CR OSSED PRODUCT S BY PR OPER ACTIONS 49 [14] S. Ech terhoﬀ. O n induc ed c ovariant systems . Pr oc. Amer. Math. Soc. 108 (1990 ), no. 3, 703–706. [15] S. Ech terhoﬀ. Cr osse d pr o ducts, the Macke y-Rieﬀel-Gr e en machine and applic ations. Preprint: arXiv:1006.4975v1 [math.O A] [16] S. Ech terhoﬀ, H. Emerson, H-J. Kim. KK-the or etic duality for pr op er twiste d actions. Math . Ann. 340 (200 8), no. 4, 839873. [17] S. Ec h terhoﬀ, H. Emerson, H-J. Kim. A L efschetz ﬁxed -p oint formula for c ertain orbifold C*-algebr as J. Noncomm ut. Geom. 4 (2010) (T o app ear). [18] S. Ec ht erhoﬀ, W. L¨ uc k, N.C. Phill ips, S. W alters. The structur e of cr osse d pr o ducts of irr a- tional r otation algebr as by ﬁnite sub gr oups of SL 2 ( Z ). J. Reine Angew. M ath. 639 (2010), 173–221. [19] S. Ech terhoﬀ, O. Pfante. Equivariant K -the ory of nite dimensiona l r ea l ve ctor sp ac es . M ¨ unst er J. of Math. 2 (2 009), 6594. [20] S. Ech terhoﬀ, D.P . Williams. I nducing primitive ide als. T rans. Amer. Math. So c. 360 (2008 ), no. 11, 611 3–6129. [21] H. Emerson. L o ca lization te chniques in cir cle-e quivariant KK -the ory . arX iv preprint 0024882. [22] H. Emerson, R. Meyer. Equivariant Lefschetz maps for simplicial c omplexes and smo oth manifolds . Math. Ann., 345 ( 2009), no. 3, pp. 599–630 . [23] H. Em erson, R. Mey er. Equivariant embe dding the or ems and top olo gic al index maps . arXiv preprint 0908.1465. [24] H. Emerson, R. Meye r. Equivariant r epr esentable K - the ory. J. T op ol . 2 (2009 ), no. 1, 123–156. [25] B. Ev ans. C* -bund les and c omp act tra nsformation gr oups. Memoirs Amer. M ath. Soc. 39 no 269 (198 2). [26] P . B . Gilkey . The eta inv ariant and ˜ K O of lens sp ac es. Math. Z. 194 (198 7), 309–320. [27] J. Glimm. L o c al ly c omp act tr ansformation gr oups . T rans. Amer. Math. Soc. 101 (1961) 124– 138. [28] J. Glimm. F amilies of induc e d r epr esent ations . Paciﬁc J. Math. 12 (1962), 885–911. [29] P . Green, C ∗ -algebr as of tr ansformation gr oups with smo oth orbit sp ac e. P aciﬁc J. Math. 72 (1977), no. 1, 71 –97. [30] P . Green, The lo c al structur e of twiste d c ovarianc e algebr as. Act a Math. 140 (1978), no. 3-4, 191–250. [31] E.C. Gootman, J. R osenberg. The structur e of cr osse d pr o duct C ∗ -algebr as: a pr o of of the gener alize d Eﬀr os-H ahn co nje ctur e. In v en t. M ath. 52 (1979), no. 3, 283–2 98. [32] J.M.G. F ell. We ak c ontainment and induc e d r epr esentations of gr oups . II. T rans. Amer. Math. Soc. 11 0 (1964), 424–447. [33] P . Julg. K -th´ eorie ´ equivariante et pr o duits cr ois ´ es . C. R. Acad. Sci. Paris S ´ er. I Math. 9 2 (1981), no. 13, 6 29–632. [34] M. Karoubi. Equivariant K -the ory of r e al ve ctor sp ac e s and r e al pr oje ctiv e sp ac es. T opology Appl. 122 (2002) , no. 3, 531–546. [35] G. Kasparov. Equivariant KK -the ory and the Novikov c onje c tur e , Inv ent. Mat h. 91 , 147-201 (1988). [36] G. Kasparov, G. Sk andalis. Gr oups acting pr op erly on “ b olic” sp ac e s and the Novikov co n- je c tur e. Annals of Math. 158 (20 03), 165–2 06. [37] W. L ¨ uc k and B. Oliver. The c ompletion the or em in K - the ory for pr op e r acti ons of a discr ete gr oup . T op ology 40 (2001), no. 3, 585–6 16. [38] W. L ¨ uc k and R. Stamm. Computations of K - and L -t he ory of c o c omp act planar gr oups. K -theory 21 (200 0), 249–292. [39] R. Meyer. Gener alize d ﬁxe d-p oint algebr as and squar e int e gr able r epr esentations. J. F unct. Anal. 186 , (20 01), 167–195. [40] R.S. P alais. On the existence of slices for actions of non-c omp act Lie gr oups. Ann. of Math. (2) 73 (1961) 29 5–323. [41] I. Raeburn, D.-P . W illi ams. Pul l-b acks of C ∗ -algebr as and cr osse d pr o ducts by c ertain diag- onal act i ons. T rans. Amer. Math. Soc. 28 7 ( 1985), no. 2, 755–777. [42] I. Raeburn, D.-P . Williams. Morita e quivalenc e and c ontinuous-tr ac e C ∗ -algebr as . Mathe- matical Surveys and M onographs, 60. Ameri can Mathemat ical Societ y , Pro vi dence, RI, 1998. xiv+327 pp. ISBN: 0-821 8-0860-5 [43] M.A. Ri eﬀel. Pr op er actions of gr oups on C ∗ -algebr as . Mappings of operator al gebras (Philadelphia, P A, 1988) , 141–182, Progr. Math., 84, Birkhuser Boston, Bost on, M A , 1 990. [44] M.A. Rieﬀel. Inte g ra ble and pr op er actions on C ∗ -algebr as, and squar e-inte gr able r epr esen- tations of gr oups . Expo. Math. 22 (2004), no. 1, 1–5 3. [45] M.A. Rieﬀel. Actions of ﬁnite groups on C ∗ -algebras. M ath. Scand. 47 (198 0), no. 1, 157–176. [46] M.A. Rieﬀel. On the uniqueness of the Heisenb er g c ommutation r elations . Duk e Math. J. 39 (1972), 745–75 2. 50 ECHTERHOFF AND EMERSON [47] G. Segal. Equiv ariant K -the ory . Inst. Hautes ´ Etudes Sci. Publ. M ath. No. 34 (1968), 129–151. [48] D.P . Willi ams. Cr ossed products of C*-algebras. M athematica l Su rveys and Monographs, V ol 134. AMS 2007 . [49] D.P . Williams . The structur e of cr osse d pr o ducts by smo oth actions . J. Austral. Math. So c. Ser. A 47 ( 1989), no. 2, 226–235. [50] M. Y ang. Cr osse d pr o ducts by ﬁnite gr oups acting on low dimensional c omplexes and appli- c ations. Ph.D Thesis, Universit y of Sask etchew an, Sask ato on, 1997. Westf ¨ alische Wilhelms-Universit ¨ at M ¨ unster, Ma them a tisches Institut, Einsteinstr. 62 D-48149 M ¨ unster, German y E-mail addr ess : echters@u ni-muenster .de Dep ar tment of Ma thema tics and St a tistics, University of Victo ria, PO BO X 30 45 STN CSCVictoria, B.C.Canada E-mail addr ess : hemerson@ math.uvic.c a

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