The Pimsner-Voiculescu sequence for coactions of compact Lie groups

The Pimsner-V oiculescu sequence for coactions of compact Lie groups Magn us Goffeng Departmen t of Mathematical Sciences, Division o f Mathematics Chalmers unive rsit y of T ec hnology and Univ ersit y o f Gothenburg Abstract The Pimsner-V oiculescu sequence is generalized to a Pimsner-V oiculescu to wer describing the K K -categ ory equ iv ariant with resp ect to coactions of a compact Lie group satisfying the Ho dgkin co ndition. A du al Pimsner- V oiculescu to wer is used to sho w that coactions of a co mpact Ho dgkin-Lie group satisf y the Baum-Connes prop erty . 2000 Mathematics Subje ct Classific ation 46L80, 19K35, 46L55 In tro duction When G is a s econd countable, lo c a lly compact group and A is a separa ble C ∗ - algebra with a contin uo us G -action, the Ba um-Connes conjecture states that the K -theory of the reduced cross ed pr o duct A ⋊ r G can b e calculated by means of geometric a nd representation theo retical prop erties o f G and A , s ee mor e in [4]. T o b e more pr ecise, the Baum- C o nnes co njectur e states that the a s sembly mapping µ A : K G ∗ ( E G ; A ) → K ∗ ( A ⋊ r G ) is a n isomo rphism. The space E G is the universal prop er G -space and K G ∗ ( E G ; A ) is the pro pe r equiv ariant K -homolog y with co efficien ts in A . There a re known counterexamples when µ A is not an isomorphism, so it is more natural to speak of groups having the Ba um- C o nnes prop erty . In [1 0], the equiv aria nt K -ho mology with co efficients in A w as proved to be the left derived functor of F ( A ) = K ∗ ( A ⋊ r G ) a nd the assembly mapping being the natur a l trans fo rmation from L F to F . The a pproach to the Baum- Connes pr op erty using tria ng ulated categor ies can be genera lized to discr ete quantum groups, see [9 ], which indica tes that geometr ic techniques such as universal prop er G - spaces can b e ge ne r alized to dis crete qua nt um gr oups. The gener alization of the Baum-Connes pr op erty to quantum g roups has bee n studied in for insta nce [1 1] and [17]. The case studied in [11] is that o f quantum gro up a ctions of the dual o f a c ompact Lie group which corresp ond to coactions of the Lie g roup. In [11] duals of compact Lie groups were shown to satisfy the str ong Baum-Connes prop erty , i.e. the embedding of the triangulated category gener ated by prop er coactions, the C ∗ -algebra s that ar e Ba a j-Sk andalis dual to trivial G -ac tions, in to the K K -ca tegory equiv a riant with resp ect to coac- tions is essentially sur jective. In this pap er we cons tr uct an analog ue of the Pimsner-V oiculescu sequence for coa ctions of a co mpact Ho dgkin- Lie group G that describ es how the K K -categor y equiv a r iant with resp ect to coactions of G 1 is built up fro m the C ∗ -algebra s with coactions of G which are prop er in the sense o f [11]. The starting p oint is to express the Pimsner-V oiculescu seq uence for Z - actions in terms of a pr op erty of the repr esentation ring o f a rank one tor us . Using the Universal Co efficient Theorem, the Pimsner-V o iculescu sequence can be cons tructed from a Ko szul complex 0 → R ( T ) α − → R ( T ) → 0, where α is defined as multiplication by 1 − t under the isomorphis m R ( T ) ∼ = Z [ t , t − 1 ] . When A has a coaction of T , i.e. a Z -action, the tensor pro duct ov er R ( T ) b etw een this Koszul co mplex and K T ∗ ( A ⋊ r Z ) giv es the Pimsner-V oiculesc u sequence. In the g e ne r alization to higher rank, when T is a to rus o f rank n we consider the Koszul co mplex 0 → ∧ n R ( T ) n → ∧ n − 1 R ( T ) n → . . . → ∧ 2 R ( T ) n → R ( T ) n → R ( T ) → 0. The bo undary mappings in this complex are defined from interior multiplication with the elemen t P ( 1 − t i ) e ∗ i ∈ Hom R ( T ) ( R ( T ) n , R ( T )) . If G is a compact Hodg kin- Lie group with maximal torus T , the r epresentation ring R ( T ) is a fre e R ( G ) - mo dule by [1 5], so the ge neralization from a torus to compact Ho dgkin-L ie groups go es in a straightforw ard fashio n. Just as w hen the rank is 1 , the Koszul complex ab ov e ca n b e used to produce sequence of dis tinguished triangles which is the a na logue of a P imsner-V o iculescu sequence for the K -theor y of crossed pro ducts by coactio ns of G . W e will give a geometric description of a sequence of distinguished triangles in the K K -categ ory equiv ariant with resp ect to coa ctions of G that corr esp onds to the a bove K oszul complex under the Universal Co efficient Theo rem. As for the Pimsner- V oiculescu sequence for Z we will obtain a pr o jective reso lution of the cros sed pro duct b y a c o action in the sens e of triang ulated categories rather than exact seq uences. Using suitable tensor pr o ducts we pro duce in Theore m 3.4 a sequence of distinguished tria ngles in the K K -catego ry equiv ariant with resp ect to coactions of G that we call the generalized Pimsner-V oic ulescu to w er for A : C w ⊗ A / / Σ n D n − 1 ( A ) / / | | z z z z z z z z z Σ n D n − 2 ( A ) / / { { w w w w w w w w w w · · · Σ C w n ⊗ A ◦ @ @ @ @ ` ` @ @ @ @ Σ 2 C w k n − 1 ⊗ A ◦ H H H H H c c H H H H H ◦ o o · · · ◦ : : : : \ \ : : : : ◦ o o · · · / / Σ n D 2 ( A )           / / Σ n D 1 ( A ) | | x x x x x x x x x / / t ( A ⋊ r ˆ G ) } } { { { { { { { { { · · · Σ n − 1 C w k 2 ⊗ A ◦ F F F F F b b F F F F ◦ o o Σ n C w ⊗ A ◦ B B B B ` ` B B B B ◦ o o . Here t ( A ⋊ r ˆ G ) denotes the C ∗ -algebra A ⋊ r ˆ G equipp ed with the triv ial ˆ G -actio n and the ter ms D i ( A ) can b e ex plicitly desc r ib ed a s a bra ided tensor pro duct. T aking K -theory of the low e r row will give a co mplex similar to the Kos zul complex that in a sens e for ms a pr o jective resolution of the K - theory of A ⋊ ˆ G . The dual P imsner-V o ic ule s cu gives a mo re pr ecise description of the results of 2 [11] by a sequence of distinguished triang le s in K K G that descr ibe s the cros sed pro duct A ⋊ r ˆ G in terms of G − C ∗ -algebra s with trivial G -action, thus giving a direct ro ute to the stro ng Baum-Connes pr op erty of ˆ G . The pap er is orga nized a s follows; the first section consists of a review of K K -theory of actio ns and coactions. In particular we g ather some kno wn r esults ab out the braided tensor pr o duct and the Drinfeld double which pla ys a may or role in constructing the dual P ims ner-V oic ule s cu tower. The main references of this sectio n are [1], [2], [3], [7], [10], [12] and [1 6]. In the second section a geometric construc tio n o f the Pimsner-V o iculescu sequence for Z -actions is presented and generalized to higher rank via a Koszul co mplex . In the third section the restriction functor for coactions is use d to g e ne r alize the Pimsner- V oiculescu sequence to coactions of compact Ho dgkin-Lie g roups G . As an example o f this we calculate the K -theor y of so me compa ct homog e neous spaces. By similar metho ds, a dual P ims ne r -V oivules cu tow er is constructed in K K G , following the ideas of [10]. At the e nd of the pap er we discuss so me po ssible generaliza tions to duals of W oronowicz deformations. Ac knowledgment s The author would lik e to thank Ryszard Nest for po sing the question on how to explicitly constr uct the crosse d pro duct o f a ˆ G − C ∗ - algebra fr om trivia l actions a nd for m uch inspiration in the wr iting pro c ess. 1 Actions and coactions of compact groups The standard approach to equiv ar iant K -theory is to in tro duce equiv ariant K K - theory . If A and B are tw o s e parable C ∗ -algebra s with a contin uo us ac tion o f a lo cally c ompact gr oup G , the equiv ar iant K K -gro up K K G ( A , B ) is defined as the set of homotopy cla sses of G -equiv ar ia nt A − B - K asparov mo dules which forms an ab elian gro up under direct sum. The K K -gro ups ca n be equipp ed with a pro duct such that if C is a third sepa r able C ∗ -algebra with a contin uous G - action ther e is an additive pairing ca lled the Kaspar ov pro duct K K G ( A , B ) × K K G ( B , C ) → K K G ( A , C ) . F ollowing the standar d co nstruction, we let K K G denote the additive catego ry of a ll separable C ∗ -algebra s with a contin uous G - action a nd a morphism in K K G from A to B is an element of K K G ( A , B ) . The co mp os ition of tw o K K G -morphisms is defined to be their Kas parov pro duct. The g roup K K G ( C , A ) coincides with the equiv ar iant K -theory of A . In particular, if G is compact K K G ( C , C ) = R ( G ) , the representation ring o f G . The a ction of R ( G ) on equiv ariant K -theory gener alizes to an R ( G ) -mo dule structure on the biv aria nt g roups K K G ( A , B ) . The category K K G can be equipp ed with a triangulated str ucture with a map- ping cone coming from the ma pping cone construction o f a ∗ -ho momorphism. The triangulated structure on K K G is universal in the sense that a n y homotopy inv aria nt, s table, split-exa ct functor on the categor y of C ∗ -algebra s with a con- tin uous G -a c tion defines a homo logical functor on K K G . The construction o f the triangula ted structure and its universality ar e thoroughly explained in [10]. Let us just recall the basics of the construction of the tria ngulated structure on K K G . The susp ensio n Σ A of a G − C ∗ -algebra is defined by C 0 ( R ) ⊗ A . By Bott per io dicity Σ 2 ∼ = id . A distinguished triangle in K K G is a tria ngle isomor phic to 3 one of the for m C ( f ) / / A f          B , ◦ 5 5 5 Z Z 5 5 5 where C ( f ) is the mapping cone of the equiv aria nt ∗ -homomo rphism f : A → B . In particular , if f : A → B is a sur jection and admits an equiv a riant completely po sitive splitting the natura l mapping ker ( f ) → C ( f ) defines an equiv ariant K K -isomorphis m, so under suitable a ssumptions a distinguished triangle is is o- morphic to a sho rt exa ct sequence. How to cons tr uct K K -theory of coactions o f gr oups is easiest seen in the simpler case when G is an ab elian gro up. If A is a C ∗ -algebra equipp ed with an action α of the ab elia n gro up G , the cross e d pro duct A ⋊ r G car ries a natura l action of the Pontry agin dual ˆ G . This actio n is called the dual action o f ˆ G . Since abelian gro ups ar e exact, the cr o ssed pr o duct b y an abelian group defines a triangulated functor K K G → K K ˆ G . The crossed pro duct by the dual a ction is describ ed by T akesaki-T ak ai duality which states that ther e is an equiv aria nt isomorphism A ⋊ r G ⋊ r ˆ G ∼ = A ⊗ K ( L 2 ( G ) ) , where A ⋊ r G ⋊ r ˆ G is equipp ed with the dual a ction of G a nd the G -a ction on A ⊗ K ( L 2 ( G ) ) is defined as α ⊗ A d . T akesaki-T a k ai duality implies that the crossed pr o duct defines a triangulated e quiv alence K K G → K K ˆ G . An action α o f a group G on A defines a ∗ -homomorphis m ∆ α : A → M ( A ⊗ C 0 ( G ) ) by letting ∆ α ( a ) b e the function g 7→ α g ( a ) . When G is ab elian there is a natura l isomorphism C 0 ( ˆ G ) ∼ = C ∗ r ( G ) and a ˆ G - action cor resp onds to a no n- degenerate ∗ -homomo rphism ∆ A : A → M ( A ⊗ mi n C ∗ r ( G ) ) satisfying cer tain con- ditions. The first instance of a coaction of a group G is on C ∗ r ( G ) . Using the universal prop erty of C ∗ r ( G ) , o ne can co nstruct a non- degenerate mapping ∆ : C ∗ r ( G ) → M ( C ∗ r ( G ) ⊗ mi n C ∗ r ( G ) ) called the comultiplication and is induced from the diagonal homomorphism G → G × G . Clear ly , the mapping ∆ s atisfies: (∆ ⊗ id )∆ = ( id ⊗ ∆)∆ , so we say that ∆ is coasso ciative. S ince ∆ 21 = ∆ the comultiplication ∆ is co commutativ e, so if we in terpret C ∗ r ( G ) as the functions on a reduced lo cally compact quantum gr oup ˆ G then ˆ G c a n be thought of a s ab elian, see mo re in [7]. With the ab elian setting as motiv atio n, w e say that a sepa rable C ∗ -algebra A has a coa ction of the lo cally co mpact second countable group G if there is non-degenera te ∗ -homomo rphism ∆ A : A → M ( A ⊗ mi n C ∗ r ( G ) ) satisfying the tw o conditions that ∆ A ( A ) · 1 A ⊗ mi n C ∗ r ( G ) is a dense subspac e o f A ⊗ mi n C ∗ r ( G ) and that ∆ A is coa sso ciative in the sense that (∆ A ⊗ id C ∗ r ( G ) )∆ A = ( id A ⊗ ∆)∆ A . (1) A s eparable C ∗ -algebra eq uipp ed with a co action of G will be ca lled a ˆ G − C ∗ - algebra. So metimes w e will abuse the notation and call a coa ction of G a ˆ G - action. An ex ample of a coactio n is th e dua l co action on C ∗ -algebra s of the form A = B ⋊ r G , for some G − C ∗ -algebra B . When G is discr e te w e can decomp ose B ⋊ r G b y means of the dense subspa ce ⊕ g ∈ G B λ g and the dual coa ction is defined 4 by ∆ A ( b λ g ) : = b λ g ⊗ λ g . In the g eneral setting, the cons truction of the dual coaction go es analogo usly and we refer the reader to [1]. Much o f the theory for group actions also hold for g roup coa ctions, the crossed pr o duct will as for ab elian g roups b e a stepping stone back and for th betw een actions and coac tions. In [1], the K K -theor y equiv a riant with resp ect to a bi- C ∗ -algebra s and the co rresp onding Kaspar ov pro duct was co nstructed. In [12] it was proved that the K K -theor y equiv a riant with resp ect to a lo cally compact quantum gr o up has a tria ngulated s tructure defined in the same fashion as for a group. Let us explain the setting of [1] more explicitly in the case of co a ctions of a gro up. An A − B -Hilbert bimo dule E is called ˆ G -equiv ariant if there is a coaction δ E : E → L B ⊗ mi n C ∗ r ( G ) ( B ⊗ mi n C ∗ r ( G ) , E ⊗ C ∗ r ( G ) ) satisfying a coasso ciativity condition similar to (1) a nd δ E should comm ute with the A -action and B -action in the o bvious ways. By P rop osition 2.4 of [1], th e coaction δ E is uniquely determined b y a unitary V E ∈ L ( E ⊗ ∆ B ( B ⊗ mi n C ∗ r ( G ) ) , E ⊗ C ∗ r ( G ) ) via the equa tio n δ E ( x ) y = V E ( x ⊗ ∆ B y ) for x ∈ E a nd y ∈ B ⊗ mi n C ∗ r ( G ) . A ˆ G - equiv ar iant A − B -Ka sparov mo dule is a n A − B -K asparov mo dule ( E , F ) such that E is a ˆ G - equiv ar iant A − B - Hilber t mo dule and the o per ator F commut es with the unitar y V E up to a co mpact op er a tor. The gr o up K K ˆ G ( A , B ) is defined as the homotopy classes of ˆ G - equiv ar iant A − B -Ka sparov mo dules. The additive category K K ˆ G is defined b y taking the ob jects to b e separ a ble ˆ G − C ∗ -algebra s and the gr oup of morphisms from A to B is K K ˆ G ( A , B ) . The co mpo sition in K K ˆ G is Kaspar ov pro duct of ˆ G - equiv ar iant Kas parov modules. T o a c lo sed subgroup H of G , the restriction of a G -action to H defines a restriction functor Re s G H : K K G → K K H and its rig ht a djo int is the induction functor I nd G H : K K H → K K G . How ever the res triction go es in the o ther direction for coactio ns. When H is a close d subgr o up o f G , there is a non-degener ate embedding C ∗ ( H ) ⊆ M ( C ∗ ( G ) ) so a coactio n of H can b e restricted to a c o action of G . This c onstruction defines a triang ulated functor Re s ˆ H ˆ G : K K ˆ H → K K ˆ G . The crossed pro duct B 7→ B ⋊ r G sends a G − C ∗ -algebra to a ˆ G − C ∗ -algebra and if G is exact the crossed pro duct induces a triangulated functor K K G → K K ˆ G . In order to construct a dua lity similar to T akesaki-T ak ai duality one introduces the cr ossed pro duct by a coaction. If A is a ˆ G − C ∗ -algebra we define A ⋊ r ˆ G : = [∆ A ( A ) · 1 A ⊗ C 0 ( G ) ] ⊆ M ( A ⊗ K ( L 2 ( G ) )) . It follows fro m Lemma 7.2 of [2] that A ⋊ r ˆ G fo r ms a C ∗ -algebra . F or a thoroug h int ro duction to cr ossed pro ducts by coa ctions s ee [13]. The C ∗ -algebra A ⋊ r ˆ G carries a contin uous G -action defined in the dense subspace ∆ A ( A ) · 1 A ⊗ C 0 ( G ) by g . (∆ A ( a ) · 1 A ⊗ f ) : = ∆ A ( a ) · 1 A ⊗ g . f . Similarly to the a be lia n setting, T akesaki-T ak ai dualit y holds so ther e are equiv- ariant isomorphisms B ⋊ r G ⋊ r ˆ G ∼ = B ⊗ K ( L 2 ( G ) ) and A ⋊ r ˆ G ⋊ r G ∼ = A ⊗ K ( L 2 ( G ) ) which ensures that the crossed pro duct defines an equiv alence o f tria ngulated categorie s kno wn a s Baa j-Sk a ndalis duality . The tensor pr o duct on the catego ry o f G − C ∗ -algebra s is well defined. If A and B hav e a ctions α r esp ectively β of G the tensor pro duct A ⊗ mi n B can b e equipp e d with the action α ⊗ β : G → A u t ( A ⊗ mi n B ) . How ever, for a non-ab elian group G the co nstruction of a tensor pro duct o f ˆ G − C ∗ -algebra s can no t b e done 5 by just taking tenso r pr o ducts of the C ∗ -algebra s. The tensor pro duct r elev ant for ˆ G − C ∗ -algebra s is the braided tensor pro duct over ˆ G whic h requir es one further structure. Supp os e that A is a ˆ G - algebra with a contin uo us G -action α . If the action α satisfies that ∆ A ◦ α g = ( α g ⊗ Ad ( g ))∆ A (2) we say that A is a Y etter-Drinfeld alg ebra. An example of a Y etter- Drinfeld algebra is C ∗ r ( G ) with G -action defined by the adjoint action G → A u t ( G ) . It is muc h easier to co nstruct a Y etter-Drinfeld a lgebra from a G − C ∗ -algebra , if A is a G − C ∗ -algebra we can in a functor ial wa y define a co action o f G on A by setting ∆ A ( a ) : = a ⊗ 1 . When A is a Y etter-Drinfeld algebr a, the C ∗ -algebra A ⋊ r ˆ G is also a Y etter-Drindeld alge bra s ince the morphism ∆ A is cov aria nt with resp ect to the G -action and ∆ A extends to a coaction of G o n A ⋊ r ˆ G , see more in [12]. This construc tio n is functorial and the crossed pro duct can b e seen as a functor on the catego ry of Y etter-Drinfeld algebras . When A is a Y etter-Drinfeld algebr a and B is a ˆ G − C ∗ -algebra w e define the mappings ι A : A → M ( A ⊗ mi n B ⊗ K ( L 2 ( G ) )) , ι ( a ) : = ∆ α ( a ) 13 ι B : B → M ( A ⊗ mi n B ⊗ K ( L 2 ( G ) )) , ι ( b ) : = ∆ B ( b ) 23 . F ollowing [12], the braided tensor pro duct A ⊠ ˆ G B is defined as the clos ed linear span of ι A ( A ) · ι B ( B ) . By Prop o sition 8.3 of [1 6], A ⊠ ˆ G B forms a ∗ -suba lgebra of M ( A ⊗ mi n B ⊗ K ( L 2 ( G ) )) so the braided tensor pro duct is a C ∗ -algebra . The coaction of G on A ⊠ ˆ G B is defined by ∆ A ⊠ ˆ G ∆ B ( ι A ( a ) · ι B ( b )) : = ( ι A ⊗ id )(∆ A ( a )) · ( ι B ⊗ id )(∆ B ( b )) . Observe that since C ∗ r ( G ) is co commutativ e, the a djoint ˆ G - action is tr ivial and a similar constr uction of a braided tensor pro duct over G b etw een G − C ∗ -algebra s with trivial ˆ G - actions co incide s with the us ual tenso r pr o duct. In general, the braided tensor pr o duct over G do es not need to coincide with the usual tensor pro duct. By Lemma 3.5 of [12] there is a G -e quiv aria nt is omorphism ( A ⊠ ˆ G B ) ⋊ r ˆ G ∼ = ( A ⋊ r ˆ G ) ⊠ ˆ G B (3) where the G - c oaction on t he r ight hand side is the trivial o ne on B . More generally , this identit y holds for any qua nt um g roup and in particular also for braided tensor pr o ducts over G . W e will prov e this statement in sp ecia l case of braided tensor pro ducts over a compact group G with C ( G ) b elow in Lemma 3.3. If we interpret the structur e of a Y etter-Drinfeld algebra as tw o actions of the quantum groups G and ˆ G sa tisfying a cer ta in co cy c le re la tion, the co c y cle defines a quantum group by means of a double cr ossed pro duct s uch that Y etter - Drinfeld a lgebras are pr ecisely the C ∗ -algebra s w ith an action of this double crossed pro duct. The right quantum gr oup to lo ok a t is the Dr infeld do uble D ( G ) . Using the notations of qua nt um groups, the algebra of functions on D ( G ) is C 0 ( G , C ∗ r ( G ) ) = C 0 ( G ) ⊗ C ∗ r ( G ) with the o bvious a ction and coaction of G . The action a nd coactio n define a comultiplication ∆ D ( G ) : C 0 ( D ( G )) → M ( C 0 ( D ( G )) ⊗ C 0 ( D ( G ))) 6 by ∆ D ( G ) : = σ 23 A d ( W 23 )(∆ C 0 ( G ) ⊗ ∆ C ∗ r ( G ) ) wher e W ∈ B ( L 2 ( G ) ⊗ L 2 ( G ) ) is the m ultiplicative unitar y of G defined by W f ( g 1 , g 2 ) = f ( g 1 , g 1 g 2 ) . The co mu lti- plication ∆ D ( G ) makes D ( G ) into a quantum group by Theo rem 5.3 of [3]. A Y etter-Drinfeld a lgebra A with the actio n α and co action ∆ A corres p o nd to a D ( G ) − C ∗ -algebra by defining the D ( G ) -coaction ∆ D ( G ) A : = (∆ α ⊗ id )∆ A : A → M ( A ⊗ mi n C 0 ( D ( G ))) , see mor e in Pr op osition 3.2 of [12]. Therefo r e we can consider the braided tensor pro duct as a tensor pro duct b etw e e n D ( G ) − C ∗ -algebra s and ˆ G − C ∗ -algebra s. The br aided tensor pro duct induces a biadditive functor ⊠ ˆ G : K K D ( G ) × K K ˆ G → K K ˆ G . Much o f the theor y of coactions can b e do ne without int ro ducing a ny quan tum groups, but in order to construct the Pimsner-V oicules cu sequence for co actions of compact Ho dgkin-L ie g roups we will need the braided tensor pr o duct as a biadditive functor b etw e e n K K -catego ries. 2 The Pimsner-V oiculescu sequence from the view- p oin t of represen tation rings In this section we will study the P ims ne r -V oicules cu sequence for Z and gener- alize to a Pimsner-V oiculescu tow er for Z n . W e will use representation theory to calculate all the mappings explicitly . These ca lculations will in a surpris- ingly straightforward wa y give a na tural route to a Pimsner - V oiculescu tow er for co actions of c ompact Lie gr oups. Consider the ev alua tion mapping l : C 0 ( R ) → C 0 ( Z ) . This mapping fits in to a Z -equiv ar iant short exact s equence 0 → Σ C 0 ( Z ) → C 0 ( R ) l − → C 0 ( Z ) → 0. (4) The Z -equiv ar iant Dirac op erato r D / on R defines a Z -equiv ar iant o dd un- bo unded K -homology cla ss, thus an ele ment [ D / ] ∈ K K Z ( C 0 ( R ) , Σ C ) . While R is the universal prop er Z -space the ele ment [ D / ] is the Dir ac element o f Z and the str ong B a um-Connes pro p er ty of Z implies tha t [ D / ] is a K K Z -isomorphis m. The exact sequence (4) induces a disting uished triangle in K K Z which after using the iso morphism C 0 ( R ) ∼ = Σ C and rotation 4 steps to the left b ecomes C 0 ( Z ) / / C 0 ( Z )          C . ◦ 3 3 3 Y Y 3 3 3 (5) In a certain sense, the distinguished triangle (5) captures the entire b ehavior of the Pimsner-V o iculescu sequence. If A is a Z − C ∗ -algebra we can apply Baa j- Sk andalis duality to (5 ) and tensor with A ⋊ r Z . If we apply B a a j-Sk andalis 7 duality again, we obtain a distinguished triangle in K K Z : A / / A          A ⋊ r Z , ◦ 3 3 3 Y Y 3 3 3 where A ⋊ r Z is giv en the trivial Z -action. T aking K -theory of this distinguished triangle gives bac k the classical Pimsner-V oic ule s cu s equence due to the follow- ing lemma: Prop ositi on 2.1. When T is a torus of r ank 1 and the element κ ∈ K K T ( C , C ) is define d u sing the isomorphisms K K T ( C , C ) ∼ = Hom R ( T ) ( R ( T ) , R ( T )) and R ( T ) ∼ = Z [ t , t − 1 ] as κ f ( t , t − 1 ) = ( 1 − t ) f ( t , t − 1 ) , the K K -morphism κ is Baaj-Skandali s dual to the K K -morphism C 0 ( Z ) → C 0 ( Z ) define d by (4) . Observe that the K -theory of the exact s equence (4) is descr ib e d from the exact se q uence: 0 → R ( T ) 1 − t − → R ( T ) → Z → 0, by Pr op osition 2.1. The first terms in this ex act sequenc e is the Kos zul complex defined by 1 − t ∈ Hom R ( T ) ( R ( T ) , R ( T )) and Z is the co homology of the K oszul complex. Pr o of. Let κ 0 ∈ Hom R ( T ) ( R ( T ) , R ( T )) deno te the Ba a j-Sk andalis dual o f the K K - morphism induced from (4 ). It follows directly from the co nstruction that the mapping R ( T ) → Z induced fr om Σ C 0 ( Z ) → C 0 ( R ) is the augmentation mapping Z [ t , t − 1 ] → Z onto the generator of K 1 ( C 0 ( R )) . Therefo re the image of κ 0 is the ideal gener a ted by either 1 + t of 1 − t so κ 0 is of the form u · ( 1 ± t ) fo r some unit u ∈ Z [ t , t − 1 ] . The sign and u = 1 is found by either a dir ect calculation or by considering the Pimsner-V oiculescu sequence for C 0 ( Z ) . W e will r eturn to the K oszul complexes later on. First we will construct a geometric interpretation of the higher rank situation. Assume that T is a torus of r ank n and co nsider the semi-op en unit cub e I = [ 0, 1 [ n ⊆ R n . F or i = 1, . . . , n we define ˜ X i as the set o f o pe n i − 1 -dimensional faces o f I . The union sa tis fie s ∪ n i = 1 ˜ X i = ∂ I ∩ I . W e let k i , for i = 1, 2, . . . n , deno te the integers k i : =  n i − 1  . The set ˜ X i has k i connected co mpo nents so if w e c ho ose a homeomo r phism ] 0, 1 [ ∼ = R there ar e homeo mo rphisms ˜ X i ∼ = k i a j = 1 R i − 1 for i = 1, 2, . . . , n , (6) where w e interpret R 0 as the one-p oint spa ce. W e take X i to b e the Z n -translates of ∪ j ≤ i ˜ X j and define Y i : = R n \ X i for 1 = 1, 2, . . . , n and Y 0 : = R n . 8 Prop ositi on 2.2. F or i = 1, 2, . . . , n ther e ar e Z n -e qu ivariant isomorphi sms C 0 ( Y i − 1 ) / C 0 ( Y i ) ∼ = C k i ⊗ Σ i − 1 C 0 ( Z n ) . Pr o of. By equation (6) there is a Z n -equiv ar iant homeomorphism Y i − 1 \ Y i ∼ = a m ∈ Z n   k i a j = 1 R i − 1   , where Z n acts by tra nslation on the first disjoint union. Therefore C 0 ( Y i − 1 ) / C 0 ( Y i ) ∼ = C 0 ( Y i − 1 \ Y i ) ∼ = C 0   a m ∈ Z n   k i a j = 1 R i − 1     ∼ = ∼ = C k i ⊗ C 0 ( Z n × R i − 1 ) ≡ C k i ⊗ Σ i − 1 C 0 ( Z n ) . Consider the cla ssifying spa ce R n for prop er ac tio ns o f Z n . Since Z n has the strong Baum-Connes prop erty , the Dirac element [ D / ] induces a K K Z n - isomorphism C 0 ( R n ) ∼ = Σ n C . An alterna tive approach to constr ucting this iso- morphism is the Julg theorem which implies that for any T − C ∗ -algebra A there is an isomorphis m K T ∗ ( A ) ∼ = K ∗ ( A ⋊ r T ) . Therefore K T ∗ (Σ n C ⋊ Z n ) ∼ = K T ∗ ( C 0 ( R n ) ⋊ Z n ) and the statement follows fr o m the Universal Co efficient Theorem for the com- pact Ho dgkin- Lie gr o up T , s ee mor e in [14]. F or i = 1, 2, . . . , n , Pro po sition 2 .2 implies that ther e is a Z n -equiv ar iant s hort exact se q uence 0 → C 0 ( Y i ) → C 0 ( Y i − 1 ) → C k i ⊗ Σ i − 1 C 0 ( Z n ) → 0. (7) W e will b y κ i ∈ K K Z n ( C k i ⊗ C 0 ( Z n ) , C k i + 1 ⊗ C 0 ( Z n )) deno te the Z n -equiv ar iant K K - morphism defined in s uch a wa y that the extensio n class defined by (7) comp osed with the res triction ma pping C 0 ( Y i ) → C k i + 1 ⊗ Σ i C 0 ( Z n ) coincides with Σ i − 1 κ i . Notice that Y n = Z n × ] 0, 1 [ n and Y 0 = R n so we have that C 0 ( Y n ) = Σ n C 0 ( Z n ) and C 0 ( Y 0 ) = C 0 ( R n ) , the latter b eing K K Z n -isomorphic to Σ n C . Thus we get a sequence o f distinguished triangles in K K Z n : Σ n C 0 ( Z n ) / / C 0 ( Y n − 1 ) / / z z u u u u u u u u u u · · · | | x x x x x x x x x x C n ⊗ Σ n − 1 C 0 ( Z n ) ◦ J J J J J Σ n κ n d d J J J J J C k n − 1 ⊗ Σ n − 2 C 0 ( Z n ) ◦ J J J J J e e J J J J J ◦ Σ n − 1 κ n − 1 o o · · · ◦ 4 4 4 4 Z Z 4 4 4 4 ◦ o o (8) · · · / / C 0 ( Y 2 ) / /           C 0 ( Y 1 ) | | y y y y y y y y y / / Σ n C           · · · C n ⊗ Σ C 0 ( Z n ) ◦ Σ 2 κ 2 o o ◦ E E E E b b E E E E C 0 ( Z n ) ◦ : : : : ] ] : : : : ◦ Σ κ 1 o o A seq uence of distinguished triangles of this type will be ca lled a tower . The tow er (8) in K K Z n is the higher ra nk analogue of the distinguished tria ngle (5). The tower (8) can b e genera lized to contain any co efficient ring. 9 T o find a b etter de s cription of the morphisms κ i let us recall the notio n of a Koszul complex. Let R deno te a commutativ e ring and E an R -mo dule. F or simplicity we will assume that E is free a nd finitely genera ted, let us say of rank N . F or an element v ∈ Hom R ( E , R ) , the Kos z ul complex o f E with resp ect to v is the complex 0 → ∧ N E ∂ 1 − → ∧ N − 1 E ∂ 2 − → . . . ∂ N − 2 − − → ∧ 2 E ∂ N − 1 − − → E v − → R → 0, where each ∂ k is defined as interior m ultiplication by v . Since we ha ve assumed E to b e fr e e, we may write ν = P ν i e ∗ i for so me ν 1 , ν 2 , . . . ν N ∈ R and the dua l basis e ∗ i of a ba sis e i , i = 1, 2, . . . N , of E . If the sequence ν 1 , ν 2 , . . . , ν N is a regular seq ue nc e the Koszul complex is exa ct except at R . The cohomolo gy o f the K oszul complex is in this case R / v ( E ) at R . See mo r e in [5]. The Koszul complex of interest to us is c o nstructed from the mo dule E : = R ( T ) n ov er the representation r ing of the torus T whic h has the following form: R ( T ) ∼ = Z [ t ± 1 1 , . . . , t ± 1 n ] . Observe that Baa j-Sk andalis dua lity and the Universal Co efficient Theorem im- plies that K K Z n ( C k i ⊗ C 0 ( Z n ) , C k i + 1 ⊗ C 0 ( Z n )) ∼ = K K T ( C k i , C k i + 1 ) ∼ = Hom R ( T ) ( R ( T ) k i , R ( T ) k i + 1 ) . W e hav e that R ( T ) k i ∼ = ∧ n − i + 1 E so the low er r ow in (8) hav e the right ranks for coinciding with a Ko szul complex . Let f i ∈ Hom R ( T ) ( ∧ n − i + 1 E , ∧ n − i E ) denote the image o f κ i under the isomor phisms ab ov e. T o simplify notations , we will by ( e i ) n i = 1 denote the R ( T ) - basis of E coming from the isomorphism E ∼ = R ( T ) ⊗ Z Z n and by ( e ∗ i ) n i = 1 denote the dual ba s is. Theorem 2.3. Under the isomorphisms R ( T ) k i ∼ = ∧ n − i + 1 E t he mappings f i c o- incide with interior mult iplic ation by t he element v : = P n i = 1 ( 1 − t i ) e ∗ i . Ther efor e the se quenc e 0 → ∧ n E f 1 − → ∧ n − 1 E f 2 − → . . . f n − 2 − − → ∧ 2 E f n − 1 − − → E f n − → R ( T ) → 0 defines a c omplex isomorphic t o the Koszul c omplex of E whose c ohomolo gy at R ( T ) is Z . Pr o of. While b oth f i and the ma pping defined by in terior multiplication by v are R ( T ) -linea r it is sufficient to prove that f i ( u ) = v ¬ u fo r element s of the form u = e m 1 ∧ · · · ∧ e m n − i + 1 ∈ ∧ n − i + 1 E , wher e m 1 , . . . , m n − i + 1 ∈ { 1, 2, . . . , n } . Let ( m p ) n p = n − i + 1 be a n enumeration of all j = 1, 2, . . . , n such tha t j / ∈ ( m p ) n − i + 1 p = 1 . If we v iew Z n as a subset of R n we can define X u ⊆ ˜ X i as the op en face in R n spanned by the vectors e m n − i + 1 , e m n − i + 2 , . . . e m n . Under the isomorphism ∧ n − i + 1 E ∼ = K i − 1 ( C k i ⊗ Σ i − 1 C 0 ( Z n )) the element u corres p o nds to a K -theory class on ˜ X i which is trivial except o n the face X u . Therefore there exists s equences of num b er s ( a j ) n − i + 1 j = 1 , ( b j ) n − i + 1 j = 1 ⊆ Z such that f i ( u ) = n − i + 1 X j = 1 ( a j + b j t j ) e m j ¬ u . 10 If j = 1, 2 . . . , n − i + 1 , we will let X u , j denote the op en face spanned by the vectors e m j , e m n − i + 1 , e m n − i + 2 , . . . e m n . It follows from re stricting to X u , j that a j = 1 s ince Bott p erio dicity implies that the index mapping K i − 1 ( C 0 ( X u )) → K i ( C 0 ( X u , j )) is an isomo rphism. In a similar fashion it follows that b i = − 1 . While v ( E ) is the ideal g enerated by the regular sequence 1 − t 1 , 1 − t 2 , . . . , 1 − t n , the coho mology of the Kos z ul complex is R ( T ) / v ( E ) = Z and the quotient mapping R ( T ) → Z coincides with the a ugmentation mapping. Consider the tow er Baa j-Sk anda lis dual to (8). Given A , B ∈ K K T we c an apply the homolo gical functor K K T ( A , − ⊗ mi n B ) to this tow er. This functor is only homo logical on the b o otstrap categor y if B is not exact, but all o b jects in the tower Baa j-Sk andalis dual to (8) are in the bo o tstrap c a tegory . The lowest row o f the corres p o nding tower in the categ ory of R ( T ) - mo dules is a Ko szul complex: 0 → ∧ n Z n ⊗ K K T ∗ ( A , B ) v A ¬ − → ∧ n − 1 Z n ⊗ K K T ∗ ( A , B ) v A ¬ − → . . . (9) . . . v A ¬ − → Z n ⊗ K K T ∗ ( A , B ) v A ¬ − → K K T ∗ ( A , B ) → 0 where v A : = n X i = 1 ( 1 − β i ∗ ) e ∗ i ∈ Hom R ( T ) ( K K T ∗ ( A , B ) n , K K T ∗ ( A , B )) and ( β i ) n i = 1 are the commuting equiv ariant auto morphisms of A that are Ba a j- Sk andalis to the Z n -action on B ⋊ r T . The cohomo logy of this Kos zul complex can be calcula ted from K K T ∗ ( A , B ) . W e will return to this sub ject in the next section in t he more g eneral case of Ho dgkin-Lie groups and explain this pro cedur e further. 3 The generalized Pimsner-V oiculescu-to w ers As mentioned in the intro duction, the representation r ing R ( T ) is free ov er R ( G ) when G is a Ho dg kin-Lie group, so the step to coactions of a compact Hodgkin- Lie gr o up will not be to o large. W e w ill thro ug hout this section assume that G is a c o mpact Ho dg kin-Lie group of rank n with maximal torus T . Recall tha t a g roup satisfies the Ho dgkin condition if it is connected a nd the fundamental group is torsio n-free. The embedding T ⊆ G induces a restrictio n functor K K ˆ T → K K ˆ G . Using the isomorphism ˆ T ∼ = Z n , the tow er (8) ca n b e restricted to a K K ˆ G -tow er: Σ n C ∗ ( T ) / / C 0 ( Y n − 1 ) / / z z v v v v v v v v v v · · · | | y y y y y y y y y y C n ⊗ Σ n − 1 C ∗ ( T ) ◦ H H H H H Σ n κ n d d H H H H H C k n − 1 ⊗ Σ n − 2 C ∗ ( T ) ◦ J J J J J d d J J J J J ◦ Σ n − 1 κ n − 1 o o · · · ◦ 4 4 4 4 Z Z 4 4 4 4 ◦ o o · · · / / C 0 ( Y 2 ) / /           C 0 ( Y 1 ) } } z z z z z z z z z / / C 0 ( R n )           · · · C n ⊗ Σ C ∗ ( T ) ◦ Σ 2 κ 2 o o ◦ D D D D a a D D D D C ∗ ( T ) ◦ 9 9 9 9 \ \ 9 9 9 9 ◦ Σ κ 1 o o 11 In order to work with this K K ˆ G -tow er we need to describ e the terms C ∗ ( T ) in the se c ond row. Lemma 3. 1. If G is a c omp act Ho dgkin-Lie gr oup with Weyl gr oup of or der w ther e is an isomorphism C ∗ ( T ) ∼ = C w ⊗ C ∗ ( G ) in K K ˆ G . Observe tha t the co ndition on G to b e a Ho dg kin gr oup is equiv a lent to ˆ G be- ing a torsion-free quantum gro up in the sens e of Mey e r, see [9]. The torsion-free quantum gr oups are the only no n-classica l discrete quantum groups for which there is a g eneral for mu lation o f the Baum-Connes pr op erty in terms of trian- gulated categor ie s. In [11], coactions of compact non-Ho dgkin Lie gr oups were considered and the ” torsion” turned out to be the torsion elemen ts o f H 2 ( G , S 1 ) . The less precise sta tement C ( G / T ) ∼ = C k in K K G for so me k is stated and pro ved in [11]. An explicit calcula tion that k = | W | ca n be fo und in [15]. W e will re- view the conceptually imp ortant part of the pr o of of a Prop ositio n in [11] which prov es Lemma 3 .1 aside from the calculation of k . Pr o of. By [1 5], the representation r ing R ( T ) is free of r ank w over the re pre- sentation ring R ( G ) if π 1 ( G ) is torsio n-free. If we let S denote the lo ca lizing sub c ategory of K K G generated by C and C ( G / T ) , L e mma 11 of [10] states tha t for A ∈ S the natural homomo r phism R ( T ) ⊗ R ( G ) K K G ( A , C ) → K K T ( A , C ) is an isomor phism. Thus the representable functor on S A → K K G ( A , C w ) ∼ = R ( T ) ⊗ R ( G ) K K G ( A , C ) coincides with the r epresentable functor A → K K G ( A , C ( G / T )) ∼ = K K T ( A , C ) . The last isomor phism ex ists as a co nsequence of the fa ct that the induction functor I n d G T is the rig ht adjoin t of the restriction functor from G to T . So the Y oneda lemma implies that C ( G / T ) ∼ = C w in S a nd therefore in K K G . Applying Baa j-Sk anda lis duality it follows that there is an equiv aria nt K K - is omorphism C ∗ ( T ) ∼ = C w ⊗ C ∗ ( G ) . Using Lemma 3.1 the tow e r (8) takes the form: Σ n C w ⊗ C ∗ ( G ) / / C 0 ( Y n − 1 ) / / z z u u u u u u u u u u · · · Σ n − 1 C w n ⊗ C ∗ ( G ) ◦ M M M M M M f f M M M M M M · · · ◦ 7 7 7 7 [ [ 7 7 7 7 ◦ o o (10) · · · / / C 0 ( Y 2 ) / /           C 0 ( Y 1 ) | | y y y y y y y y y / / Σ n C   ~ ~ ~ ~ ~ ~ ~ ~ ~ · · · Σ C w n ⊗ C ∗ ( G ) ◦ o o ◦ E E E E b b E E E E C w ⊗ C ∗ ( G ) ◦ B B B B a a B B B B ◦ o o 12 W e will call this K K ˆ G -tow er the fundamen tal G − PV-tower. The dua l fundamen- tal G − PV-tow er is defined to be the K K G -tow er which is Baa j-Sk a ndalis dual to the fundamental G − PV-tow er: Σ n C w / / D n − 1 / /           · · ·           Σ n − 1 C w n ◦ = = = = ^ ^ = = = = Σ n − 2 C w k n − 1 ◦ ? ? ? ? _ _ ? ? ? ? ◦ o o · · · ◦ - - - - V V - - - - ◦ o o (11) · · · / / D 2 / /          D 1          / / Σ n C ( G )          · · · Σ C w n ◦ o o ◦ 3 3 3 Y Y 3 3 3 C w ◦ / / / W W / / / ◦ o o where D i : = C 0 ( Y i ) ⋊ r ˆ G . As mentioned a b ov e, if A is a G − C ∗ -algebra , the trivial coaction of G o n A makes A in to a Y etter- Drinfeld algebra. This follows from that C ( G ) is commu- tative so we can extend a G -action via the D ( G ) -equiv ar ia nt ∗ -monomo rphism C ( G ) → M ( C 0 ( D ( G ))) . Clearly , a G -equiv ariant mapping is equiv ariant in this new D ( G ) -action. F ur thermore, since mapping co nes do es not dep end on the action, the tr iv ial extension of a G -action to a D ( G ) -action is functorial and resp ects ma pping cones . The following prop osition follows from univ ersa lit y . Prop ositi on 3.2 . If G is a lo c al ly c omp act gr oup, the fun ctor m apping a G − C ∗ - algebr a to a G -Y ett er- Drinfeld algebr a with trivial ˆ G - action defines a t r iangulate d functor K K G → K K D ( G ) . Using the triangulated functor of Prop os itio n 3.2, we may co nsider the tower (11) as a tow er in K K D ( G ) . Applying a cross e d pr o duct by G we obtain that also the to wer (10) is a tow er in K K D ( G ) . F or a C ∗ -algebra B we will use the notation t ( B ) for the ˆ G − C ∗ -algebra with trivial co a ction, or in the context of G − C ∗ - algebras t ( B ) will deno te the G − C ∗ -algebra with trivia l a ction. Let us state and prove the corres po nding version of (3) in a simple ca se of a braided tensor pro duct over G with C ( G ) , a more g eneral pro of can b e found in [12]. Lemma 3.3. Whe n B has a c ontinuous G -action, ther e is a ˆ G - e quivariant Morita e quivalenc e ( C ( G ) ⊗ B ) ⋊ r G ∼ M t ( B ) . Pr o of. By Baa j-Sk a ndalis dua lity , it suffices to prov e that there is a ˆ G - equiv ar iant isomorphism ( C ( G ) ⊗ B ) ⋊ r G ∼ = ( C ( G ) ⋊ r G ) ⊗ t ( B ) . Denote the G -action on B by β and define the equiv ar iant mapping ϕ 0 : L 1 ( G , C ( G , B )) → ( C ( G ) ⋊ r G ) ⊗ t ( B ) by setting ϕ 0 ( f ) ( g 1 , g 2 ) : = β g − 1 1 f ( g 1 , g 2 ) . The linear mapping ϕ 0 is a ∗ -homomor phism when L 1 ( G , C ( G , B )) is equippe d with the conv o lution twisted b y the G - action on C ( G ) ⊗ B . It is str aightforw ard to verify that ϕ 0 is b o unded in C ∗ -norm so we ca n define ϕ : ( C ( G ) ⊗ B ) ⋊ r G → ( C ( G ) ⋊ r G ) ⊗ B by contin uity . The ∗ -homomo r phism ϕ is an eq uiv aria nt isomorphism since an inv erse ca n b e constructed by extending ϕ − 1 ( f ⊗ b )( g 1 , g 2 ) : = f ( g 1 , g 2 ) β g 1 ( b ) to a ∗ -homomo rphism ϕ − 1 : ( C ( G ) ⋊ r G ) ⊗ t ( B ) → ( C ( G ) ⊗ B ) ⋊ r G . 13 Theorem 3.4 (The Pimsner-V o iculescu tow er) . L et G b e a c omp act Ho dgkin- Lie gr oup of r ank n and Weyl gr oup of or der w . F or any sep ar able ˆ G − C ∗ -algebr a A ther e is a K K ˆ G -tower C w ⊗ A / / Σ n D n − 1 ( A ) / / } } z z z z z z z z z Σ n D n − 2 ( A ) / / { { w w w w w w w w w w · · · Σ C w n ⊗ A ◦ @ @ @ @ _ _ @ @ @ @ Σ 2 C w k n − 1 ⊗ A ◦ G G G G G c c G G G G G ◦ o o · · · ◦ : : : : \ \ : : : : ◦ o o (12) · · · / / Σ n D 2 ( A )           / / Σ n D 1 ( A ) | | x x x x x x x x x / / t ( A ⋊ r ˆ G ) } } | | | | | | | | | · · · Σ n − 1 C w k 2 ⊗ A ◦ F F F F F b b F F F F ◦ o o Σ n C w ⊗ A ◦ A A A A ` ` A A A A ◦ o o wher e D i ( A ) : = ( C 0 ( Y i ) ⊗ K ( L 2 ( G ) )) ⊠ G ( A ⋊ r ˆ G ) and is e quipp e d with the ˆ G - action induc e d fr om the diagonal ˆ G - action on C 0 ( Y i ) ⊗ K ( L 2 ( G ) ) . Observe that the D ( G ) -actions on the C ∗ -algebra s C 0 ( Y i ) ⊗ K ( L 2 ( G ) ) is de- fined to come fro m those on their Baa j-Sk a ndalis duals C 0 ( Y i ) ⋊ r ˆ G , which are D ( G ) − C ∗ -algebra s in the dual G -actions on the crossed pr o ducts and the trivial ˆ G - actions. So in gener al, D i ( A ) is not the tensor pro duct of C 0 ( Y i ) ⊗ K ( L 2 ( G ) ) and A ⋊ r ˆ G . Pr o of. By Lemma 3.3 the ˆ G − C ∗ -algebra A admits the equiv ar iant Mo r ita equiv - alence: ( C ( G ) ⊗ ( A ⋊ r ˆ G ) ) ⋊ r G ∼ M t ( A ⋊ r ˆ G ) . (13) F urthermor e, the is omorphism of equatio n (3) ho lds for br aided tensor pro ducts ov er G so while the ˆ G - actions on D i = C 0 ( Y i ) ⋊ r ˆ G are trivia l there are equiv ariant isomorphisms ( D i ⊗ ( A ⋊ r ˆ G ) ) ⋊ r G ∼ = (( C 0 ( Y i ) ⋊ r ˆ G ) ⊠ G ( A ⋊ r ˆ G ) ) ⋊ r G ∼ = (14) ∼ = ( C 0 ( Y i ) ⊗ K ( L 2 ( G ) )) ⊠ G ( A ⋊ r ˆ G ) . Thu s if w e tenso r the dual fundamental G − PV-tow er (11) by the G − C ∗ -algebra A ⋊ r ˆ G we obtain a new K K G -tow er which b ecomes the Pimsner-V oiculescu tower of A after applying Baa j-Sk andalis duality , using the Morita equiv alence (13) and the iso morphisms (14). The Pimsner -V oiculesc u tow er (12) is the generaliza tio n of the r esolution in (9) to co mpact Ho dgkin-Lie gro ups. Applying the cohomolo gical functor K K ( − , B ) to the Pimsner- V oiculescu tower we obtain a similar resolution of K K ∗ ( A ⋊ r ˆ G , B ) in terms o f K K ∗ ( A , B ) a s in (9). Similarly , the homolo gical functor K K ( B , − ) applied to the Pimsner-V oiculescu tower gives a reso lution of K K ( B , A ⋊ ˆ G ) in terms of K K ( B , A ) . Observe that since A has a ˆ G -actio n, the g roups K K ( C w ⊗ A , B ) a nd K K ( B , C w ⊗ A ) will always hav e an R ( G ) -mo dule structure a nd since R ( T ) is free ov er R ( G ) also an R ( T ) -mo dule structure. As an example o f this, we will use the Pimsner-V oiculescu tow er to cal- culate the K -theory of the homogeneous space G / H when H ⊆ G is a Lie sub- group. Mo re genera lly , this technique can b e used to calcula te K ∗ ( A ⋊ r ˆ G ) for any ˆ G − C ∗ -algebra A when one knows K ∗ ( A ) and its R ( G ) -mo dule structur e c oming 14 from the Julg isomorphism K ∗ ( A ) ∼ = K G ∗ ( A ⋊ r ˆ G ) . T o calculate K ∗ ( G / H ) , consider the C ∗ -algebra A : = C ∗ ( H ) equipp e d with the ˆ G - action induced from the natu- ral ma pping C ∗ ( H ) → M ( C ∗ ( G ) ) . Green’s imprimitivity theo rem implies that C ∗ ( H ) ⋊ ˆ G is K K - equiv alent with C ( G / H ) . T hus, if we tak e the K - theory of the Pimsner-V oiculescu tow er of C ∗ ( H ) we obtain a tow er of a be lia n gro ups of the form R ( T ) ⊗ R ( G ) R ( H ) / / K ∗− n ( D n − 1 ( C ∗ ( H ))) / / w w o o o o o o o o o o o o o · · · Σ R ( T ) n ⊗ R ( G ) R ( H ) ◦ N N N N N N Σ v ⊗ 1 f f N N N N N N · · · ◦ E E E E E b b E E E E E ◦ Σ v ⊗ 1 o o (15) · · · / / K ∗− n ( D 1 ( C ∗ ( H ))) } } { { { { { { { { { { / / K ∗ ( G / H ) y y t t t t t t t t t t t · · · Σ n R ( T ) ⊗ R ( G ) R ( H ) ◦ N N N N N N g g N N N N N N ◦ Σ v ⊗ 1 o o W e use Σ to deno te deg ree s hift in the ca tegory o f Z / 2 Z -graded abelian groups. Here w e hav e use d tha t R ( T ) is a free R ( G ) -mo dule o f rank w so K ∗ ( C w ⊗ C ∗ ( H )) ∼ = R ( T ) ⊗ R ( G ) R ( H ) . Th us the low est r ow is the tensor pro duct o f R ( H ) with the Koszul complex of R ( T ) that is a sso ciated with the re gular sequence 1 − t 1 , 1 − t 2 , . . . , 1 − t n under the isomor phism R ( T ) ∼ = Z [ t ± 1 1 , t ± 1 2 , . . . , t ± 1 n ] . If w e restrict o ur attention to simple compact Lie g roups w e can p erfor m an explicit calc ulation of all the groups in (15). Assume that G = G n is a simple com- pact Ho dgkin- L ie group in the classical A , B , C - or D -series of rank n and assume that H = G k ⊆ G n is a simple simply connected compact Lie gr oup in the sa me classical se r ie b eing of rank k < n . W e may take a maximal to rus T n ⊆ G n such that T k : = T n ∩ G k is a maximal torus in G k . In this case we may consider R ( T k ) as an ideal in R ( T n ) and R ( T n ) ⊗ R ( G n ) R ( G k ) ∼ = R ( T k ) as R ( T n ) -mo dules. Under th e isomorphisms R ( T k ) ∼ = Z [ t ± 1 1 , t ± 1 2 , . . . , t ± 1 k ] and R ( T n ) ∼ = Z [ t ± 1 1 , t ± 1 2 , . . . , t ± 1 n ] , the Koszul v ector v is iden tified with P k i = 1 ( 1 − t i ) e ∗ i ∈ Hom ( R ( T k ) n , R ( T k )) . Th us we arrive at the tow er R ( T k ) / / K ∗− n ( D n − 1 ( C ∗ ( G k )))) / / x x p p p p p p p p p p p p · · · Σ Z n ⊗ Z R ( T k ) ◦ B B B B Σ ∂ n ` ` B B B B ∧ 2 Z n ⊗ Z R ( T k ) ◦ N N N N N N g g N N N N N N ◦ Σ ∂ n − 1 o o · · · ◦ o o · · · / / K ∗− n ( D 1 ( C ∗ ( G k ))) | | z z z z z z z z z z / / K ∗ ( G n / G k ) y y s s s s s s s s s s s · · · Σ n ∧ n Z n ⊗ Z R ( T k ) ◦ O O O O O O g g O O O O O O ◦ Σ ∂ 1 o o Let us use the notation E ∗ for the complex ∧ n −∗ Z n ⊗ R ( T k ) equipp ed with the Koszul differential from the vector P k i = 1 ( 1 − t i ) e ∗ i which we as ab ove denote by ∂ l : E l − 1 → E l . After some simpler calcula tio ns in this Koszul complex we a rrive 15 at the conclusion that K ∗− n ( D l ( C ∗ ( G k ))) ∼ = ker ( ∂ l + 1 ) ⊕ n + 1 M j = l + 2 Σ n − j H j ( E ∗ ) . Hence we obtain the isomo rphism K ∗ ( G n / G k ) ∼ = L n + 1 j = 1 Σ n − j H j ( E ∗ ) . These coho - mology groups are calcula ted in Co rollary 17.10 of [5] and H j ( E ∗ ) is a fre e g roup of rank k ( j ) : = ( n − k ) ! / ( n − j ) ! ( n − j − k ) ! if 0 ≤ j ≤ n − k and 0 o ther wise. Therefore K ∗ ( G n / G k ) ∼ = n − k M j = 0 Σ n − j Z k ( j ) = Z 2 n − k − 1 ⊕ Σ Z 2 n − k − 1 . Theorem 3.5 (The dual Pims ner-V oic ule s cu tow er) . Under the assumptions of The or em 3.4 t her e is a K K G -tower C w ⊗ t ( A ) / / Σ n ˜ D n − 1 ( A ) / / { { x x x x x x x x x x · · · Σ C w n ⊗ t ( A ) ◦ F F F F F b b F F F F F Σ 2 C w k n − 1 ⊗ t ( A ) ◦ I I I I I d d I I I I I ◦ o o · · · ◦ o o (16) · · · / / Σ n ˜ D 2 ( A )           / / Σ n ˜ D 1 ( A ) { { v v v v v v v v v v / / A ⋊ r ˆ G ~ ~ | | | | | | | | | · · · Σ n − 1 C w k 2 ⊗ t ( A ) ◦ H H H H H c c H H H H H ◦ o o Σ n C w ⊗ t ( A ) ◦ D D D D b b D D D D ◦ o o wher e ˜ D i ( A ) : = D i ⊠ ˆ G A . F or a homological f unctor F : K K ˆ G → Ab , the dual Pimsner-V o iculescu tow er of A allows us to calcula te F ( A ) in terms of the ob jects F ( C ∗ r ( G ) ⊗ t ( A )) . As we shall see b elow, ˆ G − C ∗ -algebra s of the form C ∗ r ( G ) ⊗ t ( A ) b ehav es s imilarly to pr op er actions. Compar e this result to Theorem 4.4 of [8]. Pr o of. Consider the br a ided tensor pro duct by Σ n A and the tow er (10): C w ⊗ C ∗ ( G ) ⊠ ˆ G A / / Σ n C 0 ( Y n − 1 ) ⊠ ˆ G A / / w w o o o o o o o o o o o o · · · Σ C w n ⊗ C ∗ ( G ) ⊠ ˆ G A ◦ O O O O O O g g O O O O O O · · · ◦ C C C C C a a C C C C C ◦ o o · · · / / Σ n C 0 ( Y 1 ) ⊠ ˆ G A } } { { { { { { { { { / / A | | y y y y y y y y y y · · · Σ n C w ⊗ C ∗ ( G ) ⊠ ˆ G A ◦ N N N N N N f f N N N N N N ◦ o o T aking cr ossed pro duct b etw e en this to wer and ˆ G implies the Theo rem since the following equiv ariant Morita equiv alences follows from (3) ( C ∗ ( G ) ⊠ ˆ G A ) ⋊ r ˆ G ∼ M t ( A ) and ( C 0 ( Y i ) ⊠ ˆ G A ) ⋊ r ˆ G ∼ M ( C 0 ( Y i ) ⋊ r ˆ G ) ⊠ ˆ G A = D i ⊠ ˆ G A . 16 One of the main motiv ations b ehind this pa p er was to give a precis e de- scription o f the B aum-Connes pr op erty of duals of Ho dgk in- Lie gro ups. The Baum-Connes prop erty for coactions of compac t Lie g roups was g iven meaning to and was proved to hold in [11]. Mor e g enerally , this fits into the progr am of gener alizing the Baum-Co nnes prop erty to quantum groups. So far , it is not known what a suitable prop erty the Ba um-Connes pro p er ty should b e for a gen- eral lo cally co mpact q uantum group. F o r discrete q uantum groups which are torsion-fr e e , in the sense of [9], there is a formulation and as men tioned ab ov e duals of compact Ho dg k in-Lie gr oups are to rsion-free. The problem that ar ises when o ne tries to define the Baum-Connes as s embly mapping for a quantu m group is that there is no natura l notion of a prop er action and ther e are in g eneral to o many quantum homogeneo us spa ces. It is m uch ea sier to generalize certain no tio ns of free actions than pr op er a ctions of a quantum group by just saying tha t an actio n of a discrete quantum group Γ on a C ∗ -algebra A is truly free if there is a C ∗ -algebra A 0 and a n equiv ariant ∗ -isomorphism A ∼ = A 0 ⊗ mi n C 0 (Γ) with Γ only acting on the second leg. In the case of a gr oup, there a re many free actio ns tha t a re not truly free but this stronger notion of a free action will suffice fo r our purp oses. Restricting one’s atten tion to generalizing the Baum-Connes prop erty o f the simpler cla ss of to rsion-free discre te g roups to the qua ntum setting, when prop er actions are free, Mey er in tro duced a class of quan tum groups kno wn as torsion- free in [9]. F ollowing [9 ], we sa y that a discrete quan tum group Γ is torsion-free if e very coaction o f the co mpact qua ntu m g roup ˆ Γ on a finite-dimensio nal C ∗ - algebra is Morita equiv alent to a trivia l coactio n on a direct sum of C :s. This fact implies that any finite-dimensional pro jectiv e repre s entation of the dual compact quantum group is equiv alent to a r epresentation. If Γ is a discrete group, coactions of the dual compact quantum group on finite-dimens io nal C ∗ - algebras that are not Mor ita equiv a lent to a trivia l coa ction on a direc t sum of C :s corresp ond to finite subgroups so a discrete gr oup is torsion-free if a nd only if it is to r sion-free in the s e nse of [9]. F or a t ors io n-free quantum group a prop er action s hould corre s po nd to a free action. Under Baa j-Sk andalis duality , a truly free Γ − C ∗ -algebra cor resp onds to a trivial ˆ Γ -action. Let C I ˆ Γ denote the imag e of t : K K → K K ˆ Γ . The triangula ted category 〈C I ˆ Γ 〉 is defined as the lo calizing sub categor y gener ated by C I ˆ Γ . F ollowing the for mulation of [9], Γ is sa id to s a tisfy the strong Baum-Connes prop erty if the embedding of triangulated catego ries 〈C I ˆ Γ 〉 → K K ˆ Γ is esse ntially surjective. The stro ng Baum-C o nnes pro p er ty of Γ is eq uiv alent to that a ny Γ − C ∗ -algebra is in the lo calizing ca tegory g enerated by all truly fr e e actio ns. So regar dle s s of what notion of a prop er a c tio n we cho ose, the stro ng Baum- Connes conjecture will imply that the lo calizing category genera ted b y all such prop er actions will b e K K Γ . The quantum group is said to satisfy the Baum- Connes pr o p erty if the same sta temen t holds after lo ca lizing with resp e ct to the kernel of equiv ariant K - theory . In [11] the Baum-Connes pro p e r ty was formulated in the slightly more gen- eral setting of duals of compac t Lie g r oups. The finite-dimensio nal pro jective representations o f a co mpact Lie group G c o rresp ond to the tors ion classes of H 2 ( G , S 1 ) , which can b e thought of as the torsio n of ˆ G . When G is Ho dgkin, H 2 ( G , S 1 ) is torsion- free so ˆ G is torsion-fr ee. In this case a ” prope r” ac tion is an ob ject of the additive catego r y genera ted by ˆ G - algebras that are Baa j-Sk andalis 17 dual to A 0 ⊗ C ω , with C ω denoting the endomorphisms of a pro jective repres en- tation ω a nd A 0 having trivial G -action. So the substitute in the setting of [11] for prop er actions is the category o f tensor pro ducts betw een Baa j-Sk a ndalis duals of co a ctions on finite-dimensiona l C ∗ -algebra s and trivia l actions, just as the truly free actions form a substitute for prop er actions of torsio n-free quan- tum groups. The Ba um-Connes pr op erty of coactions o f a compact Ho dgkin-Lie group is a direct consequence of Theorem 3.5. The method of pro of of Prop osi- tion 2.1 of [11] can b e used to generalize b oth Theor em 3 .4 and Theorem 3 .5 to arbitrar y compact Lie g roup. Finally , let us mention a pr o mising gener alization of Theo rem 3.5 to W o ronow- icz deforma tions. It w as proved in [12] that the compac t quan tum gro up S U q ( 2 ) satisfies that C ( S U q ( 2 ) / T ) is K K D ( S U q ( 2 )) -isomorphic to C 2 for q ∈ ] 0, 1 [ . So if we apply the induction functor I nd S U q ( 2 ) T : K K T → K K S U q ( 2 ) to the disting uished triangle Baa j-Sk a nda lis dual to (5) and use the isomo r phism o f Nest-V oigt we arrive at the distinguished tria ngle in K K D ( S U q ( 2 )) : C 2 / / C 2           C ( S U q ( 2 )) . ◦ ; ; ; ; ] ] ; ; ; ; Using the techn ique fro m the pro of of Theor em 3.5 any A ∈ K K Ú S U q ( 2 ) fits into a distinguished tria ngle C 2 ⊗ t ( A ) / / C 2 ⊗ t ( A ) } } { { { { { { { { { A ⋊ Ú S U q ( 2 ) . ◦ C C C C a a C C C C This distinguished tr iangle gives an alternative pro of of the strong Baum-Connes prop erty for Ú S U q ( 2 ) , a r esult first proved in [1 7]. The in ter esting part ab out this pro of is that it only r elies on the iso morphism C ( G q / T ) ∼ = C w in K K D ( G q ) . 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