THE PIMSNER-VOICULESCU SEQUENCE FOR COACTIONS OF COMPACT

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THE PIMSNER-VOICULESCU SEQUENCE FOR COACTIONS OF COMPACT

LIE GROUPS

MAGNUS GOFFENG

Abstract

The Pimsner-Voiculescu sequence is generalized to a Pimsner-Voiculescu tower describing the KK-category equivariant with respect to coactions of a compact Lie group satisfying the Hodgkin condition. A dual Pimsner-Voiculescu tower is used to show that coactions of a compact Hodgkin- Lie group satisfy the Baum-Connes property.

Introduction

WhenGis a second countable, locally compact group andAis a separableC^∗- algebra with a continuousG-action, the Baum-Connes conjecture states that theK-theory of the reduced crossed productArGcan be calculated by means of geometric and representation theoretical properties ofGandA, see more in [4]. To be more precise, the Baum-Connes conjecture states that the assembly mappingμ_A : K_∗^G(EG;A) → K_∗(Ar G)is an isomorphism. The space EGis the universal properG-space andK_∗^G(EG;A)is the proper equivariant K-homology with coefficients inA. There are known counterexamples when μ_Ais not an isomorphism, so it is more natural to speak of groups having the Baum-Connes property. In [10], the equivariantK-homology with coefficients inAwas proved to be the left derived functor ofF (A) = K_∗(Ar G)and the assembly mapping being the natural transformation fromLF to F. The approach to the Baum-Connes property using triangulated categories can be generalized to discrete quantum groups, see [9], which indicates that geometric techniques such as universal properG-spaces can be generalized to discrete quantum groups.

The generalization of the Baum-Connes property to quantum groups has been studied in for instance [11] and [17]. The case studied in [11] is that of quantum group actions of the dual of a compact Lie group which corres- pond to coactions of the Lie group. In [11] duals of compact Lie groups were shown to satisfy the strong Baum-Connes property, i.e., the embedding of

Received 29 October 2010, in final form 4 January 2011.

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the triangulated category generated by proper coactions, theC^∗-algebras that are Baaj-Skandalis dual to trivialG-actions, into theKK-category equivariant with respect to coactions is essentially surjective. In this paper we construct an analogue of the Pimsner-Voiculescu sequence for coactions of a compact Hodgkin-Lie groupG that describes how theKK-category equivariant with respect to coactions ofGis built up from theC^∗-algebras with coactions ofG which are proper in the sense of [11].

The starting point is to express the Pimsner-Voiculescu sequence for Z- actions in terms of a property of the representation ring of a rank one torus.

Using the Universal Coefficient Theorem, the Pimsner-Voiculescu sequence can be constructed from a Koszul complex

0−→R(T )−→^α R(T )−→0,

whereαis defined as multiplication by 1−t under the isomorphismR(T )∼= Z[t, t⁻¹]. When A has a coaction of T, i.e., aZ-action, the tensor product overR(T )between this Koszul complex andK_∗^T(Ar Z)gives the Pimsner- Voiculescu sequence. In the generalization to higher rank, whenT is a torus of ranknwe consider the Koszul complex

0−→ⁿR(T )ⁿ−→ⁿ⁻¹R(T )ⁿ−→ · · ·

−→²R(T )ⁿ−→R(T )ⁿ−→R(T )−→0.

The boundary mappings in this complex are defined from interior multiplication with the element(1−t_i)e_i^∗∈HomR(T )(R(T )ⁿ, R(T )). IfGis a compact Hodgkin-Lie group with maximal torusT, the representation ringR(T ) is a freeR(G)-module by [15], so the generalization from a torus to compact Hodgkin-Lie groups goes in a straightforward fashion. Just as when the rank is 1, the Koszul complex above can be used to produce sequence of distinguished triangles which is the analogue of a Pimsner-Voiculescu sequence for theK-theory of crossed products by coactions ofG.

We will give a geometric description of a sequence of distinguished triangles in theKK-category equivariant with respect to coactions ofGthat corresponds to the above Koszul complex under the Universal Coefficient Theorem. As for the Pimsner-Voiculescu sequence for Z we will obtain a projective resolution of the crossed product by a coaction in the sense of triangulated categories rather than exact sequences. Using suitable tensor products we produce in The- orem 3.4 a sequence of distinguished triangles in theKK-category equivariant with respect to coactions ofGthat we call the generalized Pimsner-Voiculescu

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tower forA:

C^w A ⁿD_n₋1(A) ⁿD_n₋2(A)

C^wn A ²C^wkⁿ⁻¹ A

ⁿD₂(A) ⁿD₁(A) t (Ar Gˆ)

ⁿ⁻¹C^wk² A ⁿC^w A

Here t (Ar G)ˆ denotes the C^∗-algebra Ar Gˆ equipped with the trivial ˆ

G-action and the termsD_i(A)can be explicitly described as braided tensor products. TakingK-theory of the lower row will give a complex similar to the Koszul complex that in a sense forms a projective resolution of theK-theory ofAG. The dual Pimsner-Voiculescu gives a more precise description of theˆ results of [11] by a sequence of distinguished triangles inKK^Gthat describes the crossed productArGˆ in terms ofG–C^∗-algebras with trivialG-action, thus giving a direct route to the strong Baum-Connes property ofG.ˆ

The paper is organized as follows; the first section consists of a review of KK-theory of actions and coactions. In particular we gather some known results about the braided tensor product and the Drinfeld double which plays a mayor role in constructing the dual Pimsner-Voiculescu tower. The main references of this section are [1], [2], [3], [7], [10], [12] and [16]. In the second section a geometric construction of the Pimsner-Voiculescu sequence forZ-actions is presented and generalized to higher rank via a Koszul complex. In the third section the restriction functor for coactions is used to generalize the Pimsner- Voiculescu sequence to coactions of compact Hodgkin-Lie groupsG. As an example of this we calculate the K-theory of some compact homogeneous spaces. By similar methods, a dual Pimsner-Voivulescu tower is constructed inKK^G, following the ideas of [10]. At the end of the paper we discuss some possible generalizations to duals of Woronowicz deformations.

Acknowledgments: The author would like to thank Ryszard Nest for posing the question on how to explicitly construct the crossed product of aG–ˆ C^∗-algebra from trivial actions and for much inspiration in the writing process.

The author also wishes to thank the referee for several valuable suggestions.

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1. Actions and coactions of compact groups

The standard approach to equivariantK-theory is to introduce equivariantKK- theory. IfAandBare two separableC^∗-algebras with a continuous action of a locally compact groupG, the equivariantKK-groupKK^G(A, B)is defined as the set of homotopy classes ofG-equivariantA−B-Kasparov modules which forms an abelian group under direct sum. The KK-groups can be equipped with a product such that ifCis a third separableC^∗-algebra with a continuous G-action there is an additive pairing called the Kasparov product

KK^G(A, B)×KK^G(B, C)−→KK^G(A, C).

Following the standard construction, we letKK^Gdenote the additive category of all separableC^∗-algebras with a continuousG-action and a morphism in KK^GfromAtoBis an element ofKK^G(A, B). The composition of twoKK^G- morphisms is defined to be their Kasparov product. The group KK^G(C, A) coincides with the equivariantK-theory ofA. In particular, ifGis compact KK^G(C,C) = R(G), the representation ring of G. The action of R(G)on equivariantK-theory generalizes to anR(G)-module structure on the bivariant groupsKK^G(A, B).

The categoryKK^Gcan be equipped with a triangulated structure with a mapping cone coming from the mapping cone construction of a∗-homomorphism.

The triangulated structure onKK^Gis universal in the sense that any homotopy invariant, stable, split-exact functor on the category ofC^∗-algebras with a con- tinuousG-action defines a homological functor onKK^G. The construction of the triangulated structure and its universality are thoroughly explained in [10].

Let us just recall the basics of the construction of the triangulated structure on KK^G. The suspensionAof aG–C^∗-algebra is defined byC₀(R)⊗A. By Bott periodicity²∼=id. A distinguished triangle inKK^Gis a triangle isomorphic to one of the form

C(f ) A

B,

f

where C(f ) is the mapping cone of the equivariant ∗-homomorphismf : A→B. In particular, iff :A→Bis a surjection and admits an equivariant completely positive splitting the natural mapping ker(f )→C(f )defines an equivariantKK-isomorphism, so under suitable assumptions a distinguished triangle is isomorphic to a short exact sequence.

How to constructKK-theory of coactions of groups is easiest seen in the simpler case whenGis an abelian group. IfAis aC^∗-algebra equipped with an

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actionαof the abelian groupG, the crossed productArGcarries a natural action of the Pontryagin dualG. This action is called the dual action ofˆ G. Sinceˆ abelian groups are exact, the crossed product by an abelian group defines a triangulated functorKK^G →KK^G^ˆ. The crossed product by the dual action is described by Takesaki-Takai duality which states that there is an equivariant isomorphism

ArGr Gˆ ∼=A⊗K(L²(G)),

whereAr Gr Gˆ is equipped with the dual action ofGand theG-action onA⊗K(L²(G))is defined asα⊗Ad. Takesaki-Takai duality implies that the crossed product defines a triangulated equivalenceKK^G→KK^G^ˆ.

An action α of a groupGon Adefines a∗-homomorphism_α : A → M(A⊗C₀(G))by letting_α(a)be the functiong →α_g(a). WhenGis abelian there is a natural isomorphismC₀(G)ˆ ∼=C_r^∗(G)and aG-action corresponds toˆ a non-degenerate∗-homomorphism_A:A→M(A⊗^minC^∗_r(G))satisfying certain conditions. The first instance of a coaction of a groupGis onC_r^∗(G).

Using the universal property ofC^∗_r(G), one can construct a non-degenerate mapping : C_r^∗(G)→ M(C_r^∗(G)⊗min C_r^∗(G))called the comultiplication and is induced from the diagonal homomorphismG→G×G. Clearly, the mappingsatisfies:

(⊗id)=(id⊗),

so we say thatis coassociative. Since₂₁ = the comultiplicationis cocommutative, so if we interpretC_r^∗(G)as the functions on a reduced locally compact quantum groupGˆ thenGˆ can be thought of as abelian, see more in [7].

With the abelian setting as motivation, we say that a separableC^∗-algebraA has a coaction of the locally compact second countable groupGif there is non- degenerate∗-homomorphism_A :A→M(A⊗minC_r^∗(G))satisfying the two conditions that_A(A)·1A⊗^minC_r^∗(G)is a dense subspace ofA⊗^minC_r^∗(G) and that_Ais coassociative in the sense that

(1) (_A⊗idC_r^∗(G))_A=(id_A⊗)_A.

A separable C^∗-algebra equipped with a coaction of Gwill be called aG–ˆ C^∗-algebra. Sometimes we will abuse the notation and call a coaction ofG aG-action. An example of a coaction is the dual coaction onˆ C^∗-algebras of the formA = Br G, for some G–C^∗-algebra B. WhenGis discrete we can decomposeBr Gby means of the dense subspace

g∈GBλ_g and the dual coaction is defined by_A(bλ_g):=bλ_g⊗λ_g. In the general setting, the construction of the dual coaction goes analogously and we refer the reader to [1].

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Much of the theory for group actions also hold for group coactions, the crossed product will as for abelian groups be a stepping stone back and forth between actions and coactions. In [1], theKK-theory equivariant with respect to a bi-C^∗-algebras and the corresponding Kasparov product was constructed.

In [12] it was proved that theKK-theory equivariant with respect to a locally compact quantum group has a triangulated structure defined in the same fashion as for a group.

Let us explain the setting of [1] more explicitly in the case of coactions of a group. An A− B-Hilbert bimoduleE is calledG-equivariant if thereˆ is a coactionδ_E : E →LB⊗minC_r^∗(G)(B⊗min C_r^∗(G),E ⊗C_r^∗(G))satisfying a coassociativity condition similar to (1) and δ_E should commute with the A-action and B-action in the obvious ways. By Proposition 2.4 of [1], the coaction δ_E is uniquely determined by a unitary V_E ∈ L(E ⊗B (B ⊗min

C_r^∗(G)),E ⊗C_r^∗(G))via the equationδ_E(x)y=V_E(x⊗B y)forx∈E and y ∈ B⊗min C_r^∗(G). AG-equivariantˆ A−B-Kasparov module is anA−B- Kasparov module(E, F )such thatEis aG-equivariantˆ A−B-Hilbert module and the operatorFcommutes with the unitaryV_Eup to a compact operator. The groupKK^G^ˆ(A, B)is defined as the homotopy classes ofG-equivariantˆ A−B- Kasparov modules. The additive categoryKK^G^ˆ is defined by taking the objects to be separableG–Cˆ ^∗-algebras and the group of morphisms fromAtoB is KK^G^ˆ(A, B). The composition inKK^G^ˆ is Kasparov product ofG-equivariantˆ Kasparov modules.

To a closed subgroupH ofG, the restriction of aG-action toH defines a restriction functor Res^G_H : KK^G → KK^H and its right adjoint is the induction functor Ind^G_H : KK^H → KK^G. However the restriction goes in the other direction for coactions. WhenH is a closed subgroup ofG, there is a non-degenerate embeddingC^∗(H ) ⊆ M(C^∗(G))so a coaction ofH can be restricted to a coaction ofG. This construction defines a triangulated functor Res^H_G_ˆ^ˆ :KK^H^ˆ →KK^G^ˆ.

The crossed product B → B r G sends a G–C^∗-algebra to a G–Cˆ ^∗- algebra and ifGis exact the crossed product induces a triangulated functor KK^G→KK^G^ˆ. In order to construct a duality similar to Takesaki-Takai duality one introduces the crossed product by a coaction. IfAis aG–Cˆ ^∗-algebra we define

ArGˆ :=[_A(A)·1A⊗C0(G)]⊆M(A⊗K(L²(G))).

It follows from Lemma 7.2 of [2] that Ar Gˆ forms a C^∗-algebra. For a thorough introduction to crossed products by coactions see [13]. The C^∗- algebraAr Gˆ carries a continuousG-action defined in the dense subspace

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_A(A)·1A⊗C₀(G)by

g.(_A(a)·1A⊗f ):=_A(a)·1A⊗g.f.

Similarly to the abelian setting, Takesaki-Takai duality holds so there are equivariant isomorphismsBrGrGˆ ∼=B⊗K(L²(G))andArGˆrG∼= A⊗K(L²(G))which ensures that the crossed product defines an equivalence of triangulated categories known as Baaj-Skandalis duality.

The tensor product on the category ofG–C^∗-algebras is well defined. IfA andB have actionsα respectivelyβ ofGthe tensor productA⊗min B can be equipped with the actionα⊗β : G → Aut(A⊗min B). However, for a non-abelian groupGthe construction of a tensor product ofG–Cˆ ^∗-algebras can not be done by just taking tensor products of theC^∗-algebras. The tensor product relevant forG–Cˆ ^∗-algebras is the braided tensor product overGˆ which requires one further structure. Suppose thatAis aG-algebra with a continuousˆ G-actionα. If the actionαsatisfies that

(2) _A◦α_g=(α_g⊗Ad(g))_A

we say thatAis a Yetter-Drinfeld algebra. An example of a Yetter-Drinfeld algebra isC_r^∗(G)withG-action defined by the adjoint actionG→Aut(G). It is much easier to construct a Yetter-Drinfeld algebra from aG–C^∗-algebra, if Ais aG–C^∗-algebra we can in a functorial way define a coaction ofGonAby setting_A(a):=a⊗1. WhenAis a Yetter-Drinfeld algebra, theC^∗-algebra ArGˆ is also a Yetter-Drinfeld algebra since the morphism_Ais covariant with respect to theG-action and_A extends to a coaction ofGonAr G,ˆ see more in [12]. This construction is functorial and the crossed product can be seen as a functor on the category of Yetter-Drinfeld algebras.

WhenAis a Yetter-Drinfeld algebra andB is aG–Cˆ ^∗-algebra we define the mappings

ι_A:A−→M(A⊗minB⊗K(L²(G))), ι(a):=_α(a)₁₃ ι_B :B−→M(A⊗minB⊗K(L²(G))), ι(b):=_B(b)₂₃. Following [12], the braided tensor productAGˆ B is defined as the closed linear span ofι_A(A)· ι_B(B). By Proposition 8.3 of [16], AGˆ B forms a

∗-subalgebra ofM(A⊗minB⊗K(L²(G)))so the braided tensor product is aC^∗-algebra. The coaction ofGonAGˆ Bis defined by

_AGˆ _B(ι_A(a)·ι_B(b)):=(ι_A⊗id)(_A(a))·(ι_B⊗id)(_B(b)).

Observe that since C_r^∗(G) is cocommutative, the adjoint G-action is trivialˆ and a similar construction of a braided tensor product over Gbetween G–

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C^∗-algebras with trivialG-actions coincides with the usual tensor product. Inˆ general, the braided tensor product overGdoes not need to coincide with the usual tensor product. By Lemma 3.5 of [12] there is aG-equivariant isomorphism

(3) (AGˆ B)r Gˆ ∼=(ArG)ˆ Gˆ B

where the G-coaction on the right hand side is the trivial one on B. More generally, this identity holds for any quantum group and in particular also for braided tensor products over G. We will prove this statement in special case of braided tensor products over a compact groupGwithC(G)below in Lemma 3.3.

If we interpret the structure of a Yetter-Drinfeld algebra as two actions of the quantum groups G and Gˆ satisfying a certain cocycle relation, the cocycle defines a quantum group by means of a double crossed product such that Yetter-Drinfeld algebras are precisely theC^∗-algebras with an action of this double crossed product. The right quantum group to look at is the Drinfeld doubleD(G). Using the notations of quantum groups, the algebra of functions onD(G)isC0(G, C_r^∗(G)) = C0(G)⊗C_r^∗(G)with the obvious action and coaction ofG. The action and coaction define a comultiplication

_D(G) :C₀(D(G))−→M(C₀(D(G))⊗C₀(D(G)))

by_D(G) :=σ₂₃Ad(W₂₃)(_C₀_(G)⊗_Cr∗(G))whereW ∈B(L²(G)⊗L²(G)) is the multiplicative unitary ofGdefined byWf (g₁, g₂)= f (g₁, g₁g₂). The comultiplication_D(G)makesD(G)into a quantum group by Theorem 5.3 of [3]. A Yetter-Drinfeld algebraAwith the actionαand coaction_Acorrespond to aD(G)−C^∗-algebra by defining theD(G)-coaction

^D(G)_A :=(_α⊗id)_A:A−→M(A⊗minC₀(D(G))),

see more in Proposition 3.2 of [12]. Therefore we can consider the braided tensor product as a tensor product betweenD(G)−C^∗-algebras andG–Cˆ ^∗- algebras. The braided tensor product induces a biadditive functor

Gˆ :KK^D(G)×KK^G^ˆ −→KK^G^ˆ.

Much of the theory of coactions can be done without introducing any quantum groups, but in order to construct the Pimsner-Voiculescu sequence for coactions of compact Hodgkin-Lie groups we will need the braided tensor product as a biadditive functor betweenKK-categories.

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2. The Pimsner-Voiculescu sequence from the viewpoint of representation rings

In this section we will study the Pimsner-Voiculescu sequence forZand generalize to a Pimsner-Voiculescu tower forZⁿ. We will use representation theory to calculate all the mappings explicitly. These calculations will in a surpris- ingly straightforward way give a natural route to a Pimsner-Voiculescu tower for coactions of compact Lie groups.

Consider the evaluation mappingl : C0(R) → C0(Z). This mapping fits into aZ-equivariant short exact sequence

(4) 0−→C₀(Z)−→C₀(R)−→^l C₀(Z)−→0.

TheZ-equivariant Dirac operatorD/onRdefines aZ-equivariant odd unbounded K-homology class, thus an element [D/] ∈ KK^Z(C₀(R), C). WhileRis the universal properZ-space the element [D/] is the Dirac element of Zand the strong Baum-Connes property ofZimplies that [D/] is a KK^Z-isomorphism.

The exact sequence (4) induces a distinguished triangle inKK^Zwhich after using the isomorphismC₀(R)∼=Cand rotation 4 steps to the left becomes

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C₀(Z) C₀(Z)

C.

In a certain sense, the distinguished triangle (5) captures the entire behavior of the Pimsner-Voiculescu sequence. IfAis aZ−C^∗-algebra we can apply Baaj- Skandalis duality to (5) and tensor withAr Z. If we apply Baaj-Skandalis duality again, we obtain a distinguished triangle inKK^Z:

A

A A

r Z,

where Ar Z is given the trivial Z-action. Taking K-theory of this distin- guished triangle gives back the classical Pimsner-Voiculescu sequence due to the following proposition:

Proposition 2.1. When T is a torus of rank 1 and the element κ ∈ KK^T(C,C)is defined using the isomorphisms KK^T(C,C) ∼=HomR(T )(R(T ), R(T ))andR(T )∼=^Z[t, t⁻¹]as

κf (t, t⁻¹)=(1−t)f (t, t⁻¹),

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the KK-morphismκ is Baaj-Skandalis dual to the KK-morphism C₀(Z) → C₀(Z)defined by(4).

Observe that theK-theory of the exact sequence (4) is described from the exact sequence:

0−→R(T )−→¹⁻^t R(T )−→^Z−→0,

by Proposition 2.1. The first terms in this exact sequence is the Koszul complex defined by 1−t ∈HomR(T )(R(T ), R(T ))andZis the cohomology of the Koszul complex.

Proof. Letκ₀∈HomR(T )(R(T ), R(T ))denote the Baaj-Skandalis dual of theKK-morphism induced from (4). It follows directly from the construction that the mappingR(T )→^Zinduced fromC0(Z)→C0(R)is the augmentation mappingZ[t, t⁻¹]→^Zonto the generator ofK₁(C₀(R)). Therefore the image ofκ₀is the ideal generated by either 1+tof 1−t soκ₀is of the form u·(1±t)for some unitu∈^Z[t, t⁻¹]. The sign andu=1 is found by either a direct calculation or by considering the Pimsner-Voiculescu sequence for C₀(Z).

We will return to the Koszul complexes later on. First we will construct a geometric interpretation of the higher rank situation. Assume that T is a torus of ranknand consider the semi-open unit cubeI = [0,1[ⁿ⊆ ^Rⁿ. For i=1, . . . , nwe defineX˜_ias the set of openi−1-dimensional faces ofI. The

union satisfies _n

i=1

˜

X_i =∂I ∩I.

We letk_i, fori = 1,2, . . . n, denote the integersk_i := n

i−1

. The setX˜_i hask_i connected components so if we choose a homeomorphism ]0,1[ ∼= ^R there are homeomorphisms

(6) X˜_i ∼=

ki

j=1

Rⁱ⁻¹ for i =1,2, . . . , n,

where we interpretR⁰as the one-point space. We takeX_ito be theZⁿ-translates of _j_≤_iX˜_j and defineY_i :=^Rⁿ\X_i for 1=1,2, . . . , nandY₀:=^Rⁿ.

Proposition2.2.Fori = 1,2, . . . , nthere areZⁿ-equivariant isomorph- isms

C₀(Y_i₋₁)/C₀(Y_i)∼=^C^kⁱ ⊗ⁱ⁻¹C₀(Zⁿ).

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Proof. By equation (6) there is aZⁿ-equivariant homeomorphism Y_i₋₁\Y_i ∼=

m∈^Zⁿ

^ki

j=1

Rⁱ⁻¹

,

whereZⁿacts by translation on the first disjoint union. Therefore C₀(Y_i₋₁)/C₀(Y_i)∼=C₀(Y_i₋₁\Y_i)∼=C₀

m∈Zⁿ

^ki

j=1

Rⁱ⁻¹

∼=^C^kⁱ ⊗C₀(Zⁿ×^Rⁱ⁻¹)≡^C^kⁱ ⊗ⁱ⁻¹C₀(Zⁿ).

Consider the classifying spaceRⁿ for proper actions ofZⁿ. SinceZⁿ has the strong Baum-Connes property, the Dirac element [D/] induces a KK^Zⁿ- isomorphism C₀(Rⁿ) ∼= ⁿC. An alternative approach to constructing this isomorphism is the Julg theorem which implies that for anyT−C^∗-algebraA there is an isomorphismK_∗^T(A)∼=K_∗(ArT ). ThereforeK_∗^T(ⁿC^Zⁿ)∼= K_∗^T(C₀(Rⁿ)^Zⁿ)and the statement follows from the Universal Coefficient Theorem for the compact Hodgkin-Lie groupT, see more in [14].

Fori = 1,2, . . . , n, Proposition 2.2 implies that there is aZⁿ-equivariant short exact sequence

(7) 0−→C0(Y_i)−→C0(Y_i₋1)−→^C^kⁱ ⊗ⁱ⁻¹C0(Zⁿ)−→0.

We will byκ_i ∈KK^Zⁿ(C^kⁱ⊗C0(Zⁿ),C^kⁱ⁺¹⊗C0(Zⁿ))denote theZⁿ-equivariant KK-morphism defined in such a way that the extension class defined by (7) composed with the restriction mappingC₀(Y_i)→^C^kⁱ⁺¹⊗ⁱC₀(Zⁿ)coincides with ⁱ⁻¹κ_i. Notice that Y_n = ^Zⁿ ×]0,1[ⁿ and Y₀ = ^Rⁿ so we have that C₀(Y_n)=ⁿC₀(Zⁿ)andC₀(Y₀)=C₀(Rⁿ), the latter beingKK^Zⁿ-isomorphic toⁿC. Thus we get a sequence of distinguished triangles inKK^Zⁿ:

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nC₀(Zⁿ) C₀(Y_n−1)

Cⁿ ⁿ

−1C₀(Zⁿ)

ⁿκn

C^kⁿ⁻¹ ⁿ⁻²C₀(Zⁿ)

ⁿ⁻¹κ_n−1

C₀(Y₂) C₀(Y₁) ⁿC

Cⁿ C₀(Zⁿ)

²κ2

C₀(Zⁿ)

κ1

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A sequence of distinguished triangles of this type will be called atower. The tower (8) inKK^Zⁿis the higher rank analogue of the distinguished triangle (5).

The tower (8) can be generalized to contain any coefficient ring.

To find a better description of the morphismsκ_i let us recall the notion of a Koszul complex. LetRdenote a commutative ring andEanR-module. For simplicity we will assume thatE is free and finitely generated, let us say of rank N. For an elementv ∈ HomR(E, R), the Koszul complex of E with respect tovis the complex

0−→^NE−→^∂¹ ^N⁻¹E−→ · · ·^∂² −→^∂^N−2 ²E−→^∂^N−1 E−→^v R−→0, where each∂_kis defined as interior multiplication byv. Since we have assumed Eto be free, we may writev=ν_ie^∗_i for someν₁, ν₂, . . . , ν_N ∈Rand the dual basis e_i^∗of a basis ei,i=1,2, . . . , NofE. If the sequenceν₁, ν₂, . . . , ν_N is a regular sequence the Koszul complex is exact except atR. The cohomology of the Koszul complex is in this caseR/v(E)atR. See more in [5].

The Koszul complex of interest to us is constructed from the moduleE:= R(T )ⁿ over the representation ring of the torus T which has the following form:

R(T )∼=^Z[t₁^±¹, . . . , t_n^±¹].

Observe that Baaj-Skandalis duality and the Universal Coefficient Theorem implies that

KK^Zⁿ(C^kⁱ ⊗C₀(Zⁿ),C^kⁱ⁺¹ ⊗C₀(Zⁿ))∼=KK^T(C^kⁱ,C^kⁱ⁺¹)

∼=HomR(T )(R(T )^kⁱ, R(T )^kⁱ⁺¹).

We have thatR(T )^kⁱ ∼=ⁿ⁻ⁱ⁺¹Eso the lower row in (8) have the right ranks for coinciding with a Koszul complex. Letf_i ∈HomR(T )(ⁿ⁻ⁱ⁺¹E,ⁿ⁻ⁱE) denote the image ofκ_i under the isomorphisms above. To simplify notations, we will by(e_i)ⁿ_i₌₁denote theR(T )-basis ofEcoming from the isomorphism E∼=R(T )⊗ZZⁿand by(e_i^∗)ⁿ_i₌₁denote the dual basis.

Theorem2.3.Under the isomorphismsR(T )^kⁱ ∼=ⁿ⁻ⁱ⁺¹Ethe mappings f_i coincide with interior multiplication by the elementv:= n

i=1(1−t_i)e^∗_i. Therefore the sequence

0−→ⁿE−→^f¹ ⁿ⁻¹E−→ · · ·^f² −→^fⁿ⁻² ²E−→^fⁿ⁻¹ E−→^fⁿ R(T )−→0 defines a complex isomorphic to the Koszul complex ofEwhose cohomology atR(T )isZ.

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Proof. While bothf_i and the mapping defined by interior multiplication byvareR(T )-linear it is sufficient to prove thatf_i(u) = v¬ufor elements of the form u = em1

· · ·em_n−i+1 ∈ ⁿ⁻ⁱ⁺¹E, where m₁, . . . , m_n₋_i₊₁ ∈ {1,2, . . . , n}. Let(m_p)_pⁿ₌_n₋_i₊₁be an enumeration of allj =1,2, . . . , nsuch thatj /∈(m_p)_pⁿ⁻₌ⁱ₁⁺¹. If we viewZⁿas a subset ofRⁿwe can defineX_u⊆ ˜X_i as the open face inRⁿspanned by the vectors em_n−i+1,em_n−i+2, . . . ,emn.

Under the isomorphismⁿ⁻ⁱ⁺¹E∼=K_i₋₁(C^kⁱ ⊗ⁱ⁻¹C₀(Zⁿ))the element ucorresponds to aK-theory class onX˜_iwhich is trivial except on the faceX_u. Therefore there exists sequences of numbers(a_j)_jⁿ₌⁻₁ⁱ⁺¹, (b_j)_jⁿ₌⁻₁ⁱ⁺¹ ⊆ ^Z such that

f_i(u)=

n−i+1 j=1

(a_j +b_jt_j)e_m_j¬u.

Ifj = 1,2, . . . , n−i+1, we will letX_u,j denote the open face spanned by the vectors emj,em_n−i+1,em_n−i+2, . . . ,emn. It follows from restricting toX_u,jthat a_j =1 since Bott periodicity implies that the index mappingK_i₋1(C0(X_u))→ K_i(C₀(X_u,j))is an isomorphism. In a similar fashion it follows thatb_i = −1.

While v(E) is the ideal generated by the regular sequence 1 − t₁,1 − t₂, . . . ,1−t_n, the cohomology of the Koszul complex isR(T )/v(E)=^Zand the quotient mappingR(T )→^Zcoincides with the augmentation mapping.

Consider the tower Baaj-Skandalis dual to (8). GivenA, B∈KK^T we can apply the homological functorKK^T(A,−⊗minB)to this tower. This functor is only homological on the bootstrap category ifBis not exact, but all objects in the tower Baaj-Skandalis dual to (8) are in the bootstrap category. The lowest row of the corresponding tower in the category ofR(T )-modules is a Koszul complex:

(9) 0−→ⁿ^Zⁿ⊗KK^T_∗(A, B)−→^v^A^¬ ⁿ⁻¹^Zⁿ⊗KK^T_∗(A, B)−→^v^A^¬ . . .

vA¬

−→^Zⁿ⊗KK^T_∗(A, B)−→^v^A^¬ KK^T_∗(A, B)−→0 where

v_A := n

i=1

(1−β_i_∗)e^∗_i ∈HomR(T )(KK^T_∗(A, B)ⁿ,KK^T_∗(A, B))

and(β_i)ⁿ_i₌₁are the commuting equivariant automorphisms ofAthat are Baaj- Skandalis to theZⁿ-action onBrT. The cohomology of this Koszul complex can be calculated fromKK^T_∗(A, B). We will return to this subject in the next section in the more general case of Hodgkin-Lie groups and explain this pro- cedure further.

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3. The generalized Pimsner-Voiculescu-towers

As mentioned in the introduction, the representation ringR(T ) is free over R(G)whenGis a Hodgkin-Lie group, so the step to coactions of a compact Hodgkin-Lie group will not be too large. We will throughout this section assume thatGis a compact Hodgkin-Lie group of ranknwith maximal torusT. Recall that a group satisfies the Hodgkin condition if it is connected and the fundamental group is torsion-free.

The embeddingT ⊆Ginduces a restriction functorKK^T^ˆ →KK^G^ˆ. Using the isomorphismTˆ ∼=^Zⁿ, the tower (8) can be restricted to aKK^G^ˆ-tower:

C₀(Y_n−1)

ⁿκn

ⁿ⁻¹κ_n−1

C₀(Y₂) C₀(Y₁)

²κ2 κ1

ⁿC^∗(T )

Cⁿ ⁿ⁻¹C^∗(T ) C^kⁿ⁻¹ ⁿ⁻²C^∗(T )

C₀(Rⁿ)

Cⁿ C^∗(T ) C^∗(T )

In order to work with thisKK^G^ˆ-tower we need to describe the termsC^∗(T )in the second row.

Lemma3.1.IfGis a compact Hodgkin-Lie group with Weyl group of order wthere is an isomorphism

C^∗(T )∼=^C^w⊗C^∗(G) in KK^G^ˆ.

Observe that the condition onGto be a Hodgkin group is equivalent toGˆ being a torsion-free quantum group in the sense of Meyer, see [9]. The torsion- free quantum groups are the only non-classical discrete quantum groups for which there is a general formulation of the Baum-Connes property in terms of triangulated categories. In [11], coactions of compact non-Hodgkin Lie groups were considered and the “torsion” turned out to be the torsion elements of H²(G, S¹). The less precise statement C(G/T ) ∼= ^C^k in KK^G for some k is stated and proved in [11]. An explicit calculation that k = |W|can be found in [15]. We will review the conceptually important part of the proof of a Proposition in [11] which proves Lemma 3.1 aside from the calculation ofk.

Proof. By [15], the representation ringR(T )is free of rank wover the representation ring R(G) if π₁(G) is torsion-free. If we let S denote the

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localizing subcategory ofKK^G generated byC andC(G/T ), Lemma 11 of [10] states that forA∈S the natural homomorphism

R(T )⊗R(G)KK^G(A,C)−→KK^T(A,C) is an isomorphism. Thus the representable functor onS

A−→KK^G(A,C^w)∼=R(T )⊗R(G)KK^G(A,C) coincides with the representable functor

A−→KK^G(A, C(G/T ))∼=KK^T(A,C).

The last isomorphism exists as a consequence of the fact that the induction functor Ind^G_T is the right adjoint of the restriction functor fromG to T. So the Yoneda lemma implies thatC(G/T ) ∼= ^C^w inS and therefore in KK^G. Applying Baaj-Skandalis duality it follows that there is an equivariantKK- isomorphismC^∗(T )∼=^C^w⊗C^∗(G).

Using Lemma 3.1 the tower (8) takes the form:

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C0(Yn−1)

C₀(Y₂) C₀(Y₁) ⁿC^w C^∗(G)

ⁿ⁻¹C^wn C^∗(G)

ⁿC

C^wn C^∗(G) C^w C^∗(G)

We will call thisKK^G^ˆ-tower the fundamentalG–PV-tower. The dual funda- mentalG–PV-tower is defined to be theKK^G-tower which is Baaj-Skandalis dual to the fundamentalG–PV-tower:

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ⁿC^w D_n−1

ⁿ⁻¹C^wn ⁿ⁻²C^wkⁿ⁻¹

D2 D1 ⁿC(G)

C^wn C^w

whereD_i :=C₀(Y_i)rG.ˆ

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As mentioned above, if A is a G–C^∗-algebra, the trivial coaction of G onA makesAinto a Yetter-Drinfeld algebra. This follows from thatC(G) is commutative so we can extend a G-action via the D(G)-equivariant ∗- monomorphismC(G)→ M(C₀(D(G))). Clearly, aG-equivariant mapping is equivariant in this new D(G)-action. Furthermore, since mapping cones does not depend on the action, the trivial extension of aG-action to aD(G)- action is functorial and respects mapping cones. The following proposition follows from universality.

Proposition3.2.IfGis a locally compact group, the functor mapping a G–C^∗-algebra to aG-Yetter-Drinfeld algebra with trivialG-action defines aˆ triangulated functor KK^G→KK^D(G).

Using the triangulated functor of Proposition 3.2, we may consider the tower (11) as a tower inKK^D(G). Applying a crossed product byGwe obtain that also the tower (10) is a tower inKK^D(G). For aC^∗-algebraBwe will use the notationt (B)for theG–Cˆ ^∗-algebra with trivial coaction, or in the context of G–C^∗-algebrast (B)will denote theG–C^∗-algebra with trivial action. Let us state and prove the corresponding version of (3) in a simple case of a braided tensor product overGwithC(G), a more general proof can be found in [12].

Lemma3.3.WhenBhas a continuousG-action, there is aG-equivariantˆ Morita equivalence

(C(G)⊗B)rG∼M t (B).

Proof. By Baaj-Skandalis duality, it suffices to prove that there is a G-ˆ equivariant isomorphism(C(G)⊗B)rG∼=(C(G)rG)⊗t (B). Denote the G-action onBbyβand define the equivariant mappingϕ₀:L¹(G, C(G, B))

→(C(G)rG)⊗t (B)by setting

ϕ₀(f )(g₁, g₂):=β_g−1

1 f (g₁, g₂).

The linear mappingϕ₀is a∗-homomorphism whenL¹(G, C(G, B))is equipped with the convolution twisted by theG-action onC(G)⊗B. It is straight- forward to verify thatϕ0is bounded inC^∗-norm so we can defineϕ :(C(G)⊗ B)r G→(C(G)r G)⊗Bby continuity. The∗-homomorphismϕis an equivariant isomorphism since an inverse can be constructed by extending

ϕ⁻¹(f ⊗b)(g₁, g₂):=f (g₁, g₂)β_g₁(b)

to a∗-homomorphismϕ⁻¹:(C(G)r G)⊗t (B)→(C(G)⊗B)r G.