Equivariant local cyclic homology and the equivariant Chern-Connes character

We define and study equivariant analytic and local cyclic homology for smooth actions of totally disconnected groups on bornological algebras. Our approach contains equivariant entire cyclic cohomology in the sense of Klimek, Kondracki and Lesniewski as a special case and provides an equivariant extension of the local cyclic theory developped by Puschnigg. As a main result we construct a multiplicative Chern-Connes character for equivariant KK-theory with values in equivariant local cyclic homology.Comment: 38 page

On the structure of quantum automorphism groups

We compute the K-theory of quantum automorphism groups of finite dimensional C∗-algebras in the sense of Wang. The results show in particular that the C∗-algebras of functions on the quantum permutation groups S+n are pairwise non-isomorphic for different values of n. Along the way we discuss some general facts regarding torsion in discrete quantum groups. In fact, the duals of quantum automorphism groups are the most basic examples of discrete quantum groups exhibiting genuine quantum torsion phenomena

Cyclic cohomology and Baaj-Skandalis duality

We construct a duality isomorphism in equivariant periodic cyclic homology analogous to Baaj-Skandalis duality in equivariant Kasparov theory. As a consequence we obtain general versions of the Green-Julg theorem and the dual Green-Julg theorem in periodic cyclic theory. Throughout we work within the framework of bornological quantum groups, thus in particular incorporating at the same time actions of arbitrary classical Lie groups as well as actions of compact or discrete quantum groups. An important ingredient in the construction of our duality isomorphism is the notion of a modular pair for a bornological quantum group, closely related to the concept introduced by Connes and Moscovici in their work on cyclic cohomology for Hopf algebras.Comment: 23 page

Quantum SU(2) and the Baum-Connes conjecture

We review the formulation and proof of the Baum-Connes conjecture for the dual of the quantum group $SU_q(2)$ of Woronowicz. As an illustration of this result we determine the $K$-groups of quantum automorphism groups of simple matrix algebras.Comment: 14 pages, contribution to the Proceedings of the Conference in Honour of the seventieth birthday of S. L. Woronowic

Complex quantum groups and a deformation of the Baum-Connes assembly map

We define and study an analogue of the Baum-Connes assembly map for complex semisimple quantum groups, that is, Drinfeld doubles of $q$-deformations of compact semisimple Lie groups. Our starting point is the deformation picture of the Baum-Connes assembly map for a complex semisimple Lie group $G$, which allows one to express the $K$-theory of the reduced group $C^*$-algebra of $G$ in terms of the $K$-theory of its associated Cartan motion group. The latter can be identified with the semidirect product of the maximal compact subgroup $K$ acting on $\mathfrak{k}^*$ via the coadjoint action. In the quantum case the role of the Cartan motion group is played by the Drinfeld double of the classical group $K$, whose associated group $C^*$-algebra is the crossed product of $C(K)$ with respect to the adjoint action of $K$. Our quantum assembly map is obtained by varying the deformation parameter in the Drinfeld double construction applied to the standard deformation $K_q$ of $K$. We prove that the quantum assembly map is an isomorphism, thus providing a description of the $K$-theory of complex quantum groups in terms of classical topology. Moreover, we show that there is a continuous field of $C^*$-algebras which encodes both the quantum and classical assembly maps as well as a natural deformation between them. It follows in particular that the quantum assembly map contains the classical Baum-Connes assembly map as a direct summand.Comment: 26 page