Bivariant KKKK-Theory and the Baum-Connes conjecure

This is a survey on Kasparov's bivariant $KK$-theory in connection with the Baum-Connes conjecture on the $K$-theory of crossed products $A\rtimes_rG$ by actions of a locally compact group $G$ on a C*-algebra $A$. In particular we shall discuss Kasparov's Dirac dual-Dirac method as well as the permanence properties of the conjecture and the "Going-Down principle" for the left hand side of the conjecture, which often allows to reduce $K$-theory computations for $A\rtimes_rG$ to computations for crossed products by compact subgroups of $G$. We give several applications for this principle including a discussion of a method developed by Cuntz, Li and the author for explicit computations of the $K$-theory groups of crossed products for certain group actions on totally disconnected spaces. This provides an important tool for the computation of $K$-theory groups of semi-group C*-algebras.Comment: Some minor correction

Full duality for coactions of discrete groups

Using the strong relation between coactions of a discrete group G on C*-algebras and Fell bundles over G, we prove a new version of Mansfield's imprimitivity theorem for coactions of discrete groups. Our imprimitivity theorem works for the universally defined full crossed products and arbitrary subgroups of G, as opposed to the usual theory which uses the spatially defined reduced crossed products and normal subgroups of G. Moreover, our theorem factors through the usual one by passing to appropriate quotients. As applications we show that a Fell bundle over a discrete group is amenable in the sense of Exel if and only if the double dual action is amenable in the sense that the maximal and reduced crossed products coincide. We also give a new characterization of induced coactions in terms of their dual actions.Comment: 18 page

Principal noncommutative torus bundles

In this paper we study continuous bundles of C*-algebras which are non-commutative analogues of principal torus bundles. We show that all such bundles, although in general being very far away from being locally trivial bundles, are at least locally trivial with respect to a suitable bundle version of bivariant K-theory (denoted RKK-theory) due to Kasparov. Using earlier results of Echterhoff and Williams, we shall give a complete classification of principal non-commutative torus bundles up to equivariant Morita equivalence. We then study these bundles as topological fibrations (forgetting the group action) and give necessary and sufficient conditions for any non-commutative principal torus bundle being RKK-equivalent to a commutative one. As an application of our methods we shall also give a K-theoretic characterization of those principal torus-bundles with H-flux, as studied by Mathai and Rosenberg which possess "classical" T-duals.Comment: 33 pages, to appear in the Proceedings of the London Mathematical Societ