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We introduce a new method for studying the Baum-Connes conjecture, which we call the direct splitting method. The method can simplify and clarify proofs of some of the known cases of the conjecture. In a separate paper, with J. Brodzki, E. Guentner and N. Higson, a similar idea will be used to give a finite-dimensional proof of the Baum-Connes conjecture for groups which act properly and co-compactly on a finite-dimensional CAT(0)-cubical space.
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-Conne
We give a decomposition of the equivariant Kasparov category for discrete quantum group with torsions. As an outcome, we show that the crossed product by a discrete quantum group in a certain class preserves the UCT. We then show that quasidiagonalit
We introduce the notion of proper Kasparov cycles for Kasparovs G-equivariant KK-theory for a general locally compact, second countable topological group G. We show that for any proper Kasparov cycle, its induced map on K-theory factors through the l
Let $left( 1to N_nto G_nto Q_nto 1 right)_{nin mathbb{N}}$ be a sequence of extensions of finitely generated groups with uniformly finite generating subsets. We show that if the sequence $left( N_n right)_{nin mathbb{N}} $ with the induced metric fro
We compute the Chern-Connes character (a map from the $K$-theory of a C$^*$-algebra under the action of a Lie group to the cohomology of its Lie algebra) for the $L^2$-norm closure of the algebra of all classical zero-order pseudodifferential operato