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Bivariant KK-Theory and the Baum-Connes conjecure

Siegfried Echterhoff,
published 2017-03-31

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 x_r G 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 x_r G 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.

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