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Twisted equivariant $$KK$$-theory and the Baum-Connes conjecture for group extensions. (English) Zbl 1010.19004
Let $$G$$ be a locally compact group and $$B$$ a $$C^{\ast}$$-algebra equipped with a strongly continuous action of $$G.$$ Let $${\mathcal E}(G)$$ be the classifying space for the proper actions of $$G,$$ and denote by $$K_{\ast}^{\text{ top}}(G,B)$$ the $$G$$-equivariant $$K$$-homology of $${\mathcal E}(G)$$ with compact support and with coefficients in $$B.$$ P. Baum, A. Connes and N. Higson gave a construction of an assembly map $$\mu _{G,B}:K_{\ast}^{\text{ top}}(G,B)\to K_{\ast}(B\times _r G)$$ where $$B\times _r G$$ is the reduced $$C^{\ast}$$-crossed product. The Baum-Connes conjecture (with coefficients) asserts that $$\mu _{G,B}$$ is an isomorphism for all $$G$$ and for every $$G-C^{\ast}$$-algebra $$B.$$ If $$G=N \triangleleft H$$ is the semi-direct product of the groups $$N$$ and $$H,$$ then the crossed product $$B\times _r G$$ decomposes as an iterated crossed product $$B\times _r N\times _r H.$$ Using this decomposition Chabert constructed a partial assembly map $$\mu _{B,N}^H:K_{\ast}^{\text{ top}}(G,B)\to K_{\ast}^{\text{ top}}(H,B\times _r N)$$ such that $$\mu _{H,B \times _r N} \cdot \mu _{B,N}^H =\mu _{G,B}$$ and also showed that there are several interesting situations in which $$\mu _{B,N}^H$$ is indeed an isomorphism [J. Chabert, J. Reine Angew. Math. 521, 161-184 (2000; Zbl 0959.46049)]. Thus, if the partial assembly map $$\mu _{B,N}^H$$ is an isomorphism for all $$B$$ it follows that $$G$$ satisfies the Baum-Connes conjecture if $$H$$ satisfies the Baum-Connes conjecture.
The main purpose of the paper is to extend Chabert’s results to arbitrary (not necessarily split) extensions of second countable locally compact groups. In particular for this it is necessary to extend Kasparov’s equivariant $$KK$$-theory to an equivariant theory for twised actions of groups on $$C^{\ast}$$-algebras.

##### MSC:
 19K35 Kasparov theory ($$KK$$-theory) 46L80 $$K$$-theory and operator algebras (including cyclic theory)
Zbl 0959.46049
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