The Connes-Kasparov conjecture for almost connected groups and for linear $p$-adic groups

Chabert, Jérôme; Echterhoff, Siegfried; Nest, Ryszard

Publications Mathématiques de l'IHÉS, Tome 97 (2003), pp. 239-278.

Let G be a locally compact group with cocompact connected component. We prove that the assembly map from the topological K-theory of G to the K-theory of the reduced C${}^{*}$-algebra of G is an isomorphism. The same is shown for the groups of $k$-rational points of any linear algebraic group over a local field $k$ of characteristic zero.

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