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Baum-Connes conjecture and semi-direct products of groups. (Stabilité de la conjecture de Baum-Connes pour certains produits semi-directs de groupes.) (French) Zbl 0971.46045

Summary: “Let \(\Gamma=G\ltimes_r N\) be the semidirect product of two locally compact groups, and \(A\) a \(\Gamma\)-algebra. We build a map from topological \(K\)-theory of \(\Gamma\) with coefficients in \(A\) to the one for \(G\) with coefficients in \(N\ltimes_rA\). For \(\Gamma\) a \(p\)-adic group equipped with a \(\gamma\)-element and \(N\) an amenable group, we prove that \(\Gamma\) satisfies the Baum-Connes conjecture with coefficients when \(G\) does.”
For a locally compact topological group \(G\) the Baum-Connes conjecture for \(G\) with coefficients asserts that the assembly map \(\mu_{G,B}\colon K^{\roman{top}}_*(G;B) \rightarrow K_*(B\rtimes_r G)\) is an isomorphism for any \(G\)-algebra \(B\). The author constructs the \(H\)-equivariant assembly map \(\mu^H_{N,B}\colon K^{\roman{top}}_*(G;B) \rightarrow K^{\roman{top}}_*(H;B\rtimes_r N)\), which coincides with \(\mu_{G,B}\) in the case when \(H\) is the trivial group. The assembly maps are related by the identity \(\mu_{G,B}= \mu_{H,B\rtimes N} \circ \mu^H_{N,B}\). This identity connects the Baum-Connes conjecture for \(G\) with coefficients in \(B\) to the Baum-Connes conjecture for \(H\) with coefficients in \(B\rtimes N\). Then the author applies the Dirac/dual-Dirac method of Kasparov. The main result of the paper is that if \(N\) is amenable and the Baum-Connes conjecture holds for \(H\), then the conjecture is also true for \(G\), provided that \(G\) has a \(\gamma\)-element, and \(H\) is almost connected or has a compact-open subgroup.
Reviewer: G.A.Noskov (Omsk)

MSC:

46L85 Noncommutative topology
19K99 \(K\)-theory and operator algebras
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