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The Baum-Connes conjecture for amenable foliations. (La conjecture de Baum-Connes pour les feuilletages moyennables.) (French) Zbl 0939.19001

The Baum-Connes conjecture claims that that for a locally compact groupoid \({\mathcal G}\) (with Haar system) the assembly (or index) map \[ \mu:K_*^{top} ({\mathcal G})\to K_*(C^*_r({\mathcal G})) \] is an isomorphism. Here \(K_*^{top} ({\mathcal G})\) denotes the topological \(K\)-homology of \({\mathcal G}\) (expressed as \(K\)-homology of a certain topological space associated to \({\mathcal G})\) and \(C^*_r({\mathcal G})\) is the reduced \(C^*\)-algebra of \({\mathcal G}\) (the completion of a convolution algebra associated to \({\mathcal G})\). The conjecture is known to be false in this generality and it is an important problem to identify classes of groupoids for which it holds. One thus class is treated by the present author. He takes up the proof of the Baum-Connes conjecture for amenable groups by Higson and Kasparov and generalizes it so that it applies to groupoids of foliations with amenable holonomy.
We cite from the introduction: “We show, using the construction of Higson and Kasparov, that the Baum-Connes conjecture holds for foliations whose holonomy groupoid is Hausdorff and amenable. More generally, for every locally compact, \(\sigma\)-compact and Hausdorff groupoid \({\mathcal G}\) acting continuously and isometrically on a continuous field of affine Euclidean spaces, the Baum-Connes conjecture with coefficients is an isomorphism, and \({\mathcal G}\) amenable in \(K\)-theory. In addition, we show that \(C^*({\mathcal G})\) satisfies the universal coefficient theorem”.

MSC:

19K35 Kasparov theory (\(KK\)-theory)
46L85 Noncommutative topology
58J22 Exotic index theories on manifolds
20L05 Groupoids (i.e. small categories in which all morphisms are isomorphisms)
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