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On the Baum-Connes conjecture in the real case. (English) Zbl 1064.19003
The classical Baum-Connes conjecture for a given countable discrete group \(\Gamma\) states that the index map (see §1) \[ \mu(\Gamma): K^\Gamma_j(\underline{E}\Gamma) \to K_j(C^*_r(\Gamma)), \quad j\in \mathbb Z \pmod 2 \] is an isomorphism, where \(K_j(C^*_r(\Gamma))\) is the reduced K-theory of the reduced group C*-algebra \(C^*_r(\Gamma)\) and \(K^\Gamma_j(\underline{E}\Gamma)\) is the reduced Kasparov K-homology with \(\Gamma\)-compact support (modulo \(\Gamma\)) of the space \(\underline{E}\Gamma\). In the real number field context of equivariant real K-homology \(KO^\Gamma_j(\underline{E}\Gamma)\) the index map is \[ \mu_\mathbb R(\Gamma): KO^\Gamma_j(\underline{E}\Gamma) \to K_j(C^*_r(\Gamma; \mathbb R)), \quad j\in \mathbb Z \pmod 8 \] The main result of the paper is the theorem in §3 stating that if \(\mu(\Gamma)\) is an isomorphism then \(\mu_\mathbb R(\Gamma)\) is also an isomorphism.

MSC:
19K35 Kasparov theory (\(KK\)-theory)
46L80 \(K\)-theory and operator algebras (including cyclic theory)
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
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