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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
Keywords:
K-theory; K-homology; reduced C*-algebra
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