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On the Baum-Connes Assembly Map for Discrete Groups

  • Alain Valette
Chapter
Part of the Advanced Courses in Mathematics CRM Barcelona book series (ACMBIRK)

Abstract

In these notes, we study the Baum-Connes analytical assembly maps (or index maps) μ i Γ : RK i Γ (EΓ) → K i (C r * Γ) and \( \bar \mu _i^\Gamma \) ,for a countable group Γ. Here RK i Γ denotes the Γ-equivariant K-homology with Γ-compact supports of the universal space E Γ for proper Γ-actions, while K i (C r * ) (resp. K i (C*Γ) denotes the analytical K-theory of the reduced (resp. full) C*-algebra of Γ. As it is simple and direct, we use the definition of β i Γ suggested by Baum, Connes and Higson in Section 3 of [BCH94]. The Baum-Connes conjecture asserts that, for any group Γ, the map β i Γ is an isomorphism (i= 0, 1). The contents of this paper are as follows:
  1. 1

    We make the necessary changes for constructing \( \bar \mu _i^\Gamma \) , and give a detailed proof that β i Γ and \( \bar \mu _i^\Gamma \) provide K-theory elements of the corresponding C*-algebras.

     
  2. 2

    We carefully describe the behavior of the left-hand side of the assembly maps under group homomorphisms, and we prove that \( \bar \mu _i^\Gamma \)is natural with respect to arbitrary group homomorphisms. As a consequence, we get a new proof of the fact that, if Γ acts freely on the space X, then the equibvalent K-homology K * Γ (X) is isomorphic to the H-homology K *(Γ\X) of the orbit space.

     
  3. 3

    To illustrate the non-triviality of the assembly map, we give a direct proof of the Bauam-Connes conjecture for the group ℤ of integers, not appealing to equivariant KK-theory.

     
  4. 4

    Denote by \( \tilde \kappa _\Gamma :\Gamma \to K_1 \left( {C_r^* \Gamma } \right) \) the homomorphism induced by the canonical inclusion of Γ in the unitary group of C r Γ . We show that there exixts a homomorphism \( \bar \beta _t :\Gamma \to RK_1^\Gamma \left( {\underline E T} \right) \) such that \( \bar \kappa _\Gamma = \mu _i^\Gamma \circ \tilde \beta _t \); this extends a result of Natsume [Nat88] for Γ torsion-free.

     

Keywords

Compact Operator Discrete Group Group Homomorphism Chern Character Finite Subgroup 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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  • Alain Valette

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