On the Baum-Connes Assembly Map for Discrete Groups

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 EΓ 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: 1We 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.2We 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.3To 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.4Denote 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.