For any countable discrete group Γ (without further assumptions on it) we first construct an explicit morphism from the Left-Hand side of Baum-Connes assembly map, Ktop⁎(Γ), to periodic cyclic homology algebra, HP⁎(CΓ). This morphism, called Chern-Baum-Connes allows in particular give a proper and formulation for Chern-Connes pairingKtop⁎(Γ)×HP⁎(CΓ)⟶C. Several theorems are needed formulate map....
