The Baum–Connes assembly map, delocalization and the Chern character

Abstract

For a discrete group Γ, we explicitly describe the rational Baum–Connes assembly map μ ∗ Γ⊗id C in “homological degree ⩽2” and show that in this domain it factors through the algebraic K-theory of the complex group ring of Γ. We also state and prove a delocalization property for μ ∗ Γ , namely expressing it rationally in terms of the Novikov assembly map. Finally, we give a handicrafted construction of the delocalized equivariant Chern character (in the analytic language) and prove that it coincides with the equivariant Chern character of Lück (Invent. Math. 149 (2002) 123–152) (defined in the topological framework).