The analytical assembly map and index theory

Abstract

In this paper we study the index theoretic interpretation of the analytical assembly map that appears in the Baum–Connes conjecture. In its general form it may be constructed using Kasparov’s equivariant $KK$-theory. In the special case of a torsionfree group the domain simplifies to the usual $K$-homology of the classifying space $BG$ of $G$ and it is frequently used that in this case the analytical assembly map is given by assigning to an operator an equivariant index. We give a precise formulation of this statement and prove it.