The Analytical Assembly Map and Index Theory

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.