The Analytical Assembly Map and Index Theory

TL;DR

This paper explores the index theoretic interpretation of the analytical assembly map in the Baum-Connes conjecture, providing a precise formulation and proof, especially for torsionfree groups.

Contribution

It offers a rigorous formulation and proof of the analytical assembly map as an equivariant index in the context of torsionfree groups.

Findings

The analytical assembly map can be interpreted as an equivariant index.

A precise formulation of the assembly map in the torsionfree case is provided.

The paper proves the equivalence between the assembly map and the equivariant index in this setting.

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.