The Baum-Connes assembly map and the generalized Bass conjecture

TL;DR

This paper links the Baum-Connes assembly map to the generalized Bass conjecture by analyzing the Connes-Karoubi-Chern character's image in cyclic homology, showing that surjectivity implies the conjecture for certain group algebra completions.

Contribution

It establishes a connection between the Baum-Connes assembly map and the generalized Bass conjecture through cyclic homology analysis, providing new implications for algebra completions.

Findings

The image of the Connes-Karoubi-Chern character lies in the elliptic summand of cyclic homology.

Rational surjectivity of the Baum-Connes map implies the generalized Bass conjecture for weighted ell-1 completions.

The result bridges topological K-theory and algebraic conjectures via cyclic homology.

Abstract

We show that the image of Connes-Karoubi-Chern character, restricted to the image of the Baum-Connes assembly map in the Bott-periodized topological K-theory of the complex group algebra, lies in the elliptic summand of the (periodic) cyclic homology of the group algebra. This implies that for any (weighted) ell-1 completion of the group algebra, rational surjectivity of the Baum-Connes assembly map implies the generalized Bass conjecture for that algebra.