The Baum--Connes conjecture localised at the unit element of a discrete group

TL;DR

This paper constructs a localized Baum--Connes assembly map at the unit element of a discrete group using KK-theory, showing it implies the strong Novikov conjecture and is weaker than the classical conjecture.

Contribution

It introduces a new localized assembly map in KK-theory at the group trace, providing a weaker but still impactful version of the Baum--Connes conjecture.

Findings

The localized assembly map $ au$-Baum--Connes conjecture implies the strong Novikov conjecture.

The map is functorial with respect to the group $ au$.

The conjecture is weaker than the classical Baum--Connes conjecture.

Abstract

We construct a Baum--Connes assembly map localised at the unit element of a discrete group $\Gamma$. This morphism, called $\mu_\tau$, is defined in $KK$-theory with coefficients in $\mathbb{R}$ by means of the action of the projection $[\tau]\in KK_{\mathbb{R}}^\Gamma(\mathbb{C},\mathbb{C})$ canonically associated to the group trace of $\Gamma$. We show that the corresponding $\tau$-Baum--Connes conjecture is weaker then the classical one but still implies the strong Novikov conjecture. The right hand side of $\mu_\tau$ is functorial with respect to the group $\Gamma$.