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

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 $[\tau ]\in K{K}_{\mathbb{R}}^{\Gamma }(\mathbb{C},\mathbb{C})$ 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 $\Gamma$.

22 pages; details added, improved exposition; final version to appear on Compositio Mathematica