Orateur
Description
Let $\Gamma$ be a discrete group. In this talk, we study a variant of the Baum–Connes isomorphism conjecture which can be called ‘localised at the unit element' of $\Gamma$.
The localised assembly map is constructed in KK-theory with coefficients in $\mathbb{R}$. These KK-groups are natural receptacles of elements coming from traces on $C^*$-algebras.
We show that the localised Baum--Connes conjecture is weaker than the classical Baum—Connes conjecture but still implies the strong Novikov conjecture. Moreover, it does not see the difference between the reduced and maximal group $C^*$-algebras.
We explain these constructions and show the relation with the Novikov conjecture by explicitly comparing at the level of K-homology with real coefficients, the classifying space for free and proper actions $E\Gamma$ with the classifying space for proper actions $\underline E\Gamma$. This is joint work with Paolo Antonini and Georges Skandalis.
