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

Sara Azzali  (University of Potsdam)

10:40-11:30, June 25, 2019   Science Building A510

Abstract:

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.