Operator algebras conference in memory of Étienne Blanchard

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

11 avr. 2019, 15:15

50m

Bâtiment Sophie Germain, Amphi Turing (Université Paris Diderot)

Abstract

Orateur

Sara Azzali (Univ. Potsdam)

Description

Let $\mathrm{\Gamma }$ 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 $\mathrm{\Gamma }$.
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 $C^{\ast }$-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 $C^{\ast }$-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 $E\mathrm{\Gamma }$ with the classifying space for proper actions $\underset{―}{E}\mathrm{\Gamma }$. This is joint work with Paolo Antonini and Georges Skandalis.