Room 326

The categorical framework for the Baum-Connes conjecture

Ruben Martos

University of Paris 7

The axiomatization of compact quantum groups (in the setting of C*-algebras) made
by S. L. Woronowicz (in 1987) is being very fruitful since it’s allowed (and it still allows) to
translate a large number of classical results in group theory like the classical Peter-Weyl
theory for representations of compact groups or classical approximation properties
(Haagerup property or amenability). Works of S. Baaj, G. Skandalis and R. Vergnioux show
that we can even define an equivariant KK-theory with respect to a quantum group and
generalize some results in this context.
This is why we can think about a “quantum” formulation of the Baum-Connes
conjecture. For this purpose, the well theoretic framework is the triangulated categories
(firstly studied by Jean-Louis Verdier and A. Grothendieck).
In this talk, we are going to introduce in a simple way the tools used for being able to
formulate the Baum-Connes conjecture in the categories language. More precisely, we’ll
define a triangulated category and some objets of interest, afterwards we’ll translate these
very general constructions to the category KK and we’ll explain how to formulate the Baum-
Connes conjecture in this language. Finally, we’ll do an overview of the current situation of
the conjecture for quantum groups.