# Selcuk Barlak (Odense): Cartan subalgebras and the UCT problem

Selcuk Barlak (University of Southern Denmark at Odense) will give a talk with title: Cartan subalgebras and the UCT problem

Abstract: By recent breakthrough results, the class of separable, unital, simple, infinite dimensional C*-algebras with finite nuclear dimension and which satisfy the UCT are classified by the Elliott invariant. This truly remarkable result now boosts among others the investigation of the rather mysterious nature of the UCT. In this talk, I will present recent joint work with Xin Li on the connection between the UCT problem for separable, nuclear C*-algebras and Cartan subalgebras, that is, MASAs admitting conditional faithful expectations and generating the ambient C*-algebra in a suitable sense. By a remarkable result of Renault, a C*-algebra that admits a Cartan subalgebra can be realized as a reduced twisted groupoid C*-algebra. Applying this and Tu's striking results and techniques used in the proof of the Baum-Connes Conjecture for amenable groupoids, it is shown that a separable, nuclear C*-algebra possessing a Cartan subalgebra satisfies the UCT. By combining this with Izumi's work on finite group actions on the Cuntz algebra O_2, we obtain a characterization of the UCT problem for separable, nuclear C*-algebras isomorphic to their tensorial CAR-algebra stabilization in terms of Cartan subalgebras and order two automorphisms of O_2.