# Bartosz Kwaśniewski

## Noncommutative Cartan C*-subalgebras

Celebrated Renault's result states that commutative Cartan $C^*$-subalgebras, i.e. regular maximal abelian $C^*$-subalgebras $A\subseteq B$ with faithful conditional expectation, are in bijective corresponence with topologically free twisted groupoids. Recently, Exel generalized the notion of a Cartan inclusion $A\subseteq B$ to the case where $A$ is an arbitrary (noncommutative) $C^*$-algebra, and proved a part of a Renault's theorem. In this talk I present a number of characterisations of noncommutative Cartan $C^*$-inclusions that fully generalize Renault's Theorem, and give tools to study properties of the ambient algebra. (Based on a number of joint works with Ralf Meyer.)