Amenable group actions on C*-algebras and the weak containment problem

The notion of amenable actions by discrete groups on C*-algebras has been introduced by Claire Amantharaman-Delaroche more than thirty years ago, and has become a well understood theory with many applications. So it is somewhat surprising that an established theory of amenable actions by general locally compact groups has been missed until 2020. We now present a theory which extends the discrete case and unifies several notions of approximation properties of actions which have been discussed in the literature. We also present far reaching results towards the weak containment problem which asks wether an action $\alpha:G\to \Aut(A)$ is amenable if and only if the maximal and reduced crossed products coincide.

In this lecture we report on joint work with Alcides Buss and Rufus Willett.