Amenability, exactness and weak containment property for groupoids

Abstract: From the mid-1970s, Eberhard Kirchberg undertook a remarkable extensive study of $C^*$-algebras exactness whose applications spread out to many branches of analysis. In this review we focus on the case of groupoid $C^*$-algebras for which the notion of exactness needs to be better understood. In particular some versions of exactness play an important role in the study of the weak containment problem, that is whether the coincidence of the full and reduced groupoid $C^*$-algebras implies the amenability of the groupoid or not.