Amenability, exactness and weak containment property for groupoids

TL;DR

This paper explores the concepts of amenability, exactness, and weak containment in groupoid $C^*$-algebras, highlighting their interrelations and implications for the structure and properties of groupoids.

Contribution

It clarifies the role of various notions of exactness in the weak containment problem for groupoid $C^*$-algebras.

Findings

Different versions of exactness influence the weak containment property.

The coincidence of full and reduced $C^*$-algebras relates to amenability.

The paper provides insights into the structure of groupoid $C^*$-algebras.

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.