Groupoid exactness and the weak containment problem

TL;DR

This paper explores the concept of exactness in locally compact groupoids, especially inner amenable étale groupoids, establishing equivalences among six notions of exactness and examining their implications for C*-algebra properties.

Contribution

It extends the theory of exactness from discrete groups to a class of étale groupoids, introducing new equivalences and examples, and discusses implications for C*-algebraic structures.

Findings

Equivalence of six notions of exactness for inner amenable étale groupoids.

Examples illustrating the concepts and results.

Discussion on the relationship between groupoid amenability and C*-algebraic properties.

Abstract

Our purpose is to study in the setting of locally compact groupoids the analogues of the well-known equivalent definitions of exactness for discrete groups. Our best results are obtained for a class of \'etale groupoids that we call inner amenable. For locally compact groups this notion coincides with a classical notion of inner amenability. We give examples of such groupoids. Whether all \'etale groupoids have this property is still unknown. For inner amenable \'etale groupoids we extend what is known for discrete groups in proving the equivalence of six natural notions of exactness: (1) strong amenability at infinity; (2) amenability at infinity; (3) nuclearity of the uniform algebra of the groupoid; (4) exactness of this C^*-algebra; (5) exactness of the groupoid in the sense of Kirchberg-Wassermann; (6) exactness of the reduced C^*-algebra of the groupoid. We give several illustrations of these results. One of our motivations for this study of exactness is that it plays a crucial role in examining the relationships between the amenability of a groupoid and the fact that its full and reduced C^*-algebras coincide. This is highlighted in the review we give of the results obtained by several authors on this subject. We end our monograph with open questions and an appendix on fibrewise compactifications whose study is needed because our work requires to extend from discrete groups to any \'etale groupoid G the notion of Stone-Cech compactification on which G acts.