Groupoid exactness and the weak containment problem

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 étale 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 étale groupoids have this property is still unknown. For inner amenable étale 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 étale groupoid G the notion of Stone-Cech compactification on which G acts.