Some remarks about the weak containment property for groupoids and semigroups

Abstract

A locally compact groupoid is said to have the weak containment property if its full $C^{\ast }$-algebra coincides with its reduced one. This property is strictly weaker than amenability and is known to be equivalent to amenability for transformation groupoids relative to actions of exact discrete groups. We believe that for general étale groupoids one should have the same equivalence of the two properties under some mild exactness assumption. In this paper we try to support this statement.

Compared to the first version, a false statement and its consequences have been removed

References in corpus (7)

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• Boundaries of reduced C*-algebras of discrete groups
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Cited by in corpus (8)

• Ideal structure and pure infiniteness of ample groupoid $C^{\ast }$-algebras
• Non-amenable principal groupoids with weak containment
• On some permanence properties of exact groupoids
• The uniform Roe algebra of an inverse semigroup
• An identification of the Baum-Connes and Davis-Lück assembly maps
• C*-algebras of Boolean inverse monoids - traces and invariant means
• Purely infinite locally compact Hausdorff étale groupoids and their $C^{\ast }$-algebras
• Nuclearity for partial crossed products by exact discrete groups