A non-amenable groupoid whose maximal and reduced $C^{\ast }$-algebras are the same

Abstract

We construct a locally compact groupoid with the properties in the title. Our example is based closely on constructions used by Higson, Lafforgue, and Skandalis in their work on counterexamples to the Baum-Connes conjecture. It is a bundle of countable groups over the one point compactification of the natural numbers, and is Hausdorff, second countable and étale.

Version two has some clarifications, minor fixes, and additional background and references

References in corpus (1)

• A characterization of amenability of group actions on $C^{\ast }$-algebras

Cited by in corpus (16)

• Étale groupoids and their $C^{\ast }$-algebras
• Spectral Continuity for Aperiodic Quantum Systems I. General Theory
• Some remarks about the weak containment property for groupoids and semigroups
• A uniqueness theorem for twisted groupoid C*-algebras
• Non-amenable principal groupoids with weak containment
• $C^{\ast }$-simplicity and representations of topological full groups of groupoids
• Going-Down functors and the Künneth formula for crossed products by étale groupoids
• On some permanence properties of exact groupoids
• Amenable actions of inverse semigroups
• On the K-theory of C*-algebras for substitution tilings (a pedestrian version)
• The uniform Roe algebra of an inverse semigroup
• Characterizing groupoid C*-algebras of non-Hausdorff étale groupoids
• C*-algebras of Boolean inverse monoids - traces and invariant means
• Isotropy fibers of ideals in groupoid C${}^{\ast }$-algebras
• A note on the quasi-diagonality of inverse semigroup reduced C*-algebras
• $C^{\ast }$-uniqueness Results for Groupoids