A non-amenable groupoid whose maximal and reduced $C^*$-algebras are the same

TL;DR

This paper constructs a specific locally compact groupoid demonstrating that its maximal and reduced $C^*$-algebras are identical, challenging assumptions about non-amenability and algebraic properties.

Contribution

It provides a novel example of a non-amenable groupoid with equal maximal and reduced $C^*$-algebras, inspired by counterexamples to the Baum-Connes conjecture.

Findings

Constructed a non-amenable groupoid with identical maximal and reduced $C^*$-algebras

The example is a bundle of countable groups over the one-point compactification of natural numbers

The groupoid is Hausdorff, second countable, and étale

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 \'{e}tale.