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We construct a locally compact groupoid with the properties in the title. Ourexample is based closely on constructions used by Higson, Lafforgue, andSkandalis in their work on counterexamples to the Baum-Connes conjecture. It isa bundle of countable groups over the one point compactification of the naturalnumbers, and is Hausdorff, second countable and \'{e}tale.

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