A characterization of simplicity of reduced groupoid C*-algebras

Abstract

We show that, for a second-countable locally compact Hausdorff \'etale minimal groupoid with compact unit space, simplicity of the reduced groupoid C*-algebra implies the existence of a comeager set of unit points with C*-simple isotropy group. Combining this result with work of Christensen and Neshveyev on exotic completions of isotropy group algebras, we show that the converse implication is also true. Finally, we construct a Hausdorff \'etale minimal groupoid with an isotropy group whose induced exotic completion differs from its reduced group C*-algebra, answering a question of Christensen and Neshveyev.