Density of essential ideals and simplicity correspondence for ample groupoids

Determine whether, for every ample groupoid G, the essential ideal of the complex Steinberg algebra C G is dense in the essential ideal of the reduced C*-algebra C^*_r(G), and ascertain whether the simplicity of the Steinberg algebra C G coincides with the simplicity of the reduced C*-algebra C^*_r(G).

Background

The paper relates contracted inverse semigroup algebras and Steinberg algebras to reduced C*-algebras of ample groupoids via the tight groupoid construction. It provides criteria ensuring that, in the contracting self-similar groupoid setting, simplicity of the complex Steinberg algebra coincides with simplicity of the reduced C*-algebra.

However, beyond these special classes, the general relationship remains unsettled: it is unknown whether the essential ideal of the Steinberg algebra is dense in the essential ideal of the reduced C*-algebra, and whether simplicity of these two algebras always coincides for arbitrary ample groupoids. The authors note existing progress under additional finiteness conditions and then establish coincidence for the contracting self-similar groupoid case addressed in the paper.