Simplicity of algebras associated to non-Hausdorff groupoids

TL;DR

This paper characterizes simplicity and proves uniqueness theorems for Steinberg algebras and C*-algebras associated with non-Hausdorff groupoids, with applications to inverse semigroups, graph actions, and self-similar groups.

Contribution

It provides new criteria for simplicity and uniqueness in non-Hausdorff groupoid algebras, extending previous results to broader settings.

Findings

C*-algebra and complex Steinberg algebra of the Grigorchuk group's self-similar action are simple.

Steinberg algebra over Z_2 coefficients is not simple.

Results apply to inverse semigroup representations and group actions on graphs.

Abstract

We prove a uniqueness theorem and give a characterization of simplicity for Steinberg algebras associated to non-Hausdorff ample groupoids. We also prove a uniqueness theorem and give a characterization of simplicity for the C*-algebra associated to non-Hausdorff \'etale groupoids. Then we show how our results apply in the setting of tight representations of inverse semigroups, groups acting on graphs, and self-similar actions. In particular, we show that C*-algebra and the complex Steinberg algebra of the self-similar action of the Grigorchuk group are simple but the Steinberg algebra with coefficients in $\mathbb{Z}_2$ is not simple.