Simplicity of algebras and $C^*$-algebras of self-similar groupoids

TL;DR

This paper characterizes when the Steinberg algebra of a self-similar groupoid is simple, linking it to the simplicity of the associated reduced $C^*$-algebra, with applications to notable examples.

Contribution

It provides a new characterization of simplicity for Steinberg algebras and reduced $C^*$-algebras of self-similar groupoids, connecting algebraic and $C^*$-algebraic properties.

Findings

Simplicity of Steinberg algebra characterized via inverse semigroup algebras.

Simplicity of reduced $C^*$-algebra coincides with Steinberg algebra simplicity.

Simplicity depends on the skeleton of the groupoid in certain cases.

Abstract

Many previously studied path algebras or self-similar group algebras may be viewed as Steinberg algebras of self-similar groupoids. By way of inverse semigroup algebras, we characterize when the Steinberg algebra of a self-similar groupoid is simple. We show that the simplicity of the reduced $C^*$-algebra of a contracting self-similar groupoid coincides with the simplicity of the Steinberg algebra. As an aside, we show that simplicity of the two algebras sometimes depends only on the skeleton of the self-similar groupoid acting on a strongly connected graph. Finally, we apply our methods to examples including a self-similar groupoid akin to multispinal self-similar groups and a self-similar groupoid built from the well-known Basilica group.