Simplicity of $C^*$-algebras of contracting self-similar groups

TL;DR

This paper establishes a criterion linking the simplicity of $C^*$-algebras and complex $ ext{ast}$-algebras for contracting self-similar groups, and improves algorithms to determine algebraic simplicity.

Contribution

It proves the equivalence of simplicity between $C^*$-algebras and complex $ ext{ast}$-algebras for these groups and enhances existing algorithms for simplicity determination.

Findings

Simplicity of $C^*$-algebra iff complex $ ext{ast}$-algebra is simple

Improved algorithm for testing algebraic simplicity

Identifies a class of non-Hausdorff amenable groupoids with simple algebras

Abstract

We show that the $C^*$-algebra associated by Nekrashevych to a contracting self-similar group is simple if and only if the corresponding complex $\ast$-algebra is simple. We also improve on Steinberg and Szaka\'c's algorithm to determine if the $\ast$-algebra is simple. This provides an interesting class of non-Hausdorff amenable, effective and minimal ample groupoids for which simplicity of the $C^*$-algebra and the complex $\ast$-algebra are equivalent.