Simplicity of C*-Algebras of Contracting Self-Similar Groups

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 \(*\)-algebra is simple. We also improve on Steinberg and Szakács’s algorithm to determine if the \(*\)-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 \(*\)-algebra are equivalent.