Simplicity of $C^{\ast }$-algebras of contracting self-similar groups

Abstract

We show that the $C^{\ast }$-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 Szakać'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 $C^{\ast }$-algebra and the complex $\ast$-algebra are equivalent.

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