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arXiv · 2501.11482

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

Eusebio Gardella·Volodymyr Nekrashevych·Benjamin Steinberg·Alina Vdovina
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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.

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Eusebio Gardella, Volodymyr Nekrashevych, Benjamin Steinberg, Alina Vdovina. 2025-01-20. Simplicity of $C^*$-algebras of contracting self-similar groups. https://arxiv.org/abs/2501.11482

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