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

Eusebio Gardella, Volodymyr Nekrashevych, Benjamin Steinberg, Alina Vdovina

TL;DR

The paper addresses when Nekrashevych’s C*-algebras of contracting self-similar groups are simple and proves that simplicity of the C*-algebra $\mathcal{O}(G,\mathsf{X})$ is equivalent to simplicity of the complex *-algebra ${\mathbb{C}}(G,\mathsf{X})$ for contracting groups. It forges this link via the theory of ample groupoids, graded étale groupoids, and the essential ideals $I_{ m ess}$ and $I_{ m ess}^{\rm alg}$, showing that both forms of simplicity are controlled by whether ${\mathbb{C}}\mathcal{N}\cap I_{ m ess}^{\rm alg}(G,\mathsf{X})$ vanishes. The authors introduce a refined, exponential-time algorithm for deciding algebraic simplicity by exploiting finite subgroups $H_w$ of the nucleus and a graph-based test, improving the previous Bell-number bound. They illustrate the framework with classical contracting groups (e.g., Grigorchuk) and variants (Grigorchuk-Erschler), clarifying how simplicity depends on field characteristics and advancing practical decision procedures for non-Hausdorff amenable ample groupoids and their self-similar C*-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 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 $C^*$-algebra and the complex $\ast$-algebra are equivalent.

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

TL;DR

The paper addresses when Nekrashevych’s C*-algebras of contracting self-similar groups are simple and proves that simplicity of the C*-algebra is equivalent to simplicity of the complex *-algebra for contracting groups. It forges this link via the theory of ample groupoids, graded étale groupoids, and the essential ideals and , showing that both forms of simplicity are controlled by whether vanishes. The authors introduce a refined, exponential-time algorithm for deciding algebraic simplicity by exploiting finite subgroups of the nucleus and a graph-based test, improving the previous Bell-number bound. They illustrate the framework with classical contracting groups (e.g., Grigorchuk) and variants (Grigorchuk-Erschler), clarifying how simplicity depends on field characteristics and advancing practical decision procedures for non-Hausdorff amenable ample groupoids and their self-similar C*-algebras.

Abstract

We show that the -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 Szakać'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 -algebra and the complex -algebra are equivalent.
Paper Structure (6 sections, 13 theorems, 19 equations, 2 figures)

This paper contains 6 sections, 13 theorems, 19 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Ample groupoids and associated algebras
  3. Graded étale groupoids
  4. Preliminaries on contracting self-similar groups
  5. Simplicity of ${\mathcal{O}}(G,{\mathsf{X}})$ is equivalent to simplicity of ${\mathbb{C}}(G,{\mathsf{X}})$
  6. A simplification of the simplicity algorithm

Key Result

Lemma 3.1

Let $\mathcal{G}$ be a graded groupoid with cocycle $c$, let $a\in C^*(\mathcal{G})$, let $x\in\mathcal{G}^{(0)}$, let $\gamma_1,\gamma_2\in \mathcal{G}x$ and let $z\in{\mathbb{T}}$. Then for all $\gamma_1,\gamma_2\in\mathcal{G}x$.

Figures (2)

  • Figure 1: $\mathcal{H}$ and $\Delta$ for the Grigorchuk group
  • Figure 2: $\mathcal{H}$ and $\Delta$ for the Grigorchuk-Erschler group

Theorems & Definitions (37)

  • Definition 1
  • Definition 2
  • Definition 3
  • Remark 1
  • Definition 4
  • Lemma 3.1
  • proof
  • Proposition 1
  • proof
  • Lemma 3.2
  • ...and 27 more