Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Simplicity of algebras and $C^*$-algebras of self-similar groupoids

Josiah Aakre

Abstract

Many previously studied path algebras or self-similar group algebras may be viewed as Steinberg algebras of self-similar groupoids. By way of inverse semigroup algebras, we characterize when the Steinberg algebra of a self-similar groupoid is simple. We show that the simplicity of the reduced $C^*$-algebra of a contracting self-similar groupoid coincides with the simplicity of the Steinberg algebra. As an aside, we show that simplicity of the two algebras sometimes depends only on the skeleton of the self-similar groupoid acting on a strongly connected graph. Finally, we apply our methods to examples including a self-similar groupoid akin to multispinal self-similar groups and a self-similar groupoid built from the well-known Basilica group.

Simplicity of algebras and $C^*$-algebras of self-similar groupoids

Abstract

Many previously studied path algebras or self-similar group algebras may be viewed as Steinberg algebras of self-similar groupoids. By way of inverse semigroup algebras, we characterize when the Steinberg algebra of a self-similar groupoid is simple. We show that the simplicity of the reduced -algebra of a contracting self-similar groupoid coincides with the simplicity of the Steinberg algebra. As an aside, we show that simplicity of the two algebras sometimes depends only on the skeleton of the self-similar groupoid acting on a strongly connected graph. Finally, we apply our methods to examples including a self-similar groupoid akin to multispinal self-similar groups and a self-similar groupoid built from the well-known Basilica group.
Paper Structure (10 sections, 36 theorems, 64 equations, 8 figures)

This paper contains 10 sections, 36 theorems, 64 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Congruence-free associated inverse semigroups
  4. The tight and essential ideals of $KS_{(G,\Gamma^*)}$
  5. Collapsing self-similar groupoids
  6. Supporting $\mathcal{I}_{\textup{ess}}\setminus\mathcal{I}_{\textup{tight}}$ on the groupoid algebra.
  7. Supporting $\mathcal{I}_{\textup{ess}}\setminus\mathcal{I}_{\textup{tight}}$ when $G$ is contracting
  8. A method to determine simplicity
  9. Simplicity of ample groupoid $C^*$-algebras
  10. Examples

Key Result

Proposition 2.3

For all $g,h\in G$ with $\mathop{\operatorname{im}}\nolimits g = \mathop{\operatorname{dom}}\nolimits h$, and for all $p,q, w\in \Gamma^*$ with $\textbf{r}(p)=\textbf{s}(q)$, $pq\in \mathop{\operatorname{dom}}\nolimits g$, and $w\in \mathop{\operatorname{dom}}\nolimits g^{-1}$, we have

Figures (8)

  • Figure 1: The graphs $\Gamma$ and $\Gamma^*$.
  • Figure 2: Non-idempotent elements of $G$.
  • Figure 3: The graph $\Sigma$ of Example \ref{['Ex: example 3.10 of whittaker']}.
  • Figure 4: The Moore diagram (in black and red) and ${\mathcal{H}}$(in red) of the nucleus of $L$.
  • Figure 5: The graph $\Gamma$ of Example \ref{['Ex: collapsing a groupoid onto a group']}.
  • ...and 3 more figures

Theorems & Definitions (74)

  • Definition 2.1
  • Remark 2.2
  • Proposition 2.3: SSGroupoidsWhitaker
  • Definition 2.4
  • Remark 2.5
  • Definition 2.6
  • Definition 2.7
  • Theorem 2.8
  • Theorem 3.1
  • Proposition 3.2
  • ...and 64 more