A dichotomy for simple self-similar graph $C^\ast$-algebras

TL;DR

This paper establishes a clear dichotomy for simple self-similar graph $C^*$-algebras, showing they are either stably finite or purely infinite based on the presence of $G$-circuits, and extends results to self-similar $k$-graph $C^*$-algebras.

Contribution

It introduces a dichotomy for simple self-similar graph $C^*$-algebras and extends the analysis to self-similar $k$-graph $C^*$-algebras, linking properties to graph structures.

Findings

If no $G$-circuits, then $ ext{O}_{G,E}$ is stably finite.

Presence of $G$-circuits implies $ ext{O}_{G,E}$ is purely infinite.

For finite $ ext{O}_{G, ext{Lambda}}$, simplicity implies pure infiniteness.

Abstract

We investigate the pure infiniteness and stable finiteness of the Exel-Pardo $C^*$-algebras $\mathcal{O}_{G,E}$ for countable self-similar graphs $(G,E,\varphi)$. In particular, we associate a specific ordinary graph $\widetilde{E}$ to $(G,E,\varphi)$ such that some properties such as simpleness, stable finiteness or pure infiniteness of the graph $C^*$-algebra $C^*(\widetilde{E})$ imply that of $\mathcal{O}_{G,E}$. Among others, this follows a dichotomy for simple $\mathcal{O}_{G,E}$: if $(G,E,\varphi)$ contains no $G$-circuits, then $\mathcal{O}_{G,E}$ is stably finite; otherwise, $\mathcal{O}_{G,E}$ is purely infinite. Furthermore, Li and Yang recently introduced self-similar $k$-graph $C^*$-algebras $\mathcal{O}_{G,\Lambda}$. We also show that when $|\Lambda^0|<\infty$ and $\mathcal{O}_{G,\Lambda}$ is simple, then it is purely infinite.