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

Abstract

We investigate the pure infiniteness and stable finiteness of the Exel-Pardo $C^{\ast }$-algebras ${\mathcal{O}}_{G,E}$ for countable self-similar graphs $(G,E,\phi )$. In particular, we associate a specific ordinary graph $\tilde{E}$ to $(G,E,\phi )$ such that some properties such as simpleness, stable finiteness or pure infiniteness of the graph $C^{\ast }$-algebra $C^{\ast }(\tilde{E})$ imply that of ${\mathcal{O}}_{G,E}$. Among others, this follows a dichotomy for simple ${\mathcal{O}}_{G,E}$: if $(G,E,\phi )$ 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 $C^{\ast }$-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.

17 pages

References in corpus (2)

• Purely infinite $C^{\ast }$-algebras associated to étale groupoids
• Self-Similar $k$-Graph C*-Algebras

Cited by in corpus (1)

• On groupoids and $C^{\ast }$-algebras from self-similar actions