A dichotomy for simple self-similar graph C⁎-algebras

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We investigate the pure infiniteness and stable finiteness of the Exel-Pardo C⁎-algebras OG,E for countable self-similar graphs (G,E,φ). In particular, we associate a specific ordinary graph E˜ to (G,E,φ) such that some properties such as simpleness, stable finiteness or pure infiniteness of the graph C⁎-algebra C⁎(E˜) imply that of OG,E. Among others, this follows a dichotomy for simple OG,E: if (G,E,φ) contains no G-circuits, then OG,E is stably finite; otherwise, OG,E is purely infinite. Furthermore, Li and Yang recently introduced self-similar k-graph C⁎-algebras OG,Λ. We also show that when |Λ0|<∞ and OG,Λ is simple, then it is purely infinite.