Self-Similar $k$-Graph C*-Algebras

Abstract

In this paper, we introduce a notion of a self-similar action of a group $G$ on a $k$-graph $\Lambda$, and associate it a universal C*-algebra ${Ø}_{G,\Lambda }$. We prove that ${Ø}_{G,\Lambda }$ can be realized as the Cuntz-Pimsner algebra of a product system. If $G$ is amenable and the action is pseudo free, then ${Ø}_{G,\Lambda }$ is shown to be isomorphic to a "path-like" groupoid C*-algebra. This facilitates studying the properties of ${Ø}_{G,\Lambda }$. We show that ${Ø}_{G,\Lambda }$ is always nuclear and satisfies the Universal Coefficient Theorem; we characterize the simplicity of ${Ø}_{G,\Lambda }$ in terms of the underlying action; and we prove that, whenever ${Ø}_{G,\Lambda }$ is simple, there is a dichotomy: it is either stably finite or purely infinite, depending on whether $\Lambda$ has nonzero graph traces or not. Our main results generalize the recent work of Exel and Pardo on self-similar graphs.

28 pages; minor changes

References in corpus (2)

• C*-algebras associated to product systems of Hilbert bimodules
• A dichotomy for groupoid C*-algebras

Cited by in corpus (1)

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