Self-Similar k-Graph C*-Algebras

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