Self-similark-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$^\ast $-algebra ${{\mathcal{O}}}_{G,\Lambda }$. We prove that ${{\mathcal{O}}}_{G,\Lambda }$ can be realized as the Cuntz–Pimsner algebra of a product system. If $G$ is amenable and the action is pseudo free, then ${{\mathcal{O}}}_{G,\Lambda }$ is shown to be isomorphic to a “path-like” groupoid C$^\ast $-algebra. This facilitates studying the properties of ${{\mathcal{O}}}_{G,\Lambda }$. We show that ${{\mathcal{O}}}_{G,\Lambda }$ is always nuclear and satisfies the universal coefficient theorem; we characterize the simplicity of ${{\mathcal{O}}}_{G,\Lambda }$ in terms of the underlying action, and we prove that, whenever ${{\mathcal{O}}}_{G,\Lambda }$ 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.

KMS states of self-similar k-graph C*-algebras

“…The reason why it is inevitable to involve self-similar P -graph theory here is simply because taking quotient is not closed for self-similar k-graph C*-algebras, but lies in the realm of self-similar P -graph C*-algebras. When P = N k and Λ 0 is finite, the definition of self-similar P -graph C*-algebras in this paper coincides with the one given in [20]. We should also mention that, very recently, Exel, Pardo, and Starling in [10] provide a definition for self-similar directed graph C*-algebras by using a different approach.…”

“…Self-similar P -graph C*-algebras. In [20], we associated a C*-algebra to each self-similar k-graph whose underlying k-graph has finite vertices. In this subsection, we generalize it in two ways: (i) the underlying k-graph is replaced by a P -graph, and (ii) the vertex set of the P -graph is not required to be finite.…”

“…Let (G, Λ) be a self-similar P -graph. Completely similar to [20,Section 4], we associate a product system to a self-similar P -graph (G, Λ). In what follows, we only sketch the main ideas behind.…”

“…Let Λ be a row-finite source-free k-graph with finite vertices, and G be a (countable discrete) group acting on Λ. If the action is self-similar, then we associated (G, Λ) a universal C*-algebra O G,Λ which is called the self-similar k-graph C*-algebra of (G, Λ) ( [20]). Roughly speaking, O G,Λ is generated by a universal pair {u, s} of representations, where u is a unitary representation of G in O G,Λ and s is a Cuntz-Krieger representation of Λ in O G,Λ such that u and s are compatible with respect to the underlying self-similar action of (G, Λ).…”

“…It turns out that our C*-algebras in the case of P = N are isomorphic to theirs. For self-similar k-graph C*-algebras, we generalize a series of structure theorems from the unital case proved in [20] to the nonunital case. In Section 4, we prove that for any self-similar k-graph there is a one-to-one correspondence between the set of all G-hereditary and G-saturated subsets of Λ 0 and the set of its all gauge-invariant and diagonal-invariant ideals of the self-similar k-graph C*-algebra.…”