Abstract. Kumjian and Pask introduced an aperiodicity condition for higher rank graphs. We present a detailed analysis of when this occurs in certain rank 2 graphs. When the algebra is aperiodic, we give another proof of the simplicity of C * (F + θ ). The periodic C * -algebras are characterized, and it is shown that C

We provide a detailed analysis of atomic * -representations of rank 2 graphs on a single vertex. They are completely classified up to unitary equivalence, and decomposed into a direct sum or direct integral of irreducible atomic representations. The building blocks are described as the minimal * -dilations of defect free representations modelled on finite groups of rank 2.

In this paper, we initiate the study of endomorphisms and modular theory of the graph C*-algebras O θ of a 2-graph F + θ on a single vertex. We prove that there is a semigroup isomorphism between unital endomorphisms of O θ and its unitary pairs with a twisted property. We characterize when endomorphisms preserve the fixed point algebra F of the gauge automorphisms and its canonical masa D. Some other properties of endomorphisms are also investigated.As far as the modular theory of O θ is concerned, we show that the algebraic *-algebra generated by the generators of O θ with the inner product induced from a distinguished state ω is a modular Hilbert algebra. Consequently, we obtain that the von Neumann algebra π(O θ ) ′′ generated by the GNS representation of ω is an AFD factor of type III 1 , provided ln m ln n ∈ Q. Here m, n are the numbers of generators of F + θ of degree (1, 0) and (0, 1), respectively. This work is a continuation of [11,12] by Davidson-Power-Yang and [13] by Davidson-Yang.2000 Mathematics Subject Classification. 46L05, 46L37, 46L10, 46L40.

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.

In this paper we study characteristic theory of polynomial-like iterative equations in the general case where multiple characteristic roots may be involved. We describe such an equation in linear difference form, give relations between the characteristic solutions and other solutions, and give a construction of all continuous solutions in some cases. We also show that the equation has no continuous real solutions, if it has no real characteristic roots.