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Register a new userSelf-similar graphs, a unified treatment of Katsura and Nekrashevych C*-algebras
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Given a graph $E$, an action of a group $G$ on $E$, and a $G$-valued cocycle $phi$ on the edges of $E$, we define a C*-algebra denoted ${cal O}_{G,E}$, which is shown to be isomorphic to the tight C*-algebra associated to a certain inverse semigroup $S_{G,E}$ built naturally from the triple $(G,E,phi)$. As a tight C*-algebra, ${cal O}_{G,E}$ is also isomorphic to the full C*-algebra of a naturally occurring groupoid ${cal G}_{tight}(S_{G,E})$. We then study the relationship between properties of the action, of the groupoid and of the C*-algebra, with an emphasis on situations in which ${cal O}_{G,E}$ is a Kirchberg algebra. Our main applications are to Katsura algebras and to certain algebras constructed by Nekrashevych from self-similar groups. These two classes of C*-algebras are shown to be special cases of our ${cal O}_{G,E}$, and many of their known properties are shown to follow from our general theory.
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Given a self-similar $K$ set defined from an iterated function system $Gamma=(gamma_1,ldots,gamma_n)$ and a set of function $H={h_i:Ktomathbb{R}}_{i=1}^d$ satisfying suitable conditions, we define a generalized gauge action on Kawjiwara-Watatani algebras $mathcal{O}_Gamma$ and their Toeplitz extensions $mathcal{T}_Gamma$. We then characterize the KMS states for this action. For each $betain(0,infty)$, there is a Ruelle operator $mathcal{L}_{H,beta}$ and the existence of KMS states at inverse temperature $beta$ is related to this operator. The critical inverse temperature $beta_c$ is such that $mathcal{L}_{H,beta_c}$ has spectral radius 1. If $beta<beta_c$, there are no KMS states on $mathcal{O}_Gamma$ and $mathcal{T}_Gamma$; if $beta=beta_c$, there is a unique KMS state on $mathcal{O}_Gamma$ and $mathcal{T}_Gamma$ which is given by the eigenmeasure of $mathcal{L}_{H,beta_c}$; and if $beta>beta_c$, including $beta=infty$, the extreme points of the set of KMS states on $mathcal{T}_Gamma$ are parametrized by the elements of $K$ and on $mathcal{O}_Gamma$ by the set of branched points.
To a large class of graphs of groups we associate a C*-algebra universal for generators and relations. We show that this C*-algebra is stably isomorphic to the crossed product induced from the action of the fundamental group of the graph of groups on the boundary of its Bass-Serre tree. We characterise when this action is minimal, and find a sufficient condition under which it is locally contractive. In the case of generalised Baumslag-Solitar graphs of groups (graphs of groups in which every group is infinite cyclic) we also characterise topological freeness of this action. We are then able to establish a dichotomy for simple C*-algebras associated to generalised Baumslag-Solitar graphs of groups: they are either a Kirchberg algebra, or a stable Bunce-Deddens algebra.
We introduce P-graphs, which are generalisations of directed graphs in which paths have a degree in a semigroup P rather than a length in N. We focus on semigroups P arising as part of a quasi-lattice ordered group (G,P) in the sense of Nica, and on P-graphs which are finitely aligned in the sense of Raeburn and Sims. We show that each finitely aligned P-graph admits a C*-algebra C*_{min}(Lambda) which is co-universal for partial-isometric representations of Lambda which admit a coaction of G compatible with the P-valued length function. We also characterise when a homomorphism induced by the co-universal property is injective. Our results combined with those of Spielberg show that every Kirchberg algebra is Morita equivalent C*_{min}(Lambda) for some (N^2 * N)-graph Lambda.
We study the structure and compute the stable rank of C*-algebras of finite higher-rank graphs. We completely determine the stable rank of the C*-algebra when the k-graph either contains no cycle with an entrance, or is cofinal. We also determine exactly which finite, locally convex k-graphs yield unital stably finite C*-algebras. We give several examples to illustrate our results.
In this paper we show that every non-cycle finite transitive directed graph has a Cuntz-Krieger family whose WOT-closed algebra is $B(mathcal{H})$. This is accomplished through a new construction that reduces this problem to in-degree $2$-regular graphs, which is then treated by applying the periodic Road Coloring Theorem of Beal and Perrin. As a consequence we show that finite disjoint unions of finite transitive directed graphs are exactly those finite graphs which admit self-adjoint free semigroupoid algebras.
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