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Journal of Operator Theory

Volume 75, Issue 1, Winter 2016  pp. 119-138.

KMS states for quasi-free actions on finite-graph algebras

Authors:  Christopher Chlebovec
Author institution: UNB, Fredericton, NB, E3B 5A3, Canada

Summary: Given a graph $E$ and a labeling map $\omega$, we consider the quasi-free action $\alpha^{\omega}$ of $\mathbb{R}$ on the graph algebra $C^{\ast}(E)$. For a finite graph $E$, we give a complete characterization of all $\text{KMS}_{\beta}$ states of a graph algebra in terms of a polyhedral set in $\mathbb{R}^{E^0}$. This characterization allows us to generalize the results of an Huef, Laca, Raeburn, and Sims. We make an explicit construction of all $\text{KMS}_{\beta}$ states for $\beta$ above a critical inverse temperature $\beta_\mathrm c,$ as well as a precise description of the KMS states for graphs with a certain strongly connected subgraph. In addition, we find a correspondence between the $\text{KMS}$ states of a graph algebra and its dual-graph algebra when $E$ is a row-finite graph with no sinks.

Keywords: KMS states, graph algebras, quasi-free actions, $ C^{\ast}$-dynamical systems

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