KMS states on finite-graph C*-algebras

TL;DR

This paper investigates KMS states on finite-graph C*-algebras with sinks and sources, revealing a correspondence between sinks and extremal KMS states at high inverse temperatures, and extends existing theories to non-injective bimodule actions.

Contribution

It generalizes the construction of KMS states on Cuntz-Pimsner algebras to cases with non-injective left actions, connecting graph sinks to rational function branched points.

Findings

Extreme KMS states are parametrized by graph sinks at high inverse temperature.

Graph sinks correspond to branched points of rational functions in the KMS state framework.

The generalization applies to non-injective bimodule actions in Cuntz-Pimsner algebras.

Abstract

We study KMS states on finite-graph C*-algebras with sinks and sources. We compare finite-graph C*-algebras with C*-algebras associated with complex dynamical systems of rational functions. We show that if the inverse temperature $\beta$ is large, then the set of extreme $\beta$-KMS states is parametrized by the set of sinks of the graph. This means that the sinks of a graph correspond to the branched points of a rational funcition from the point of KMS states. Since we consider graphs with sinks and sources, left actions of the associated bimodules are not injective. Then the associated graph C*-algebras are realized as (relative) Cuntz-Pimsner algebras studied by Katsura. We need to generalize Laca-Neshevyev's theorem of the construction of KMS states on Cuntz-Pimsner algebras to the case that left actions of bimodules are not injective.