Equilibrium states on graph algebras

TL;DR

This paper classifies equilibrium (KMS) states for dynamical systems on graph algebras, providing a comprehensive understanding of their structure and temperature ranges, with applications to related operator algebras.

Contribution

It offers a complete classification of KMS states on graph algebras and demonstrates the sharpness of temperature bounds for their existence.

Findings

Complete classification of KMS states on graph algebras.

Thomsen's temperature bounds are proven to be sharp.

Applications to Cuntz-Pimsner algebras and local homeomorphisms.

Abstract

We consider operator-algebraic dynamical systems given by actions of the real line on unital $C^*$-algebras, and especially the equilibrium states (or KMS states) of such systems. We are particularly interested in systems built from the gauge action on the Toeplitz algebra and graph algebra of a finite directed graph, and we describe a complete classification of the KMS states obtained in joint work with Laca and Sims. We then discuss applications of these results to Cuntz-Pimsner algebras associated to local homeomorphisms, obtained in collaboration with Afsar. Thomsen has given bounds on the range of inverse temperatures at which KMS states may exist. We show that Thomsen's bounds are sharp.