Symmetries of the KMS simplex

TL;DR

This paper studies the structure and symmetries of KMS states in $C^*$-dynamical systems arising from groupoid homomorphisms, showing they form simplices with transitive abelian group actions, and applies this to higher rank graph algebras.

Contribution

It proves that the set of KMS states forms a simplex with a transitive abelian symmetry group for a broad class of groupoids, and describes these states for higher rank graph algebras.

Findings

$ ext{KMS states form a simplex}$

$ ext{An abelian group acts transitively on extremal KMS states}$

$ ext{Application to higher rank graph Cuntz-Krieger algebras}$

Abstract

A continuous groupoid homomorphism $c$ on a locally compact second countable Hausdorff \'etale groupoid $\mathcal{G}$ gives rise to a $C^{*}$-dynamical system in which every $\beta$-KMS state can be associated to a $e^{-\beta c}$-quasi-invariant measure $\mu$ on $\mathcal{G}^{(0)}$. Letting $\Delta_{\mu}$ denote the set of KMS states associated to such a $\mu$, we will prove that $\Delta_{\mu}$ is a simplex for a large class of groupoids, and we will show that there is an abelian group that acts transitively and freely on the extremal points of $\Delta_{\mu}$. This group can be described using the support of $\mu$, so our theory of symmetries can be used to obtain a description of all KMS states by describing the $e^{-\beta c}$-quasi-invariant measures. To illustrate this we will describe the KMS states for the Cuntz-Krieger algebras of all finite higher rank graphs without sources and a large class of continuous one-parameter groups.