Symmetries of the KMS simplex

Abstract

A continuous groupoid homomorphism $c$ on a locally compact second countable Hausdorff étale groupoid $\mathcal{G}$ gives rise to a $C^{\ast }$-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.

25 pages. Some typos and a paragraph in the introduction have been corrected. This is a pre-print of an article to appear in Communications in Mathematical Physics

References in corpus (1)

• Simplicity of C*-algebras associated to higher-rank graphs

Cited by in corpus (3)

• KMS states on Nica-Toeplitz C*-algebras
• The groupoid approach to equilibrium states on right LCM semigroup C*-algebras
• KMS states on the crossed product $C^{\ast }$-algebra of a homeomorphism