KMS states on the crossed product $C^{*}$-algebra of a homeomorphism

TL;DR

This paper investigates KMS states on crossed product C*-algebras associated with homeomorphisms, revealing complex structures and complete characterization of inverse temperatures in simple cases, linking operator algebras with ergodic theory.

Contribution

It develops a detailed analysis of KMS states for flows on crossed product C*-algebras, connecting their structure to ergodic properties of non-singular transformations, and fully characterizes inverse temperature sets.

Findings

KMS states relate to ergodic theory of non-singular transformations

Structure of KMS states can be very rich and complex

Inverse temperature set is either {0} or all of R in simple cases

Abstract

Let $\varphi:X\to X$ be a homeomorphism of a compact metric space $X$. For any continuous function $F:X\to \mathbb{R}$ there is a one-parameter group $\alpha^{F}$ of automorphisms on the crossed product $C^*$-algebra $C(X)\rtimes_{\varphi}\mathbb{Z}$ defined such that $\alpha^{F}_{t}(fU)=fUe^{-itF}$ when $f \in C(X)$ and $U$ is the canonical unitary in the construction of the crossed product. In this paper we study the KMS states for these flows by developing an intimate relation to the ergodic theory of non-singular transformations and show that the structure of KMS-states can be very rich and complicated. Our results are complete concerning the set of possible inverse temperatures; in particular, we show that when $C(X) \rtimes_{\phi} \mathbb Z$ is simple this set is either $\{0\}$ or the whole line $\mathbb R$.