The groupoid approach to equilibrium states on right LCM semigroup C*-algebras

TL;DR

This paper uses the groupoid approach to analyze equilibrium states (KMS states) on C*-algebras associated with right LCM semigroups, providing conditions for their existence and uniqueness, especially at inverse temperatures where the zeta function is finite.

Contribution

It establishes necessary and sufficient conditions for the existence and uniqueness of KMS states on these C*-algebras, extending previous results for generalized scales.

Findings

Conditions for existence and uniqueness of KMS states

Explicit correspondence between KMS states and tracial states at finite zeta function

Necessity of the sufficient condition for generalized scales

Abstract

Given a right LCM semigroup $S$ and a homomorphism $N\colon S\to[1,+\infty)$, we use the groupoid approach to study the KMS$_\beta$-states on $C^*(S)$ with respect to the dynamics induced by $N$. We establish necessary and sufficient conditions for the existence and uniqueness of KMS$_\beta$-states. As an application, we show that the sufficient condition for the uniqueness obtained for so-called generalized scales is necessary as well. Our most complete results are obtained for inverse temperatures $\beta$ at which the $\zeta$-function of $N$ is finite. In this case we get an explicit bijective correspondence between the KMS$_\beta$-states on $C^*(S)$ and the tracial states on $C^*(\operatorname{ker} N)$.