The groupoid approach to equilibrium states on right LCM semigroup C*-algebras
Sergey Neshveyev, Nicolai Stammeier
TL;DR
The paper develops a groupoid framework to study equilibrium (KMS) states on right LCM semigroup C$^*$-algebras $C^*(S)$ with dynamics induced by a scale $N$. It shows that, under mild hypotheses, KMS$_eta$-states are governed by quasi-invariant measures on the unit space and fields of isotropy traces, and concentrates the analysis on the kernel $C^*( extrm{ker} olimits N)$. In the finite-type regime $igl( ext{finite }eta ext{-zeta function }igr)$, there is a bijection between KMS$_eta$-states and tracial states on $C^*( extrm{ker} olimits N)$ with an explicit formula for the states; the paper also provides a sharp combinatorial criterion for the uniqueness of KMS$_eta$-states, showing that generalized-scale results are both necessary and sufficient, and discusses the boundary quotient and simplicity implications. Overall, the work extends the groupoid approach beyond restrictive assumptions, clarifies the role of $ extrm{ker} olimits N$, and yields explicit classifications of KMS states in a broad class of semigroup C$^*$-algebras.
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$_β$-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$_β$-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 $β$ at which the $ζ$-function of $N$ is finite. In this case we get an explicit bijective correspondence between the KMS$_β$-states on $C^*(S)$ and the tracial states on $C^*(\operatorname{ker} N)$.