# Equilibrium states on right LCM semigroup C^{*}-algebras

###### Abstract.

We determine the structure of equilibrium states for a natural dynamics on the boundary quotient diagram of C^{*}-algebras for a large class of right LCM semigroups. The approach is based on abstract properties of the semigroup and covers the previous case studies on \mathbb{N}\rtimes\mathbb{N}^{\times}, dilation matrices, self-similar actions, and Baumslag–Solitar monoids. At the same time, it provides new results for right LCM semigroups associated to algebraic dynamical systems.

## 1. Introduction

Equilibrium states have been studied in operator algebras starting with quantum systems modelling ensembles of particles, and have been the means of describing properties of physical models with C^{*}-algebraic tools, cf. [BRII]. Specifically, given a quantum statistical mechanical system, which is a pair formed of a C^{*}-algebra A that encodes the observables and a one-parameter group of automorphisms \sigma of A interpreted as a time evolution, one seeks to express equilibrium of the system via states with specific properties. A \operatorname{KMS}_{\beta}-state for (A,\sigma) is a state on A that satisfies the \operatorname{KMS}_{\beta}-condition (for Kubo-Martin-Schwinger) at a real parameter \beta bearing the significance of an inverse temperature. Gradually, it has become apparent that the study of KMS-states for systems that do not necessarily have physical origins brings valuable insight into the structure of the underlying C^{*}-algebra, and uncovers new directions of interplay between the theory of C^{*}-algebras and other fields of mathematics. A rich supply of examples is by now present in the literature, see [BC, LR2, LRRW, CDL, Nes], to mention only some.

This article is concerned with the study of KMS-states for systems whose underlying C^{*}-algebras are Toeplitz-type algebras modelling a large class of semigroups. While our initial motivation was to classify KMS-states for a specific class of examples, as had been done previously, we felt that by building upon the deep insight developed in the case-studies already present in the literature, the time was ripe for proposing a general framework that would encompass our motivating example and cover all these case-studies. We believe it is a strength of our approach that we can identify simply phrased conditions at the level of the semigroup which govern the structure of \operatorname{KMS}-states, including uniqueness of the \operatorname{KMS}_{\beta}-state for \beta in the critical interval. In all the examples we treat, these conditions admit natural interpretations. Moreover, they are often subject to feasible verification.

To be more explicit, the interest in studying KMS-states on Toeplitz-type C^{*}-algebras gained new momentum with the work of Laca and Raeburn [LR2] on the Toeplitz algebra of the ax+b-semigroup over the natural numbers \mathbb{N}\rtimes\mathbb{N}^{\times}. Further results on the KMS-state structure of Toeplitz-type C^{*}-algebras include the work on Exel crossed products associated to dilation matrices [LRR], C^{*}-algebras associated to self-similar actions [LRRW], and C^{*}-algebras associated to Baumslag–Solitar monoids [CaHR]. In all these cases the relevant Toeplitz-type C^{*}-algebra can be viewed as a semigroup C^{*}-algebra in the sense of [Li1]. Further, the semigroups in question are right LCM semigroups, see [BLS1] for dilation matrices, and [BRRW] for all the other cases. In [BLS1, BLS2], the last three named authors have studied C^{*}-algebras of right LCM semigroups associated to algebraic dynamical systems. These C^{*}-algebras often admit a natural dynamics that is built from the underlying algebraic dynamical system. The original motivation for our work was to classify KMS-states in this context.

Although there is a common thread in the methods and techniques used to prove the KMS-classification results in [LR2, LRR, LRRW, CaHR], one cannot speak of a single proof that runs with adaptations. Indeed, in each of these papers a careful and rather involved analysis is carried out using concrete properties of the respective setup. One main achievement of this paper is a general theory of KMS-classification which both unifies the classification results from [LR2, LRR, LRRW, CaHR], and also considerably enlarges the class of semigroup C^{*}-algebras for which KMS-classification is new and interesting, see Theorem LABEL:thm:KMS_results-gen and Section LABEL:sec:examples.

We expect that our work may build a bridge to the analysis of KMS-states via groupoid models and the result of Neshveyev [Nes]*Theorem 1.3. At this point we would also like to mention the recent analysis of equilibrium states on C^{*}-algebras associated to self-similar actions of groupoids on graphs that generalises [LRRW], see [LRRW1]. It would be interesting to explore connections to this line of development.

One of the keys to establishing our general theory is the insight that it pays off to work with the boundary quotient diagram for right LCM semigroups proposed by the fourth-named author in [Sta3]. The motivating example for this diagram was first considered in [BaHLR], where it was shown to give extra insight into the structure of \operatorname{KMS}-states on \mathcal{T}(\mathbb{N}\rtimes\mathbb{N}^{\times}). Especially the core subsemigroup S_{c}, first introduced in [Star], and motivated by the quasi-lattice ordered situation in [CrispLaca], and the core irreducible elements S_{ci} introduced in [Sta3] turn out to be central for our work. Recall that S_{c} consists of all the elements whose principal right ideal intersects all other principal right ideals and S_{ci} of all s\in S\setminus S_{c} such that every factorisation s=ta with a\in S_{c} forces a to be invertible in S. Via so-called (accurate) foundation sets, S_{c} and S_{ci} give rise to quotients \mathcal{Q}_{c}(S) and \mathcal{Q}_{p}(S) of C^{*}(S) of S, respectively. Together with C^{*}(S) and the boundary quotient \mathcal{Q}(S) from [BRRW], they form a commutative diagram:

(1.1) | \begin{gathered}\displaystyle\xymatrix@=17mm{C^{*}(S){}{}{}{}{}{}{}{}{}{}% \xy@@ix@{{\hbox{}\end{gathered}}}} |