Equilibrium states on right LCM semigroup C*-algebras

TL;DR

This paper characterizes equilibrium states in a broad class of right LCM semigroup C*-algebras, extending previous specific cases and introducing new results for algebraic dynamical systems.

Contribution

It provides a unified framework for analyzing equilibrium states on right LCM semigroup C*-algebras, covering many known cases and new classes.

Findings

Determined the structure of equilibrium states for a wide class of right LCM semigroup C*-algebras.

Extended previous case studies to new classes including algebraic dynamical systems.

Provided a general approach based on semigroup properties for analyzing equilibrium states.

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 large classes of right LCM semigroups, including those associated to algebraic dynamical systems.