Equilibrium states for right LCM semigroup $C^{\ast }$-algebras

Zahra Afsar, Nathan Brownlowe, Nadia S. Larsen, Nicolai Stammeier

Abstract

We determine the structure of equilibrium states for a natural dynamics on the boundary quotient diagram of $C^{\ast }$-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 $\mathbb{N}⋊{\mathbb{N}}^{\times }$, 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.