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

Abstract

Given a right LCM semigroup $S$ and a homomorphism $N:S\to [1,+\infty )$, we use the groupoid approach to study the KMS${}_{\beta }$-states on $C^{\ast }(S)$ 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 $C^{\ast }(S)$ and the tracial states on $C^{\ast }(\ker N)$.

26 pages; v3: minor changes, numbering changed for consistency with the published version; v2: a few examples added, minor corrections

References in corpus (5)

• Ground states of groupoid C*-algebras, phase transitions and arithmetic subalgebras for Hecke algebras
• KMS states on Nica-Toeplitz C*-algebras
• Equilibrium states and growth of quasi-lattice ordered monoids
• Symmetries of the KMS simplex
• $C^{\ast }$-algebras of right LCM monoids and their equilibrium states