The groupoid approach to equilibrium states on right LCM semigroup C∗$^*$‐algebras

Given a right LCM semigroup S $S$ and homomorphism N : → [ 1 , + ∞ ) $N\colon S\rightarrow [1,+\infty )$ we use the groupoid approach to study KMS β $_\beta$ -states on C ∗ ( $C^*(S)$ with respect dynamics induced by $N$ . We establish necessary sufficient conditions for existence uniqueness of -states. As an application, show that condition obtained so-called generalized scales is as well. Our most complete results are inverse temperatures $\beta$ at which ζ $\zeta$ -function finite. In this case, get explicit bijective correspondence between tracial states ker $C^*(\operatorname{ker}N)$