Let K be a number field with ring of integers R. Given a modulus m for K and a group Γ of residues modulo m, we consider the semi-direct product R Rm,Γ obtained by restricting the multiplicative part of the full ax + b-semigroup over R to those algebraic integers whose residue modulo m lies in Γ, and we study the left regular C*-algebra of this semigroup. We give two presentations of this C*-algebra and realize it as a full corner in a crossed product C*-algebra. We also establish a faithfulness criterion for representations in terms of projections associated with ideal classes in a quotient of the ray class group modulo m, and we explicitly describe the primitive ideals using relations only involving the range projections of the generating isometries; this leads to an explicit description of the boundary quotient. Our results generalize and strengthen those of Cuntz, Deninger, and Laca and of Echterhoff and Laca for the C*-algebra of the full ax + b-semigroup. We conclude by showing that our construction is functorial in the appropriate sense; in particular, we prove that the left regular C*-algebra of R Rm,Γ embeds canonically into the left regular C*-algebra of the full ax + b-semigroup. Our methods rely heavily on Li's theory of semigroup C*-algebras.

Each multiplicative real-valued homomorphism on a quasi-lattice ordered monoid gives rise to a quasi-periodic dynamics on the associated Toeplitz C*-algebra; here we study the KMS equilibrium states of the resulting C*-dynamical system. We show that, under a nondegeneracy assumption on the homomorphism, there is a critical inverse temperature βc such that at each inverse temperature β ≥ βc there exists a unique KMS state. Strictly above βc, the KMS states are generalised Gibbs states with density operators determined by analytic extension to the upper half-plane of the unitaries implementing the dynamics. These are faithful Type I states. The critical value βc is the largest real pole of the partition function of the system and is related to the clique polynomial and skew-growth function of the monoid, relative to the degree map given by the logarithm of the multiplicative homomorphism. Motivated by the study of equilibrium states, we give a proof of the inversion formula for the growth series of a quasi-lattice ordered monoid in terms of the clique polynomial as in recent work of Albenque-Nadeau and McMullen for the finitely generated case, and in terms of the skew-growth series as in recent work of Saito. Specifically, we show that e −βc is the smallest pole of the growth series and thus is the smallest positive real root of the clique polynomial. We use this to show that equilibrium states in the subcritical range can only occur at inverse temperatures that correspond to roots of the clique polynomial in the interval (e −βc , 1), but we are not aware of any examples in which such roots exist.

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