#### Document Type

Journal Article

#### Abstract

We show that the group QxQ_{+}* of orientation-preserving affine transformations of the rational numbers is quasi-lattice ordered by its subsemigroup NxN^{x}—. The associated Toeplitz C*-algebra T(NxN^{x}) is universal for isometric representations which are covariant in the sense of Nica. We give a presentation of T(NxN^{x}) in terms of generators and relations, and use this to show that the C*-algebra Q_{N }recently introduced by Cuntz is the boundary quotient of (QxQ_{+}*, NxN^{x}) in the sense of Crisp and Laca. The Toeplitz algebra T(NxN^{x}) carries a natural dynamics σ, which induces the one considered by Cuntz on the quotient Q_{N}, and our main result is the computation of the KMS_{β} (equilibrium) states of the dynamical system (T(NxN^{x}),R,σ) for all values of the inverse temperature β. For β ∈[1,2] there is a unique KMS_{β} state, and the KMS_{1} state factors through the quotient map onto Q_{N}, giving the unique KMS state discovered by Cuntz. At β=2 there is a phase transition, and for β>2 the KMS_{β} states are indexed by probability measures on the circle. There is a further phase transition at β=∞, where the KMS_{∞ }states are indexed by the probability measures on the circle, but the ground states are indexed by the states on the classical Toeplitz algebra T(N).

#### RIS ID

36765