#### RIS ID

36765

#### Abstract

We show that the group of orientation-preserving affine transformations of the rational numbers is quasi-lattice ordered by its subsemigroup *N*⋊*N*^{×}. The associated Toeplitz *C*^{∗}-algebra *T*(*N*⋊*N*^{×}) is universal for isometric representations which are covariant in the sense of Nica. We give a presentation of *T*(*N*⋊*N*^{×}) 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 in the sense of Crisp and Laca. The Toeplitz algebra *T*(*N*⋊*N*^{×}) 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*(*N*⋊*N*^{×}),*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*).

## Publication Details

Laca, M. & Raeburn, I. F. (2010). Phase transition on the Toeplitz algebra of the affine semigroup over the natural numbers. Advances in Mathematics, 225 (2), 643-688.