Phase transition on the Toeplitz algebra of the affine semigroup over the natural numbers

Abstract

We show that the group $\mathbb{Q}⋊{\mathbb{Q}}_{+}^{\ast }$ of orientation-preserving affine transformations of the rational numbers is quasi-lattice ordered by its subsemigroup $\mathbb{N}⋊{\mathbb{N}}^{\times }$. The associated Toeplitz $C^{\ast }$-algebra $\mathcal{T}(\mathbb{N}⋊{\mathbb{N}}^{\times })$ is universal for isometric representations which are covariant in the sense of Nica. We give a presentation of this Toeplitz algebra in terms of generators and relations, and use this to show that the $C^{\ast }$-algebra ${\mathcal{Q}}_{\mathbb{N}}$ recently introduced by Cuntz is the boundary quotient of $(\mathbb{Q}⋊{\mathbb{Q}}_{+}^{\ast },\mathbb{N}⋊{\mathbb{N}}^{\times })$ in the sense of Crisp and Laca. The Toeplitz algebra $\mathcal{T}(\mathbb{N}⋊{\mathbb{N}}^{\times })$ carries a natural dynamics $\sigma$, which induces the one considered by Cuntz on the quotient ${\mathcal{Q}}_{\mathbb{N}}$, and our main result is the computation of the KMS${}_{\beta }$ (equilibrium) states of the dynamical system $(\mathcal{T}(\mathbb{N}⋊{\mathbb{N}}^{\times }),\mathbb{R},\sigma )$ for all values of the inverse temperature $\beta$. For $\beta \in [1,2]$ there is a unique KMS${}_{\beta }$ state, and the KMS${}_1$ state factors through the quotient map onto ${\mathcal{Q}}_{\mathbb{N}}$, giving the unique KMS state discovered by Cuntz. At $\beta =2$ there is a phase transition, and for $\beta >2$ the KMS${}_{\beta }$ states are indexed by probability measures on the circle. There is a further phase transition at $\beta =\infty$, where the KMS${}_{\infty }$ states are indexed by the probability measures on the circle, but the ground states are indexed by the states on the classical Toeplitz algebra $\mathcal{T}(\mathbb{N})$.

38 pages