2009/07/22 by Marcelo Laca, Laca, Marcelo, Iain Raeburn +1
Mathematics · #46L55 #82B10 #Advanced Operator Algebra Research #Advanced Topics in Algebra #FOS: Mathematics #Operator Algebras (math.OA) #Random Matrices and Applications #math.OA #msc:46L55 #msc:82B10
paper · pdf · doi:10.48550/arxiv.0907.3760
38 pages
openalex publication_date 2009/07/22 · arxiv created 2009/10/19 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We show that the group \mathbb Q \rtimes \mathbb Q^*+ of orientation-preserving affine transformations of the rational numbers is quasi-lattice ordered by its subsemigroup \mathbb N \rtimes \mathbb N^×. The associated Toeplitz C^*-algebra \mathcal T(\mathbb N \rtimes \mathbb N^×) 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^*-algebra \mathcal Q_\mathbb N recently introduced by Cuntz is the boundary quotient of (\mathbb Q \rtimes \mathbb Q^*+, \mathbb N \rtimes \mathbb N^×) in the sense of Crisp and Laca. The Toeplitz algebra \mathcal T(\mathbb N \rtimes \mathbb N^×) carries a natural dynamics σ, which induces the one considered by Cuntz on the quotient \mathcal Q_\mathbb N, and our main result is the computation of the KMSβ (equilibrium) states of the dynamical system (\mathcal T(\mathbb N \rtimes \mathbb N^×), \mathbb R,σ) for all values of the inverse temperature β. For β∈ [1, 2] there is a unique KMSβ state, and the KMS1 state factors through the quotient map onto \mathcal Q_\mathbb N, giving the unique KMS state discovered by Cuntz. At β=2 there is a phase transition, and for β>2 the KMSβ 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 \mathcal T(\mathbb N).