Ergodicity of the action of the positive rationals on the group of finite adeles and the Bost-Connes phase transition theorem

Abstract

For each β\in(0,+\infty) there exists a canonical measure μ_βon the ring A_f of finite adeles. We show that the positive rationals act ergodically on (A_f,μ_β) for β\in(0,1], and then deduce from this the uniqueness of KMS_β-states for the Bost-Connes system.

arXiv commentLaTeX2e, 4 pages

Cited by in corpus (16)

• From Physics to Number Theory via Noncommutative Geometry. Part I: Quantum Statistical Mechanics of Q-lattices
• On Bost-Connes type systems for number fields
• Phase transition in the Connes-Marcolli GL2-system
• Bost-Connes systems, Hecke algebras, and induction
• Type III_1 equilibrium states of the Toeplitz algebra of the affine semigroup over the natural numbers
• Phase transitions on Hecke C*-algebras and class-field theory over Q
• Von Neumann algebras arising from Bost-Connes type systems
• Hecke C*-algebras, Schlichting completions, and Morita equivalence
• KMS states and complex multiplication
• Hecke algebras of semidirect products and the finite part of the Connes-Marcolli C*-algebra
• Phase transitions on C*-algebras from actions of congruence monoids on rings of algebraic integers
• Ergodicity of the action of K* on A_K
• Bost-Connes type systems for function fields
• On arithmetic models and functoriality of Bost-Connes systems. With an appendix by Sergey Neshveyev
• Phase transition on the Toeplitz algebra of the affine semigroup over the natural numbers
• Subquotients of Hecke C^*-algebras