Von Neumann algebras arising from Bost-Connes type systems

Abstract

We show that the KMS_beta-states of Bost-Connes type systems for number fields in the region 0<beta\le 1, as well as of the Connes-Marcolli GL_2-system for 1<beta\le 2, have type III_1. This is equivalent to ergodicity of various actions on adelic spaces. For example, the case beta=2 of the GL_2-system corresponds to ergodicity of the action of GL_2(Q) on Mat_2(A) with its Haar measure.

AMS-LaTeX, 12 pages

References in corpus (5)

• From Physics to Number Theory via Noncommutative Geometry. Part I: Quantum Statistical Mechanics of Q-lattices
• On Bost-Connes type systems for number fields
• The ratio set of the harmonic measure of a random walk on a hyperbolic group
• Phase transition in the Connes-Marcolli GL2-system
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Cited by in corpus (6)

• KMS states on the C*-algebras of non-principal groupoids
• 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 C*-algebras from actions of congruence monoids on rings of algebraic integers
• Bost-Connes systems associated with function fields
• Ergodicity of the action of K* on A_K