Von Neumann algebras arising from Bost-Connes type systems

2009/07/09 by Sergey Neshveyev, Neshveyev, Sergey
Computer Science · Mathematics · #Advanced Operator Algebra Research #Computability, Logic, AI Algorithms #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Operator Algebras (math.OA) #math.DS #math.OA

paper · pdf · doi:10.48550/arxiv.0907.1456

AMS-LaTeX, 12 pages

arxiv created 2009/07/09 · openalex publication_date 2009/07/09 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We show that the KMS_beta-states of Bost-Connes type systems for number fields in the region 0<beta≤ 1, as well as of the Connes-Marcolli GL₂-system for 1<beta≤ 2, have type III₁. This is equivalent to ergodicity of various actions on adelic spaces. For example, the case beta=2 of the GL₂-system corresponds to ergodicity of the action of GL₂(Q) on Mat₂(A) with its Haar measure.

Citations

• From Physics to Number Theory via Noncommutative Geometry. Part I: Quantum Statistical Mechanics of Q-lattices

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