Von Neumann algebras arising from Bost-Connes type systems

TL;DR

This paper demonstrates that certain Bost-Connes type systems and the Connes-Marcolli GL_2-system produce von Neumann algebras of type III_1, linking algebraic states to ergodic actions on adelic spaces.

Contribution

It establishes the type classification of KMS states for these systems and connects them to ergodic actions on adelic spaces, extending previous understanding.

Findings

KMS_beta-states for 0<beta≤1 are type III_1

KMS_beta-states for 1<beta≤2 are type III_1

Ergodicity of actions on adelic spaces is demonstrated

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.