Von Neumann algebras arising from Bost-Connes type systems
Abstract
We show that the KMSbeta-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.
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