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

Author: Sergey Neshveyev
Journal: Proc. Amer. Math. Soc. 130 (2002), 2999-3003
MSC (1991): Primary 46L55; Secondary 28D15
Published electronically: March 14, 2002
MathSciNet review: 1908923
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Abstract: We study relatively invariant measures with the multiplicators ${\mathbb Q}^*_+\ni q\mapsto q^{-\beta}$ on the space $\mathcal A_f$ of finite adeles. We prove that for $\beta\in(0,1]$ such measures are ergodic, and then deduce from this the uniqueness of KMS$_\beta$-states for the Bost-Connes system. Combining this with a result of Blackadar and Boca-Zaharescu, we also obtain ergodicity of the action of $\mathbb Q^*$ on the full adeles.

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Additional Information

Sergey Neshveyev
Affiliation: Institute for Low Temperature Physics & Engineering, 47 Lenin ave, Kharkov 310164, Ukraine
Address at time of publication: Matematisk Institutt, P.B. 1053 Blindern, 0316 Oslo, Norway

Received by editor(s): November 28, 2000
Received by editor(s) in revised form: May 11, 2001
Published electronically: March 14, 2002
Additional Notes: This research was partially supported by Award No UM1-2092 of the Civilian Research & Development Foundation
Communicated by: David R. Larson
Article copyright: © Copyright 2002 American Mathematical Society

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