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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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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
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by Sergey Neshveyev PDF
Proc. Amer. Math. Soc. 130 (2002), 2999-3003 Request permission


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
  • Email:
  • 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
  • © Copyright 2002 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 130 (2002), 2999-3003
  • MSC (1991): Primary 46L55; Secondary 28D15
  • DOI:
  • MathSciNet review: 1908923