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

Neshveyev, Sergey

doi:10.1090/S0002-9939-02-06449-3

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

Proc. Amer. Math. Soc. 130 (2002), 2999-3003

DOI: https://doi.org/10.1090/S0002-9939-02-06449-3

Published electronically: March 14, 2002

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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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