## 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

## 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.## References

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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: neshveyev@hotmail.com
- 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: https://doi.org/10.1090/S0002-9939-02-06449-3
- MathSciNet review: 1908923