Constructing number field isomorphisms from *-isomorphisms of certain crossed product C*-algebras

Bruce, C. and Takeishi, T. (2024) Constructing number field isomorphisms from *-isomorphisms of certain crossed product C*-algebras. Communications in Mathematical Physics, 405(1), 20. (doi: 10.1007/s00220-023-04927-y)

Abstract

We prove that the class of crossed product C*-algebras associated with the action of the multiplicative group of a number field on its ring of finite adeles is rigid in the following explicit sense: Given any *-isomorphism between two such C*-algebras, we construct an isomorphism between the underlying number fields. As an application, we prove an analogue of the Neukirch–Uchida theorem using topological full groups, which gives a new class of discrete groups associated with number fields whose abstract isomorphism class completely characterises the number field.

• Item Type: Articles
• Additional Information: C. Bruce was supported by a Banting Fellowship administered by the Natural Sciences and Engineering Research Council of Canada (NSERC) and has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie Grant Agreement No. 101022531 and from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant Agreement No. 817597). T. Takeishi is supported by JSPS KAKENHI Grant Number JP19K14551.
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Bruce, Dr Chris and Takeishi, Dr Takuya
• Authors: Bruce, C., and Takeishi, T.
• College/School: College of Science and Engineering > School of Mathematics and Statistics
• Journal Name: Communications in Mathematical Physics
• Publisher: Springer
• ISSN: 0010-3616
• ISSN (Online): 1432-0916
• Published Online: 29 January 2024
• Copyright Holders: Copyright © 2024 The Authors
• First Published: First published in Communications in Mathematical Physics 405(1): 20
• Publisher Policy: Reproduced under a Creative Commons License

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Project Code	Award No	Project Name	Principal Investigator	Funder's Name	Funder Ref	Lead Dept
312369		Groups, operator algebras, and dynamical systems	Xin Li	European Commission (EC)	101022531	M&S - Mathematics