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Additive conjugacy and the Bohr compactification of orthogonal representations

Chase, Zachary and Hann-Caruthers, Wade and Tamuz, Omer (2021) Additive conjugacy and the Bohr compactification of orthogonal representations. Mathematische Annalen . ISSN 0025-5831. doi:10.1007/s00208-021-02191-w. (In Press) https://resolver.caltech.edu/CaltechAUTHORS:20200124-084913765

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Abstract

We say that two unitary or orthogonal representations of a finitely generated group G are additive conjugates if they are intertwined by an additive map, which need not be continuous. We associate to each representation of G a topological action that is a complete additive conjugacy invariant: the action of G by group automorphisms on the Bohr compactification of the underlying Hilbert space. Using this construction we show that the property of having almost invariant vectors is an additive conjugacy invariant. As an application we show that G is amenable if and only if there is a nonzero homomorphism from L²(G) into R/Z that is invariant to the G-action.


Item Type:Article
Related URLs:
URLURL TypeDescription
https://doi.org/10.1007/s00208-021-02191-wDOIArticle
https://rdcu.be/cjGgtPublisherFree ReadCube access
https://arxiv.org/abs/1905.11599arXivDiscussion Paper
ORCID:
AuthorORCID
Tamuz, Omer0000-0002-0111-0418
Additional Information:© The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2021. Received 24 July 2019; Revised 09 January 2021; Accepted 21 April 2021; Published 27 April 2021. We would like to thank Todor Tsankov for suggesting some improvements to our proofs, Yehuda Shalom for suggesting to us the classification of the additive conjugacy classes of the irreducible representations of Z, and Andreas Thom for pointing out an error in an earlier version, as well as suggesting a correct proof. We would also like to thank Joshua Frisch, Eli Glasner, Alexander Kechris, Jesse Peterson, Pooya Vahidi Ferdowsi, Benjamin Weiss, and Andy Zucker for helpful discussions. Omer Tamuz was supported by a grant from the Simons Foundation (#419427), a Sloan research fellowship, a BSF award (#2018397), and an NSF CAREER award (DMS-1944153).
Funders:
Funding AgencyGrant Number
Simons Foundation419427
Alfred P. Sloan FoundationUNSPECIFIED
Binational Science Foundation (USA-Israel)2018397
NSFDMS-1944153
DOI:10.1007/s00208-021-02191-w
Record Number:CaltechAUTHORS:20200124-084913765
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20200124-084913765
Official Citation:Chase, Z., Hann-Caruthers, W. & Tamuz, O. Additive conjugacy and the Bohr compactification of orthogonal representations. Math. Ann. (2021). https://doi.org/10.1007/s00208-021-02191-w
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:100887
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:24 Jan 2020 20:00
Last Modified:11 Aug 2021 16:57

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