Additive conjugacy and the Bohr compactification of orthogonal representations

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 a...

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Bibliographic Details
Published in:Mathematische annalen 2021-10, Vol.381 (1-2), p.319-333
Main Authors: Chase, Zachary, Hann-Caruthers, Wade, Tamuz, Omer
Format: Article
Language:English
Subjects:
Automorphisms
Hilbert space
Homomorphisms
Invariants
Mathematics
Mathematics and Statistics
Representations
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Summary: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 L 2 ( G ) into R / Z that is invariant to the G -action.
ISSN:0025-5831
1432-1807
DOI:10.1007/s00208-021-02191-w