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OPERATOR ALGEBRAIC APPROACH TO INVERSE AND STABILITY THEOREMS FOR AMENABLE GROUPS

Part of: Locally compact groups and their algebras

Published online by Cambridge University Press:  30 August 2018

Marcus De Chiffre ,
Narutaka Ozawa  and
Andreas Thom
Marcus De Chiffre
Affiliation:
Institut für Geometrie, TU Dresden, 01062 Dresden, Germany email marcus_dorph.de_chiffre@tu-dresden.de
Narutaka Ozawa
Affiliation:
RIMS, Kyoto University, Kyoto 606-8502, Japan email narutaka@kurims.kyoto-u.ac.jp
Andreas Thom
Affiliation:
Institut für Geometrie, TU Dresden, 01062 Dresden, Germany email andreas.thom@tu-dresden.de
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Abstract

We prove an inverse theorem for the Gowers $U^{2}$-norm for maps $G\rightarrow {\mathcal{M}}$ from a countable, discrete, amenable group $G$ into a von Neumann algebra ${\mathcal{M}}$ equipped with an ultraweakly lower semi-continuous, unitarily invariant (semi-)norm $\Vert \cdot \Vert$. We use this result to prove a stability result for unitary-valued $\unicode[STIX]{x1D700}$-representations $G\rightarrow {\mathcal{U}}({\mathcal{M}})$ with respect to $\Vert \cdot \Vert$.

MSC classification

Primary: 22D10: Unitary representations of locally compact groups
Secondary: 39B82: Stability, separation, extension, and related topics
Type
Research Article
Information
Mathematika , Volume 65 , Issue 1 , 2019 , pp. 98 - 118
Copyright
Copyright © University College London 2018 

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