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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

Information

Type
Research Article
Information
Mathematika , Volume 65 , Issue 1 , 2019 , pp. 98 - 118
Copyright
Copyright © University College London 2018 

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References

Burger, M., Ozawa, N. and Thom, A., On Ulam stability. Israel J. Math. 193(1) 2013, 109–129.Google Scholar
Gowers, W. T., A new proof of Szemerédi’s theorem. Geom. Funct. Anal. 11(3) 2001, 465–588.Google Scholar
Gowers, W. T., A new proof of Szemerédi’s theorem for arithmetic progressions of length four. Geom. Funct. Anal. 8(3) 1998, 529–551.Google Scholar
Gowers, W. T. and Hatami, O., Inverse and stability theorems for approximate representations of finite groups. Mat. Sb. 208(12) 2017, 1784–1817.Google Scholar
Grove, K., Karcher, H. and Ruh, E. A., Group actions and curvature. Bull. Amer. Math. Soc. 81 1975, 89–92.Google Scholar
Johnson, B. E., Approximately multiplicative functionals. J. Lond. Math. Soc. (2) 34(3) 1986, 489–510.Google Scholar
Johnson, B. E., Cohomology in Banach algebras. Mem. Amer. Math. Soc. 127 1972.Google Scholar
Kadison, R. V. and Pedersen, G. K., Means and convex combinations of unitary operators. Math. Scand. 57(2) 1985, 249–266.Google Scholar
Kasparov, G. G., Hilbert C ∗ -modules: theorems of Stinespring and Voiculescu. J. Operator Theory 4(1) 1980, 133–150.Google Scholar
Kazhdan, D., On 𝜀-representations. Israel J. Math. 43(4) 1982, 315–323.Google Scholar
Rademacher, H., Zur Theorie die Modulfunktionen. J. Reine Angew. Math. 167 1932, 312–336.Google Scholar
Shtern, A. I., Roughness and approximation of quasi-representations of amenable groups. Math. Notes 65(6) 1999, 760–769.Google Scholar
Ulam, S. M., A Collection of Mathematical Problems (Interscience Tracts in Pure and Applied Mathematics 8 ), Interscience Publishers (New York, London, 1960).Google Scholar