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Triviality Theorem for Quasirepresentations in Reflexive Banach Spaces

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Abstract

It is proved that, for a sufficiently small defect of a (not necessarily bounded) quasirepresentations of an amenable group in a reflexive Banach space with dense set of bounded orbits, there is an extension of this quasirepresentation for which there is a close ordinary representation of the group in the space of this extension.

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Notes

  1. Note that, for \(f\in E^*\subset L^*\) and \(x\in L\), we certainly have the inequality

    $$ |f(\rho(g)x)|=|f(\pi(g)x)+f((\rho(g)-\pi(g))x)|\le|f(\pi(g)x)|+|f((\rho(g)-\pi(g))x)|$$
    $$+C\|\pi(g)\|_{{\mathcal L}(E)}\|x\|_E+C\|(\rho(g)-\pi(g))x\|_E,$$

    where \(\|(\rho(g)-\pi(g))x\|_E\le\|\rho(g)-\pi(g)\|_{{\mathcal L}(E)}\|x\|_E;\) however, the operator \(\rho(g)-\pi(g)\), which is bounded on \(L\), need not be bounded with respect to the norm of the space \(E\), and \(E^*\) need not be a \(\rho^*\)-invariant subspace of \(L^*\).

References

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Funding

The research was partially supported by the Moscow Center for Fundamental and Applied Mathematics.

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Authors and Affiliations

  1. Moscow Center for Fundamental and Applied Mathematics, Moscow, 119991, Russia

    A. I. Shtern

  2. Department of Mechanics and Mathematics, Moscow State University, Moscow, 119991, Russia

    A. I. Shtern

  3. Federal State Institution “Scientific-Research Institute for System Analysis of the Russian Academy of Sciences” (SRISA), Moscow, 117312, Russia

    A. I. Shtern

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  1. A. I. Shtern
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Correspondence to A. I. Shtern.

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Shtern, A.I. Triviality Theorem for Quasirepresentations in Reflexive Banach Spaces. Russ. J. Math. Phys. 29, 397–401 (2022). https://doi.org/10.1134/S1061920822030074

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  • DOI: https://doi.org/10.1134/S1061920822030074

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