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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Countable groups of isometries on Banach spaces
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by Valentin Ferenczi and Elói Medina Galego PDF
Trans. Amer. Math. Soc. 362 (2010), 4385-4431 Request permission

Abstract:

A group $G$ is representable in a Banach space $X$ if $G$ is isomorphic to the group of isometries on $X$ in some equivalent norm. We prove that a countable group $G$ is representable in a separable real Banach space $X$ in several general cases, including when $G \simeq \{-1,1\} \times H$, $H$ finite and $\dim X \geq |H|$, or when $G$ contains a normal subgroup with two elements and $X$ is of the form $c_0(Y)$ or $\ell _p(Y)$, $1 \leq p <+\infty$. This is a consequence of a result inspired by methods of S. Bellenot (1986) and stating that under rather general conditions on a separable real Banach space $X$ and a countable bounded group $G$ of isomorphisms on $X$ containing $-Id$, there exists an equivalent norm on $X$ for which $G$ is equal to the group of isometries on $X$.

We also extend methods of K. Jarosz (1988) to prove that any complex Banach space of dimension at least $2$ may be renormed with an equivalent complex norm to admit only trivial real isometries, and that any complexification of a Banach space may be renormed with an equivalent complex norm to admit only trivial and conjugation real isometries. It follows that every real Banach space of dimension at least $4$ and with a complex structure may be renormed to admit exactly two complex structures up to isometry, and that every real Cartesian square may be renormed to admit a unique complex structure up to isometry.

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Additional Information
  • Valentin Ferenczi
  • Affiliation: Departamento de Matemática, Instituto de Matemática e Estatística, Universidade de São Paulo, rua do Matão, 1010 - Cidade Universitária, 05508-090 São Paulo, SP, Brazil
  • MR Author ID: 360353
  • ORCID: 0000-0001-5239-111X
  • Email: ferenczi@ime.usp.br
  • Elói Medina Galego
  • Affiliation: Departamento de Matemática, Instituto de Matemática e Estatística, Universidade de São Paulo, rua do Matão, 1010 - Cidade Universitária, 05508-090 São Paulo, SP, Brazil
  • MR Author ID: 647154
  • Email: eloi@ime.usp.br
  • Received by editor(s): June 13, 2007
  • Received by editor(s) in revised form: March 2, 2009
  • Published electronically: March 12, 2010
  • © Copyright 2010 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 362 (2010), 4385-4431
  • MSC (2000): Primary 46B03, 46B04
  • DOI: https://doi.org/10.1090/S0002-9947-10-05034-8
  • MathSciNet review: 2608411