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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality 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 1.43.

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Involutions of $\ell ^2$ and $s$ with unique fixed points
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by Jan van Mill and James E. West PDF
Trans. Amer. Math. Soc. 373 (2020), 7327-7346 Request permission

Abstract:

Let $\sigma _{\ell ^2}$ and $\sigma _{\Bbb R^{\infty }}$ be the linear involutions of $\ell ^2$ and $\mathbb {R}^\infty$, respectively, given by the formula $x\to -x$. We prove that although $\ell ^2$ and $\Bbb R^{\infty }$ are homeomorphic, $\sigma _{\ell ^2}$ is not topologically conjugate to $\sigma _{\Bbb R^{\infty }}$. We proceed to examine the implications of this and give characterizations of the involutions that are conjugate to $\sigma _{\ell ^2}$ and to $\sigma _{\Bbb R^{\infty }}$. We show that the linear involution $x\to -x$ of a separable, infinite-dimensional Fréchet space $E$ is topologically conjugate to $\sigma _{\ell ^2}$ if and only if $E$ contains an infinite-dimensional Banach subspace and otherwise is linearly conjugate to $\sigma _{\Bbb R^{\infty }}$.
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Additional Information
  • Jan van Mill
  • Affiliation: KdV Institute for Mathematics, University of Amsterdam, Science Park 105-107, P.O. Box 94248, 1090 GE Amsterdam, The Netherlands
  • MR Author ID: 124825
  • Email: j.vanMill@uva.nl
  • James E. West
  • Affiliation: Department of Mathematics, Cornell University, Ithaca, New York 14053-4201
  • MR Author ID: 182055
  • Email: west@math.cornell.edu
  • Received by editor(s): May 27, 2019
  • Received by editor(s) in revised form: February 26, 2020
  • Published electronically: August 5, 2020
  • © Copyright 2020 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 373 (2020), 7327-7346
  • MSC (2000): Primary 57N20, 57S99
  • DOI: https://doi.org/10.1090/tran/8162
  • MathSciNet review: 4155209