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Transactions of the American Mathematical Society

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Involutions of $\ell ^2$ and $s$ with unique fixed points


Authors: Jan van Mill and James E. West
Journal: Trans. Amer. Math. Soc. 373 (2020), 7327-7346
MSC (2000): Primary 57N20, 57S99
DOI: https://doi.org/10.1090/tran/8162
Published electronically: August 5, 2020
MathSciNet review: 4155209
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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

Keywords: Anderson conjecture, involution, compact fixed-point set, Hilbert space, countable infinite product of lines, topologically conjugate
Received by editor(s): May 27, 2019
Received by editor(s) in revised form: February 26, 2020
Published electronically: August 5, 2020
Article copyright: © Copyright 2020 American Mathematical Society