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The wigner property for CL-spaces and finite-dimensional polyhedral Banach spaces

Part of: Normed linear spaces and Banach spaces; Banach lattices

Published online by Cambridge University Press:  30 April 2021

Dongni Tan  and
Xujian Huang
Dongni Tan
Affiliation:
School of Computer Science and Engineering, Tianjin University of Technology, Tianjin300384, P.R. China (tandongni0608@sina.cn)
Xujian Huang
Affiliation:
Department of Mathematics, Tianjin University of Technology, Tianjin300384, P.R. China (huangxujian86@sina.cn)
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Abstract

We say that a map $f$ from a Banach space $X$ to another Banach space $Y$ is a phase-isometry if the equality

\[ \{\|f(x)+f(y)\|, \|f(x)-f(y)\|\}=\{\|x+y\|, \|x-y\|\} \]
holds for all $x,\,y\in X$. A Banach space $X$ is said to have the Wigner property if for any Banach space $Y$ and every surjective phase-isometry $f : X\rightarrow Y$, there exists a phase function $\varepsilon : X \rightarrow \{-1,\,1\}$ such that $\varepsilon \cdot f$ is a linear isometry. We present some basic properties of phase-isometries between two real Banach spaces. These enable us to show that all finite-dimensional polyhedral Banach spaces and CL-spaces possess the Wigner property.

Keywords

the Wigner propertyphase-isometryCL-spacepolyhedral space

MSC classification

Primary: 46B20: Geometry and structure of normed linear spaces
Secondary: 46B04: Isometric theory of Banach spaces
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 64 , Issue 2 , May 2021 , pp. 183 - 199
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Copyright
Copyright © The Author(s) 2021. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society

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