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

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...

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Bibliographic Details
Published inProceedings of the Edinburgh Mathematical Society Vol. 64; no. 2; pp. 183 - 199
Main Authors Tan, Dongni, Huang, Xujian
Format Journal Article
LanguageEnglish
Published Cambridge, UK Cambridge University Press 01.05.2021
Subjects
Banach spaces
the Wigner property
46B20
phase-isometry
polyhedral space
CL-space
46B04
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Summary: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.
ISSN:0013-0915
1464-3839
DOI:10.1017/S0013091521000079