RETRACTED – The wigner property for CL-spaces and finite-dimensional polyhedral banach spaces
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...
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Published in | Proceedings of the Edinburgh Mathematical Society Vol. 64; no. 3; pp. 717 - 733 |
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Main Authors | , |
Format | Journal Article |
Language | English |
Published |
01.08.2021
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Online Access | Get full text |
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Summary: | 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. |
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ISSN: | 0013-0915 1464-3839 |
DOI: | 10.1017/S0013091521000250 |