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THE WIGNER PROPERTY OF SMOOTH NORMED SPACES
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- Journal:
- Bulletin of the Australian Mathematical Society , First View
- Published online by Cambridge University Press:
- 09 May 2024, pp. 1-9
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We prove that every smooth complex normed space X has the Wigner property. That is, for any complex normed space Y and every surjective mapping
$f: X\rightarrow Y$ satisfying 
$$ \begin{align*} \{\|f(x)+\alpha f(y)\|: \alpha\in \mathbb{T}\}=\{\|x+\alpha y\|: \alpha\in \mathbb{T}\}, \quad x,y\in X, \end{align*} $$where
$\mathbb {T}$ is the unit circle of the complex plane, there exists a function 
$\sigma : X\rightarrow \mathbb {T}$ such that 
$\sigma \cdot f$ is a linear or anti-linear isometry. This is a variant of Wigner’s theorem for complex normed spaces.
RETRACTED – The wigner property for CL-spaces and finite-dimensional polyhedral banach spaces
- Part of
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- Journal:
- Proceedings of the Edinburgh Mathematical Society / Volume 64 / Issue 3 / August 2021
- Published online by Cambridge University Press:
- 04 June 2021, pp. 717-733
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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.
The wigner property for CL-spaces and finite-dimensional polyhedral Banach spaces
- Part of
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- Journal:
- Proceedings of the Edinburgh Mathematical Society / Volume 64 / Issue 2 / May 2021
- Published online by Cambridge University Press:
- 30 April 2021, pp. 183-199
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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.
