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MIN-PHASE-ISOMETRIES ON THE UNIT SPHERE OF
$L^1$ SPACES
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society , First View
- Published online by Cambridge University Press:
- 01 April 2026, pp. 1-11
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Let
$(\Omega , \mathfrak {M}, \mu )$ be a measure space. We show that if a surjective map 
$f: S_{L^1(\mu )} \to S_Y$ between the unit spheres of real 
$L^1(\mu )$ and of an arbitrary real normed space Y satisfies 
$$ \begin{align*} \min\{\|f(x)+f(y)\|, \|f(x)-f(y)\|\}=\min\{\|x+y\|, \|x-y\|\},\quad x,y\in S_{L^1(\mu)}, \end{align*} $$then there exists a phase function
$\varepsilon : S_{L^1(\mu )} \to \{-1, 1\}$ such that 
$\varepsilon \cdot f$ is a surjective isometry from 
$S_{L^1(\mu )}$ onto 
$S_Y$, and furthermore, this isometry can be extended to a linear isometry on the whole space 
$L^1(\mu )$.
THE WIGNER PROPERTY OF SMOOTH NORMED SPACES
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 110 / Issue 3 / December 2024
- Published online by Cambridge University Press:
- 09 May 2024, pp. 545-553
- Print publication:
- December 2024
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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.
