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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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- Article
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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.
