THE WIGNER PROPERTY OF SMOOTH NORMED SPACES

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.