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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.
Suppose that X and Y are two real normed spaces. A map 
$f:X\rightarrow Y$ is called a min-phase-isometry if it satisfies 
$$ \begin{align*} \min\{\|f(x)+f(y)\|,\|f(x)-f(y)\|\}=\min\{\|x+y\|,\|x-y\|\} \quad (x,y\in X). \end{align*} $$
We present properties of min-phase-isometries in the case that Y is strictly convex and show that if a min-phase-isometry f (not necessarily surjective) fixes the origin, then it is phase-equivalent to a linear isometry, that is, 
$f(x)=\varepsilon (x)g(x)$ for 
$x\in X$, where 
$g:X\rightarrow Y$ is a linear isometry and 
$\varepsilon $ is a map from X to 
$\{-1,1\}$.
${\mathcal{L}}^{\infty }(\unicode[STIX]{x1D6E4})$-TYPE SPACES
We investigate surjective solutions of the functional equation where 
$$\begin{eqnarray}\displaystyle \{\Vert f(x)+f(y)\Vert ,\Vert f(x)-f(y)\Vert \}=\{\Vert x+y\Vert ,\Vert x-y\Vert \}\quad (x,y\in X), & & \displaystyle \nonumber\end{eqnarray}$$
$f:X\rightarrow Y$ is a map between two real 
${\mathcal{L}}^{\infty }(\unicode[STIX]{x1D6E4})$-type spaces. We show that all such solutions are phase equivalent to real linear isometries. This can be considered as an extension of Wigner’s theorem on symmetry for real 
${\mathcal{L}}^{\infty }(\unicode[STIX]{x1D6E4})$-type spaces.
