WIGNER’S THEOREM IN ${\mathcal{L}}^{∞ }(\unicode[STIX]{x1D6E4})$ -TYPE SPACES - JIA, WEIKE; TAN, DONGNI (Bulletin of the Australian Mathematical Society, 2018)
We investigate surjective solutions of the functional equation $$\begin{eqnarray}\displaystyle \{\Vert f(x)+f(y)\Vert ,\Vert f(x)-f(y)\Vert \}=\{\Vert x+y\Vert ,\Vert x-y\Vert \}\quad (x,y\in X), & & \displaystyle \nonumber\end{eqnarray}$$ where $f:X→ Y$ is a map between two real ${\mathcal{L}}^{∞ }(\unicode[STIX]{x1D6E4})$ -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 ${\mathcal{L}}^{∞ }(\unicode[STIX]{x1D6E4})$ -type spaces.