Phase-isometries on real normed spaces

We say that a mapping $f:X\to Y$ between two real normed spaces is a phase-isometry if it satisfies the functional equation \begin{eqnarray*} {|f(x)+f(y)|, |f(x)-f(y)|}={|x+y|, |x-y|} \quad (x,y\in X).\end{eqnarray*} A generalized Mazur-Ulam question is whether every surjective phase-isometry is a multiplication of a linear isometry and a map with range $\{-1,1\}$. This assertion is also an extension of a fundamental statement in the mathematical description of quantum mechanics, Wigner’s theorem to real normed spaces. In this paper, we show that for every space $Y$ the problem is solved in positive way if $X$ is a smooth normed space, an ${\mathcal{L}}^{\mathrm{\infty }}(\mathrm{\Gamma })$-type space or an ${\mathrm{\ell }}^1(\mathrm{\Gamma })$-space with $\mathrm{\Gamma }$ being an index set.