Min-phase-isometries on the unit sphere of $$\mathcal {L}^\infty (\Gamma )$$ -type spaces

We show that every surjective mapping f between the unit spheres of two real \(\mathcal {L}^\infty (\Gamma )\)-type spaces satisfies

$$\begin{aligned} \min \{\Vert f(x)+f(y)\Vert ,\Vert f(x)-f(y)\Vert \}=\min \{\Vert x+y\Vert ,\Vert x-y\Vert \}\quad (x,y\in S_X) \end{aligned}$$

if and only if f is phase-equivalent to an isometry, i.e., there is a phase-function \(\varepsilon \) from the unit sphere of the \(\mathcal {L}^\infty (\Gamma )\)-type space onto \(\{-1,1\}\) such that \(\varepsilon \cdot f\) is a surjective isometry between the unit spheres of two real \(\mathcal {L}^\infty (\Gamma )\)-type spaces, and furthermore, this isometry can be extended to a linear isometry on the whole space \(\mathcal {L}^\infty (\Gamma )\). We also give an example to show that these are not true if “min” is replaced by “max”.