We say that a map \(f:S_X \rightarrow S_Y\) between the unit spheres of two Banach spaces X and Y is a max-phase-isometry (min-phase-isometry, respectively) if it satisfies \(\begin{aligned} \max \{\Vert f(x)+f(y)\Vert , \Vert f(x)-f(y)\Vert \}=\max \{\Vert x+y\Vert , \Vert x-y\Vert \}\quad (x,y\in S_X),\\ \min \{\Vert f(x)+f(y)\Vert , \Vert f(x)-f(y)\Vert \}=\min \{\Vert x+y\Vert , \Vert x-y\Vert \}\quad (x,y\in S_X). \end{aligned}\) Let \(\Gamma ,\Delta \) be two arbitrary index sets, and let \(p>0\) and \(p\ne 1\) . Here, all \(\ell ^p(\Gamma )\) -type spaces are over the real numbers. We show that for every surjective max-phase-isometry or min-phase-isometry \(f:S_{\ell ^p(\Gamma )}\rightarrow S_{\ell ^p(\Delta )}\) , there exists a phase function \(\varepsilon : S_{\ell ^p(\Gamma )} \rightarrow \{-1, 1\}\) such that \(\varepsilon \cdot f\) is an isometry. This isometry is the restriction of a linear isometry from \(\ell ^p(\Gamma )\) onto \(\ell ^p(\Delta )\) . Furthermore, for \(p=1\) , this result is valid for min-phase-isometries but fails, in general, for max-phase-isometries.
