Mazur-Ulam property of the sum of two strictly convex Banach spaces (Q2814347)

A Banach space \(E\) has the Mazur-Ulam property provided that, for any Banach space \(F\), each surjective isometry \(T_{0}\) between the unit spheres \(S(E)\) and \(S(F)\) has a linear isometric extension \(T\) from \(E\) to \(F\). A surjective mapping \(T:E\rightarrow F\) is called a sphere isometry if \(\| T(u)-T(v)\| =\| u-v\| \) for all \(u,v\in E\) satisfying \(\| u\| =\| v\|\). Moreover, \(T\) preserves spheres if \( \| T(u)\| =\| u\| \) for all \( u\in E\). The author proves that a strictly convex Banach space \(E\) has the Mazur-Ulam property if and only if, for any Banach space \(F\), any surjective mapping \(T:E\rightarrow F\) which is a positively homogeneous sphere isometry and preserves the sphere must be linear (Theorem 2.2).NEWLINENEWLINENext, the author studies the \(l^{1}\)-sum (\(l^{\infty }\)-sum) of two strictly convex Banach spaces \(E_{1},E_{2}\) denoted by \(Z=E_{1}\oplus_{l^{1}}E_{2}\) or \(Z=E_{1}\oplus_{l^{\infty }}E_{2}\). Let \(F\) be a Banach space. The author proves that each surjective isometry \(T_{0}:S(Z)\rightarrow S(F)\) can be extended to a linear isometry on the whole space whenever NEWLINE\[NEWLINE \mathbb{R}\cdot T_{0}(S(E_{1}))\subset F\text{ and }\mathbb{R} \cdot T_{0}(S(E_{2}))\subset F NEWLINE\]NEWLINE are both subspaces (Theorem 3.7 and 4.6). Finally, it is shown that the space \(Z\) has the Mazur-Ulam property if and only if, for each Banach space \( F\) and for any surjective isometry \(T_{0}:S(Z)\rightarrow S(F),\) the respective sets NEWLINE\[NEWLINE \mathbb{R}\cdot T_{0}(S(E_{1}))\subset F\text{ and }\mathbb{R} \cdot T_{0}(S(E_{2}))\subset F NEWLINE\]NEWLINE are both subspaces (Theorem 5.2).