The Mazur--Ulam property in $\ell_\infty$-sum and $c_0$-sum of strictly convex Banach spaces

Abstract: In this paper we deal with those Banach spaces $Z$ which satisfy the Mazur--Ulam property, namely that every surjective isometry $\Delta$ from the unit sphere of $Z$ to the unit sphere of any Banach space $Y$ admits an unique extension to a surjective real-linear isometry from $Z$ to $Y$. We prove that for every countable set $\Gamma$ with $\vert \Gamma \vert \geq 2$, the Banach space $\bigoplus_{\gamma \in \Gamma}^{c_0} X_\gamma $ satisfies the Mazur--Ulam property, whenever the Banach space $X_\gamma $ is strictly convex with dim$((X_\gamma ){\mathbb{R}})\geq 2$ for every $\gamma $. Moreover we prove that the Banach space $C_0(K,X)$ satisfies the Mazur--Ulam property whenever $K$ is a totally disconnected locally compact Hausdorff space with $\vert K\vert \geq 2$, and $X$ is a strictly convex separable Banach space with dim$(X{\mathbb{R}})\geq 2$. As consequences, we obtain the following results: (1) Every weakly countably determined Banach space can be equivalently renormed so that it satisfies the Mazur--Ulam property. (2) If $X$ is a strictly convex Banach space with dim$(X_{\mathbb{R}}) \geq 2$, then $C(\mathfrak{C} ,X)$ satisfies the Mazur--Ulam property, where $ \mathfrak{C}$ denotes the Cantor set.