The Mazur--Ulam property in ${\mathrm{\ell }}_{\mathrm{\infty }}$-sum and $c_0$-sum of strictly convex Banach spaces

In this paper we deal with those Banach spaces $Z$ which satisfy the Mazur–Ulam property, namely that every surjective isometry $\mathrm{\Delta }$ 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 $\mathrm{\Gamma }$ with $\mathrm{\mid }\mathrm{\Gamma }\mathrm{\mid }\ge 2$, the Banach space ${⨁}_{\gamma \in \mathrm{\Gamma }}^{c_0}{X}_{\gamma }$ satisfies the Mazur–Ulam property, whenever the Banach space $X_{\gamma }$ is strictly convex with dim$((X_{\gamma }{)}_{\mathbb{R}})\ge 2$ 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$(X_{\mathbb{R}})\ge 2$. 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$(X_{\mathbb{R}})\ge 2$, then $C(\mathfrak{C},X)$ satisfies the Mazur–Ulam property, where $\mathfrak{C}$ denotes the Cantor set.