On extension of isometries between the positive unit spheres of \(c_0(\Gamma )\)

Let \(\Gamma , \Delta \) $\mathrm{\Gamma },\mathrm{\Delta }$ be two index sets and let \(S_{c_0(\Gamma )}^+=\{x=(x_\gamma )_{\gamma \in \Gamma }\in c_0(\Gamma ): \Vert x\Vert =1; x_\gamma \ge 0,\; \forall \;\gamma \in \Gamma \}\) $S_{c_0(\mathrm{\Gamma })}^{+}=\{x={(x_{\gamma })}_{\gamma \in \mathrm{\Gamma }}\in c_0(\mathrm{\Gamma }):\Vert x\Vert =1;x_{\gamma }\ge 0,\forall \gamma \in \mathrm{\Gamma }\}$ . In this paper, we show that every surjective isometry \(f:S_{c_0(\Gamma )}^+\rightarrow S_{c_0(\Delta )}^+\) $f:S_{c_0(\mathrm{\Gamma })}^{+}\to S_{c_0(\mathrm{\Delta })}^{+}$ can be extended to a linear surjective isometry from \(c_0(\Gamma )\) $c_0(\mathrm{\Gamma })$ onto \(c_0(\Delta )\) $c_0(\mathrm{\Delta })$ .