The Mazur–Ulam property on Banach spaces of vector-valued continuous functions

Abstract

Let K be a totally disconnected locally compact Hausdorff space with |K|≥2, and let X be a strictly convex separable Banach space with dim(XR)≥2. We prove that the Banach space Z=C0(K,X) satisfies the Mazur–Ulam property, namely that every surjective isometry Δ from the unit sphere of Z to the unit sphere of any Banach space Y admits a unique extension to a surjective real-linear isometry from Z to Y. We obtain the following result, if X is a strictly convex Banach space with dim(XR)≥2, then C(C,X) satisfies the Mazur–Ulam property, where C denotes the Cantor set.