The Mazur–Ulam property in ℓ∞-sum and c0-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 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 prove that for every countable set Γ with |Γ|≥2, the Banach space ⨁γ∈Γc0Xγ satisfies the Mazur–Ulam property, whenever the Banach space Xγ is strictly convex with dim((Xγ)R)≥2 for every γ. As a consequence, every weakly countably determined Banach space can be equivalently renormed so that it satisfies the Mazur–Ulam property.