The Mazur--Ulam property in ℓ_∞-sum and c₀-sum of strictly convex Banach spaces

2019/05/05 by Guerrero, Julio Becerra
#FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.1905.01731

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 an unique extension to a surjective real-linear isometry from Z to Y. We prove that for every countable set Γ with \vert Γ\vert ≥ 2, the Banach space \bigoplus_{γ∈ Γ}^c₀ X_γ satisfies the Mazur--Ulam property, whenever the Banach space X_γ is strictly convex with dim((X_γ)_ℝ)≥ 2 for every γ. Moreover we prove that the Banach space C₀(K,X) satisfies the Mazur--Ulam property whenever K is a totally disconnected locally compact Hausdorff space with \vert K\vert ≥ 2, and X is a strictly convex separable Banach space with dim(X_ℝ)≥ 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_ℝ) ≥ 2, then C(\mathfrakC ,X) satisfies the Mazur--Ulam property, where \mathfrakC denotes the Cantor set.

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