Conditional plasticity of the unit ball of the ℓ∞-sum of finitely many strictly convex Banach spaces

Abstract

We prove that for any ℓ∞-sum Z=⨁i∈[n]Xi of finitely many strictly convex Banach spaces (Xi)i∈[n], an extremeness preserving 1-Lipschitz bijection f:BZ→BZ is an isometry, by constraining the componentwise behavior of the inverse g=f−1 with a theorem admitting a graph-theoretic interpretation. We also show that if X,Y are Banach spaces, then a bijective 1-Lipschitz non-isometry of type BX→BY can be used to construct a bijective 1-Lipschitz non-isometry of type BX′→BX′ for some Banach space X′, and that a homeomorphic 1-Lipschitz non-isometry of type BX→BX restricts to a homeomorphic 1-Lipschitz non-isometry of type BS→BS for some separable subspace S≤X.