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

2024/03/14 by Kurik, Kaarel August · 1 citation
#05C69 #46B20 #47H09 #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2403.09245

Abstract

We prove that for any ℓ_∞-sum Z = \bigoplus_{i ∈ [n]} X_i of finitely many strictly convex Banach spaces (X_i)_{i ∈ [n]}, an extremeness preserving 1-Lipschitz bijection f\colon B_Z → B_Z 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 B_X → B_Y can be used to construct a bijective 1-Lipschitz non-isometry of type B_X' → B_X' for some Banach space X', and that a homeomorphic 1-Lipschitz non-isometry of type B_X → B_X restricts to a homeomorphic 1-Lipschitz non-isometry of type B_S → B_S for some separable subspace S ≤ X.

Cited by

• Plastic metric spaces and groups

Related

• Non-expansive bijections to the unit ball of ℓ₁-sum of strictly convex Banach spaces
• Two new examples of Banach spaces with a plastic unit ball
• Products of Lipschitz-free spaces and applications
• Geometry and volume product of finite dimensional Lipschitz-free spaces
• Rough Isometries of Lipschitz Function Spaces