Conditional plasticity of the unit ball of the $\ell_\infty$-sum of finitely many strictly convex Banach spaces

Abstract

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

Theorems & Definitions (27)

• Theorem 3.1
• proof
• Lemma 3.1
• proof
• Theorem 3.2
• proof
• Theorem 4.1
• Lemma 4.1
• proof
• Lemma 4.2