Non-expansive bijections to the unit ball of $\ell_1$-sum of strictly convex Banach spaces

TL;DR

This paper proves that any non-expansive bijection from the unit ball of a Banach space to the unit ball of an $ ext{l}_1$-sum of strictly convex Banach spaces is necessarily an isometry, extending previous results.

Contribution

It generalizes earlier findings by showing that such bijections are isometries for a broader class of Banach spaces and sums.

Findings

Non-expansive bijections are isometries in this setting.

Extends previous results to $ ext{l}_1$-sums of strictly convex Banach spaces.

Generalizes the class of spaces where non-expansive bijections are isometries.

Abstract

Extending recent results by Cascales, Kadets, Orihuela and Wingler (2016), Kadets and Zavarzina (2017), and Zavarzina (2017) we demonstrate that for every Banach space $X$ and every collection $Z_i, i\in I$ of strictly convex Banach spaces every non-expansive bijection from the unit ball of $X$ to the unit ball of sum of $Z_i$ by $\ell_1$ is an isometry.