On projectional skeletons and the Plichko property in Lipschitz-free Banach spaces

TL;DR

This paper explores the Plichko property in Lipschitz-free Banach spaces, linking it to metric space geometry, and introduces new concepts like retractional trees to extend known results.

Contribution

It characterizes the Plichko property via metric properties, studies Lipschitz-free spaces of $ ext{R}$-trees, and introduces retractional trees to generalize results.

Findings

Lipschitz-free space of $ ext{R}$-trees has the Plichko property with molecules

A metric property characterizes the Plichko property witnessed by Dirac measures

No separable subspace of $ ext{l}_ ext{infinity}$ containing $c_0$ is an $r$-Lipschitz retract for $r<2$

Abstract

We study projectional skeletons and the Plichko property in Lipschitz-free spaces, relating these concepts to the geometry of the underlying metric space. Specifically, we identify a metric property that characterizes the Plichko property witnessed by Dirac measures in the associated Lipschitz-free space. We also show that the Lipschitz-free space of all $\mathbb{R}$-trees has the Plichko property witnessed by molecules, and define the concept of retractional trees to generalize this result to a bigger class of metric spaces. Finally, we show that no separable subspace of $\ell_\infty$ containing $c_0$ is an $r$-Lipschitz retract for $r < 2$, which implies in particular that $\mathcal{F}(\ell_\infty)$ is not $r$-Plichko for $r < 2$.