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

Antonio J. Guirao; Vicente Montesinos; Andrés Quilis

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arXiv · 2305.08543

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

Antonio J. Guirao·Vicente Montesinos·Andrés Quilis

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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 $r < 2$.

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Antonio J. Guirao, Vicente Montesinos, Andrés Quilis. 2023-05-15. On projectional skeletons and the Plichko property in Lipschitz-free Banach spaces. https://arxiv.org/abs/2305.08543

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