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arXiv · 2610.08616

On the $L$-embeddability of Lipschitz-free spaces in their biduals

Bruno de Mendonça Braga·Chris Gartland·Gilles Lancien·Pavlos Motakis·Eva Pernecká·Thomas Schlumprecht
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Abstract

A Banach space $X$ is $L$-embedded in its bidual if there is a projection $P\colon X^{**}\to X$ such that $\|x\|=\|Px\|+\|x-Px\|$ for all $x\in X^{**}$. We show that, as long as $X$ has dimension at least $2$, its Lipschitz-free space, denoted by $\mathcal{F}(X)$, is not $L$-embedded in its bidual. Our methods have applications to the problem of when the $L$-embeddability of $\mathcal{F}(M)$ passes to $\mathcal{F}(A)$ for a metric space $M$ and $A\subseteq M$. This is the case when $A$ is compact or when $A$ is geodesically closed.

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Bruno de Mendonça Braga, Chris Gartland, Gilles Lancien, Pavlos Motakis, Eva Pernecká, Thomas Schlumprecht. 2026-10-06. On the $L$-embeddability of Lipschitz-free spaces in their biduals. https://arxiv.org/abs/2610.08616

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