Embeddings of Lipschitz-free spaces into l(1)

Abstract

[EN] We show that, for a separable and complete metric space $M$, the Lipschitz-free space $\mathcal{F}(M)$ embeds linearly and almost-isometrically into $\ell_1$ if and only if $M$ is a subset of an $\mathbb{R}$-tree with length measure 0. Moreover, it embeds isometrically if and only if the length measure of the closure of the set of branching points of $M$ (taken in any minimal $\mathbb{R}$-tree that contains $M$) is also 0. We also prove that, for subspaces of $L_1$ spaces, every extreme point of the unit ball is preserved; as a consequence we obtain a complete characterization of extreme points of the unit ball of $\mathcal{F}(M)$ when $M$ is a subset of an $\mathbb{R}$-tree.