Embeddings of Lipschitz-free spaces into l₁

We show that, for a separable and complete metric space M, the Lipschitz-free space F(M) embeds linearly and almost-isometrically into l(1) if and only if M is a subset of an 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 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 F(M) when M is a subset of an R-tree. (c) 2020 Elsevier Inc. All rights reserved.