Duality of Lipschitz-free spaces over ultrametric spaces

TL;DR

The paper provides a complete characterization of when the Lipschitz-free space $\mathcal{F}(M)$ over a complete separable ultrametric space $M$ is a dual Banach space, proving equivalence with spherical completeness and related 1-complementation properties. It constructs a natural predual $Y$ for $\mathcal{F}(M)$ in the separable case, and shows $Y^*=\mathcal{F}(M)$; when $M$ is proper, $Y$ coincides with $\mathrm{lip}_0^u(M)$, aligning with Dalet's predual. The results establish that $\mathrm{lip}_0^u(M)$ is an $M$-ideal in $\mathrm{Lip}_0(M)$ for ultrametric $M$, while providing a counterexample in the non-ultrametric compact case. Additionally, the work identifies strongly extreme points in $\mathrm{Lip}_0(M)$ for ultrametric spaces, implying that these Lipschitz function spaces are not LASQ and revealing intricate geometric structure of the unit ball. Overall, the work connects duality, preduals, and $M$-ideals with the ultrametric geometry of $M$, advancing understanding of Lipschitz-free spaces beyond proper spaces and answering questions about their general duality properties.

Abstract

We give a metric characterisation of when the Lipschitz-free space over a separable ultrametric space is a dual Banach space. In the case where the Lipschitz-free space has a predual, we show that this predual is M-embedded if and only if the metric space is proper. We show that for ultrametric spaces, the little Lipschitz space is always an M-ideal in the corresponding space of Lipschitz functions, and we show that this is not the case for metric spaces in general, thus answering a question posed by Werner in the negative. Finally, we show that the space of Lipschitz functions of an ultrametric space contains a strongly extreme point.

Theorems & Definitions (56)

• Definition 2.1
• Lemma 2.2
• Lemma 2.3
• proof
• Definition 2.4
• Definition 2.5
• Proposition 2.6
• Definition 2.7
• Definition 2.8
• Proposition 2.9