Octahedrality in Lipschitz free Banach spaces

TL;DR

This paper investigates octahedrality in Lipschitz-free Banach spaces and shows that under certain topological conditions, these spaces exhibit the weak-star strong diameter two property, with an example confirming the optimality of these conditions.

Contribution

It establishes conditions under which Lipschitz-free Banach spaces are octahedral and demonstrates the weak-star strong diameter two property for Lipschitz functions into dual Banach spaces.

Findings

Lipschitz-free Banach spaces are octahedral under specific topological hypotheses.

Lipschitz functions into dual Banach spaces satisfy the weak-star strong diameter two property.

An example confirms the optimality of the topological hypotheses.

Abstract

The aim of this note is to study octahedrality in vector valued Lipschitz-free Banach spaces on a metric space under topological hypotheses on it. As a consequence, we get that the space of Lipschitz functions on a metric space valued in a dual Banach space satisfies the weak-star strong diameter two property, under natural topological hipothesess on the metric space. Also, we show an example proving that these hypotheses are optimal.