Approximation and Schur properties for Lipschitz free spaces over compact metric spaces

TL;DR

This paper explores the structure of Lipschitz-free spaces over compact metric spaces, demonstrating their rich properties including containing any separable Banach space, failing the approximation property, and having the Schur property in certain cases.

Contribution

It constructs compact metric spaces homeomorphic to the Cantor space with Lipschitz-free spaces exhibiting diverse and complex properties, answering several open questions.

Findings

Lipschitz-free space over a countable compact space has the Schur property.

Existence of compact metric spaces with Lipschitz-free spaces containing any separable Banach space.

Counterexamples of Lipschitz-free spaces failing the approximation property.

Abstract

We prove that for any separable Banach space $X$, there exists a compact metric space which is homeomorphic to the Cantor space and whose Lipschitz-free space contains a complemented subspace isomorphic to $X$. As a consequence we give an example of a compact metric space which is homeomorphic to the Cantor space and whose Lipschitz-free space fails the approximation property and we prove that there exists an uncountable family of topologically equivalent distances on the Cantor space whose free spaces are pairwise non isomorphic. We also prove that the free space over a countable compact metric space has the Schur property. These results answer questions by G. Godefroy.