Lipschitz-free spaces and approximating sequences of projections

Abstract

The Lipschitz-free space \({\mathcal {F}}(M)\) has an F.D.D. when M is a separable \({\mathcal {L}}_1\)-Banach space, or when \(M\subset {\mathbb {R}}^n\) is a somewhat regular subset. The interplay between the existence of extension operators for Lipschitz maps and the \((\pi )\)-property in Lipschitz-free spaces is investigated. If M is an arbitrary metric space, then \({\mathcal {F}}(M)\) has the \((\pi )\)-property up to a universal logarithmic factor. It follows in particular that the \((\pi )\)-property up to a logarithmic factor fails to imply the approximation property. A list of commented open problems is provided.