Lipschitz-free spaces over Cantor sets and approximation properties

TL;DR

This paper investigates the approximation properties of Lipschitz-free spaces over the Cantor set with various metrics, showing that the set of metrics yielding spaces with the metric approximation property is residual, while those failing it are dense and meager.

Contribution

It characterizes the generic approximation property behavior of Lipschitz-free spaces over the Cantor set across different metrics, answering a question by Godefroy.

Findings

Metrics with Lipschitz-free spaces having the metric approximation property form a residual set.

Metrics with Lipschitz-free spaces failing the approximation property are dense and meager.

The results apply to all metrics on the Cantor set inducing its usual topology.

Abstract

Let $K=2^\mathbb{N}$ be the Cantor set, let $\mathcal{M}$ be the set of all metrics $d$ on $K$ that give its usual (product) topology, and equip $\mathcal{M}$ with the topology of uniform convergence, where the metrics are regarded as functions on $K^2$. We prove that the set of metrics $d\in\mathcal{M}$ for which the Lipschitz-free space $\mathcal{F}(K,d)$ has the metric approximation property is a residual $F_{\sigma\delta}$ set in $\mathcal{M}$, and that the set of metrics $d\in\mathcal{M}$ for which $\mathcal{F}(K,d)$ fails the approximation property is a dense meager set in $\mathcal{M}$. This answers a question posed by G. Godefroy.