Lipschitz-free spaces over Cantor sets and approximation properties

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.