Lipschitz-free spaces and Bossard's reduction argument
Richard J. Smith
TL;DR
This work advances the study of Lipschitz-free spaces ${\mathcal F}(M)$ by embedding them into a descriptive-set-theoretic framework and applying Bossard's reduction argument to derive universality results. It proves that isomorphic universality for free spaces over countable complete discrete metric spaces lifts to all separable Banach spaces, and that Lipschitz universality for those discrete spaces lifts to universality for all separable metric spaces; it also constructs countable complete discrete $M$ with ${\mathcal F}(M)$ failing the BAP, hence not dual, and determines the descriptive complexity of the approximation-property classes. The analysis blends metric-geometry constructions with sharp set-theoretic complexity (e.g., ${\boldsymbol\Pi}^{1}_{1}$-completeness) to classify when free spaces and their metric counterparts possess the AP, offering a robust framework for further exploration of universality and approximation properties in this area.
Abstract
We set up a descriptive set-theoretic framework to study Lipschitz-free spaces and use the reduction argument of Bossard to prove several results. We prove two universality results: if a separable Banach space is isomorphically universal for the class of Lipschitz-free spaces over the countable complete discrete metric spaces then it is isomorphically universal for the class of separable Banach spaces, and if a complete separable metric space is Lipschitz universal for the same class of metric spaces then it is Lipschitz universal for all separable metric spaces. We also show that there exist countable complete discrete metric spaces whose Lipschitz-free spaces fail the bounded approximation property and are thus not isomorphic to any dual Banach space. Finally, we calculate the descriptive complexity of the classes of separable Banach spaces and separable Lipschitz-free spaces having the approximation property.