Pe{\l}czy\'nski's property (V$^*$) in Lipschitz-free spaces

TL;DR

This paper investigates Pelczyński's property (V*) in Lipschitz-free spaces, establishing conditions under which it holds, especially for spaces with specific geometric or metric properties.

Contribution

It proves that property (V*) is locally determined in Lipschitz-free spaces and identifies several geometric conditions ensuring its validity.

Findings

Property (V*) is locally determined in Lipschitz-free spaces.

Spaces with certain geometric properties, like being locally compact and purely 1-unrectifiable, have property (V*).

Hilbert spaces and specific Carnot-Carathéodory spaces also satisfy property (V*).

Abstract

We prove that Pelczy\'nski's property (V$^*$) is locally determined for Lipschitz-free spaces, and obtain several sufficient conditions for it to hold. We deduce that $\mathcal{F}(M)$ has property (V$^*$) when the complete metric space $M$ is locally compact and purely 1-unrectifiable, a Hilbert space, or belongs to a class of Carnot-Carath\'eodory spaces satisfying a bi-H\"older condition, including Carnot groups.