A characterisation of $L_1$-preduals in terms of extending Lipschitz maps

TL;DR

This paper characterizes $L_1$-predual Banach spaces by their extension properties for Lipschitz compact maps, providing new insights into the structure of these spaces and their Lipschitz-free counterparts.

Contribution

It offers a novel characterization of $L_1$-preduals based on Lipschitz map extension properties, linking geometric and functional analytic aspects.

Findings

$L_1$-preduals are characterized by Lipschitz extension properties.

Extension of Lipschitz compact maps is possible with controlled norm.

Implications for Lipschitz-free spaces and their structure.

Abstract

We characterise the Banach spaces $X$ which are $L_1$-predual as those for which every Lipschitz compact mapping $f:N\longrightarrow X$ admits, for every $\varepsilon>0$ and every $M$ containing $N$, a Lipschitz (compact) extension $F:M\longrightarrow X$ so that $\Vert F\Vert\leq (1+\varepsilon)\Vert f\Vert$. Some consequences are derived about $L_1$-preduals and about Lipschitz-free spaces.