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

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.