On complementation and $L$-embedding of Lipschitz-free spaces in their biduals

Abstract

A closed subspace $E$ of a Banach space $F$ is \emph{$L$-embedded in $F$} if there is a projection $P\colon F\to E$ such that $\|x\|=\|Px\|+\|x-Px\|$ for all $x\in F$. We show that if $X$ is a proper, geodesic metric space with locally finite $\H^1$-measure, then its Lipschitz-free space, denoted by $\F(X)$, is $L$-embedded in its bidual. We also extend an argument by Cúth, Kalenda, and Kaplický, to show that if $E$ is a finite dimensional normed space and $C\ge 1$ is such that every subspace of $E$ is $C$-complemented in $E$, then $\cal F(E)$ is $C$-complemented in its bidual. In particular, for every two-dimensional normed space $E$, $\cal F(E)$ is $1$-complemented in its bidual.