Lipschitz-free spaces over separable Banach spaces and their complementability in the bidual

Abstract

We prove that the Lipschitz-free space over a separable Banach space is complemented in its bidual if and only if the underlying Banach space is isomorphic to a Hilbert space. This answers negatively a long-standing question of Godefroy, Lancien, and Zizler concerning the case of $\ell_1$. We also characterize the existence of a projection of norm one and extend both characterizations to nonempty open convex subsets. Our approach yields several further results which we believe may be useful for understanding the structure of Lipschitz-free spaces over Banach spaces. These include an isometric representation of such spaces as quotients of Bochner $L^1$ spaces constructed using Gaussian measures.