Isometries without linear shadows: failure of Figiel's theorem beyond local convexity

Abstract

A classical theorem of Figiel asserts that every isometric embedding $\phi :X\to Y$ between real Banach spaces, with $\phi (0)=0$, admits a linear contractive left inverse on the closed linear span of $\phi (X)$. We show that this phenomenon breaks down completely once local convexity is lost. More precisely, for every $0<p<1$ and every nonzero real Banach space $X$, we construct a $p$-Banach space $Y$ and an isometric embedding $\phi :X\to Y$ such that $\phi (0)=0$ and the closed linear span of $\phi (X)$ is the entire space $Y$, but there is no linear map, bounded or not, on the linear span of $\phi (X)$ which is a left inverse of $\phi$. The construction is carried out using the $p$-Banach counterpart of Lipschitz-free spaces. In fact, $Y$ may be chosen isomorphic to the Lipschitz-free $p$-space over $X$. The main ingredient is a quantitative estimate controlling the distance to elementary molecules in Lipschitz-free $p$-spaces, a phenomenon specific to the range $0<p<1$. As a further consequence, we show that a real Banach space has the isometric $p$-lifting property if and only if its dimension is at most one.