A Coarse-Lipschitz Embedding of $c_0$ into a Separable Dual Banach Space

Abstract

Let $$ G=\mathbb{Z}^{<\omega}\subset c_0 \qquad\text{and}\qquad G_R=G\cap R B_{c_0},\quad R\in\mathbb{N}, $$ where the metric $d$ is inherited from $c_0$. On each $G_R$ we construct a commuting family of retractions onto finite initial segments of a special ordering of $G_R$, with Lipschitz constant at most two. This yields a boundedly complete basis of the Lipschitz-free space $\mathcal{F}(G_R)$ whose basis constant is at most two, and $2R$-equivalent to the unit vector basis of $\ell_1$. Thus the space is $2$-isomorphic to a dual space. The constant two is sharp: the radius-two grid $G_2$ does not embed with distortion strictly less than two into a separable dual Banach space. Using Kalton's annular decomposition, we then embed $\mathcal{F}(G)$ into a fixed separable dual space with distortion at most $2(1+\varepsilon)$ for every $\varepsilon>0$. Since $G$ is an integer net in $c_0$, this gives a coarse-Lipschitz embedding of $c_0$ into a separable dual. The optimal coarse-Lipschitz distortion, understood as an infimum over all separable dual targets, is equal to two.