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

Abstract

We prove that $c_0$ admits a coarse Lipschitz embedding into a separable dual Banach space and that the optimal coarse Lipschitz distortion is equal to two. Let $$ G=\mathbb Z^{<\omega}\subset c_0, \qquad G_R=G\cap R B_{c_0}, \quad R\in\mathbb N, $$ with the metric 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. Associated to these retractions there is a boundedly complete Schauder basis of $\mathcal F(G_R)$ whose basis constant is at most two and which is $2R$-equivalent to the unit vector basis of $\ell_1$. Consequently, each $\mathcal F(G_R)$ is $2$-isomorphic to a separable dual Banach space, uniformly in $R$. Kalton's annular decomposition then gives an embedding of $\mathcal F(G)$ into a separable dual space with distortion at most $2(1+\varepsilon)$ for every $\varepsilon>0$. A decomposition result of Aliaga and Medina further shows that \[ \mathcal F(G) \cong \Big( \bigoplus_{n\geq0}\mathcal F(G_{2^n}) \Big)_{\ell_1}, \] and hence $\mathcal F(G)$ itself is isomorphic to a separable dual Banach space. The constant two is sharp: if $G_2$ embeds into $X^*$ with distortion strictly smaller than two, then $X$ contains an isomorphic copy of $\ell_1$. It follows that the infimum of the coarse Lipschitz distortions of embeddings of $c_0$ into separable dual Banach spaces is exactly two.