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 $$G={\mathbb{Z}}^{<\omega }\subset c_0,G_R=G\cap R{B}_{c_0},R\in \mathbb{N},$$ 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 $2(1+\epsilon )$ for every $\epsilon >0$. 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 $X^{\ast }$ 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.

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