Subspaces of separable $L_1$-preduals: $W_\alpha$ everywhere

Abstract

The spaces $W_\alpha$ are the Banach spaces whose duals are isometric to $\ell_1$ and such that the standard basis of $\ell_1$ is $w^*$-convergent to $\alpha\in \ell_1$. The core result of our paper proves that an $\ell_1$-predual $X$ contains isometric copies of all $W_\alpha$, where the norm of $\alpha$ is controlled by the supremum of the norms of the $w^*$-cluster points of the extreme points of the closed unit ball in $\ell_1$. More precisely, for every $\ell_1$-predual $X$ we have $$ r^*(X)=\sup\left\lbrace \left\|g^*\right\|: g^*\in \left(\mathrm{ext}\, B_{\ell_1}\right)'\right\rbrace =\sup \left\lbrace \left\| \alpha\right\|: \, \alpha \in B_{\ell_1}, \, W_\alpha \subset X\right\rbrace . $$ We also prove that, for any $\varepsilon >0$, $X$ contains an isometric copy of some space $W_\alpha$ with $\left\| \alpha\right\|>r^*(X)- \varepsilon$ which is $(1+ \varepsilon)$-complemented in $X$. From these results we obtain several consequences. First we provide a new characterization of separable $L_1$-preduals containing an isometric copy of a space of affine continuous functions on a Choquet simplex. Then, we prove that an $\ell_1$-predual $X$ contains almost isometric copies of the space $c$ of convergent sequences if and only if $X^*$ lacks the stable $w^*$-fixed point property for nonexpansive mappings.