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

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 thenorm 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 provethat, 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 severalconsequences. First we provide a new characterization of separable$L_1$-preduals containing an isometric copy of a space of affine continuousfunctions on a Choquet simplex. Then, we prove that an $\ell_1$-predual $X$contains almost isometric copies of the space $c$ of convergent sequences ifand only if $X^*$ lacks the stable $w^*$-fixed point property for nonexpansivemappings.