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 \begin{equation*} r^*(X) ≔\sup \left \lbrace \left \|g^*\right \|: g^*\in \left (ext\, B_{\ell _1}\right )’\right \rbrace =\sup \left \lbrace \left \| \alpha \right \|: \, \alpha \in B_{\ell _1}, \, W_\alpha \subset X\right \rbrace . \end{equation*} 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 outcomes. First we provide a new characterization of $\ell _1$-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.