This is an announcement for the paper “The Szlenk index of $C(K,X)$” by Ryan M. Causey.

Abstract: Given any Banach space X and any w∗-compact subset $K$ of $X^*$, we compute the Szlenk index of the $w^*$ closed, convex hull of $K$ as a function of the Szlenk index of $K$. As a consequence, for any compact, Hausdorff topological space $K$ and any Banach space $X$, we compute the the Szlenk index of $C(K, X)$ as a function of the Szlenk index of $X$ and the Cantor-Bendixson index of $K$. Also as an application, we compute the Szlenk index of any injective tensor product in terms of $S_z(X)$ and $S_z(Y)$. As another application, we give a complete characterization of those ordinals which occur as the Szlenk index of a Banach space, as well as those ordinals which occur as the Bourgain $\ell_1$ or $c_0$ index of a Banach space.

The paper may be downloaded from the archive by web browser from URL http://arxiv.org/abs/1604.07875