The Szlenk index of injective tensor products and convex hulls

Given any Banach space X and any weak⁎-compact subset K of X⁎, we compute the Szlenk index of the weak⁎-closed, convex hull of K as a function of the Szlenk index of K. Also as an application, we compute the Szlenk index of any injective tensor product of two operators. In particular, we compute the Szlenk index of an injective tensor product X⊗ˆεY in terms of Sz(X) and Sz(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 ℓ1 or c0 index of a Banach space.