This is an announcement for the paper “Power type ξ-Asymptotically uniformly smooth and ξ-asymptotically uniformly flat norms” by R. M. Causeyhttps://arxiv.org/find/math/1/au:+Causey_R/0/1/0/all/0/1.
Abstract: For each ordinal $\xi$ and each $1<p<\infty$, we offer a natural, ismorphic characterization of those spaces and operators which admit an equivalent $\xi$ -$p$ -asymptotically uniformly smooth norm. We also introduce the notion of $\xi$ -asymptotically uniformly flat norms and provide an isomorphic characterization of those spaces and operators which admit an equivalent $\xi$ -asymptotically uniformly flat norm. Given a compact, Hausdorff space $K$, we prove an optimal renormong theorem regarding the $\xi$ -asymptotic smoothness of $C(K)$ in terms of the Cantor-Bendixson index of $K$. We also prove that for all ordinals, both the isomorphic properties and isometric properties we study pass from Banach spaces to their injective tensor products. We study the classes of $\xi$ -$p$ -asymptotically uniformly smooth, $\xi$ -$p$ -asymptotically uniformly smoothable, $\xi$ -asymptotically uniformly flat, and $\xi$ -asymptotically uniformly flattenable operators. We show that these classes are either a Banach ideal or a right Banach ideal when assigned an appropriate ideal norm.
The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1705.09834