This is an announcement for the paper "Arbitrarily distortable Banach spaces of higher order" by Kevin Beanland, Ryan Causey, and Pavlos Motakis.

Abstract: We study an ordinal rank on the class of Banach spaces with bases that quantifies the distortion of the norm of a given Banach space. The rank $AD(\cdot)$, introduced by P. Dodos, uses the transfinite Schreier familes and has the property that $AD(X) < \omega_1$ if and only if $X$ is arbitrarily distortable. We prove several properties of this rank as well as some new results concerning higher order $\ell_1$ spreading models. We also compute this rank for for several Banach spaces. In particular, it is shown that class of Banach spaces $\mathfrak{X}^{\omega^\xi}_{0,1}$ , which each admit $\ell_1$ and $c_0$ spreading models hereditarily, and were introduced by S.A. Argyros, the first and third author, satisfy $AD(\mathfrak{X}^{\omega^\xi}_{0,1}) = \omega^\xi + 1$. This answers some questions of Dodos.

Archive classification: math.FA

Submitted from: CAUSEYRM@mailbox.sc.edu

The paper may be downloaded from the archive by web browser from URL

http://front.math.ucdavis.edu/1408.5065

or

http://arXiv.org/abs/1408.5065