Trees and Branches in Banach Spaces

Edward Odell, Thomas Schlumprecht

Published 2000-02-25Version 1

An infinite dimensional notion of asymptotic structure is considered. This notion is developed in terms of trees and branches on Banach spaces. Every countably infinite countably branching tree $\mathcal T$ of a certain type on a space X is presumed to have a branch with some property. It is shown that then X can be embedded into a space with an FDD $(E_i)$ so that all normalized sequences in X which are almost a skipped blocking of $(E_i)$ have that property. As an application of our work we prove that if X is a separable reflexive Banach space and for some $1<p<\infty$ and $C<\infty$ every weakly null tree $\mathcal T$ on the sphere of X has a branch C-equivalent to the unit vector basis of $\ell_p$, then for all $\epsilon>0$, there exists a finite codimensional subspace of X which $C^2+\epsilon$ embeds into the $\ell_p$ sum of finite dimensional spaces.