Trees and Branches in Banach Spaces

TL;DR

This paper introduces a new asymptotic structure in Banach spaces using trees and branches, showing how certain properties of trees imply embeddings into spaces with finite-dimensional decompositions.

Contribution

It develops a novel framework for analyzing Banach spaces via trees and branches, linking asymptotic properties to embeddings with finite-dimensional structures.

Findings

Every countably infinite tree on a Banach space has a branch with a specific property.

Separable reflexive Banach spaces with certain tree properties embed into sums of finite-dimensional spaces.

The work connects tree structures to the existence of embeddings into classical Banach spaces.

Abstract

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.