$\alpha$-minimal Banach spaces

TL;DR

This paper introduces a classification of Banach spaces based on the concept of $oldsymbol{ extit{ extalpha}}$-minimality, using Bourgain embeddability indices, and provides a dichotomy to identify when such subspaces exist.

Contribution

It establishes a dichotomy that characterizes the existence of $oldsymbol{ extalpha}$-minimal subspaces in Banach spaces, advancing the classification project initiated by Gowers.

Findings

Proves a dichotomy for $ extalpha$-minimal subspaces

Characterizes when Banach spaces contain $ extalpha$-minimal subspaces

Contributes to classifying Banach spaces by subspace structure

Abstract

A Banach space with a Schauder basis is said to be $\alpha$-minimal for some countable ordinal $\alpha$ if, for any two block subspaces, the Bourgain embeddability index of one into the other is at least $\alpha$. We prove a dichotomy that characterises when a Banach space has an $\alpha$-minimal subspace, which contributes to the ongoing project, initiated by W. T. Gowers, of classifying separable Banach spaces by identifying characteristic subspaces.