$\alpha$-minimal Banach spaces
Christian Rosendal

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 -minimal for some countable ordinal if, for any two block subspaces, the Bourgain embeddability index of one into the other is at least . We prove a dichotomy that characterises when a Banach space has an -minimal subspace, which contributes to the ongoing project, initiated by W. T. Gowers, of classifying separable Banach spaces by identifying characteristic subspaces.
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Taxonomy
TopicsAdvanced Banach Space Theory · Advanced Topology and Set Theory
