Banach Spaces from Barriers in High Dimensional Ellentuck Spaces

TL;DR

This paper constructs a hierarchy of Banach spaces using barriers in high dimensional Ellentuck spaces, revealing their structural properties and embedding relations, extending classical Banach space theory.

Contribution

It introduces a new hierarchy of Banach spaces $T_k(d,\theta)$ based on barriers in high dimensional Ellentuck spaces, with detailed structural properties and embedding relations.

Findings

Each space contains arbitrarily large copies of $\ell_\infty^n$.

Spaces are $\ell_p$-saturated, extending $\ell_p$ spaces.

Hierarchy of spaces with strict embedding relations.

Abstract

A new hierarchy of Banach spaces $T_k(d,\theta)$, $k$ any positive integer, is constructed using barriers in high dimensional Ellentuck spaces \cite{DobrinenJSL15} following the classical framework under which a Tsirelson type norm is defined from a barrier in the Ellentuck space \cite{Argyros/TodorcevicBK}. The following structural properties of these spaces are proved. Each of these spaces contains arbitrarily large copies of $\ell_\infty^n$, with the bound constant for all $n$. For each fixed pair $d$ and $\theta$, the spaces $T_k(d,\theta)$, $k\ge 1$, are $\ell_p$-saturated, forming natural extensions of the $\ell_p$ space, where $p$ satisfies $d\theta=d^{1/p}$. Moreover, they form a strict hierarchy over the $\ell_p$ space: For any $j<k$, the space $T_j(d,\theta)$ embeds isometrically into $T_k(d,\theta)$ as a subspace which is non-isomorphic to $T_k(d,\theta)$.