Banach spaces from barriers in high dimensional Ellentuck spaces

Alvaro Arias, Natasha Dobrinen, Gabriel Girón-Garnica, José G. Mijares

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 $T_k(d,\theta)$.


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DOI: https://doi.org/10.4115/jla.2018.10.5

Copyright (c) 2018 Alvaro Arias, Natasha Dobrinen, Gabriel Girón-Garnica, José G. Mijares

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Journal of Logic and Analysis ISSN:  1759-9008