Banach Spaces from Barriers in High Dimensional Ellentuck Spaces

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 $T_k(d,\theta )$.

28 pages, 6 figures; Keywords: Banach space, High dimensional Ellentuck space, barrier