Higher Order Tsirelson Spaces and their Modified Versions are Isomorphic

Abstract

We prove that for every countable ordinal $\xi$, the Tsirelson's space $T_{\xi }$ of order $\xi$, is naturally, i.e., via the identity, $3$-isomorphc to its modified version. For the first step, we prove that the Schreier family ${\mathcal{S}}_{\xi }$ is the same as its modified version ${\mathcal{S}}_{\xi }^M$, thus answering a question by Argyros and Tolias. As an application, we show that the algebra of linear bounded operators on $T_{\xi }$ has $2^{\mathfrak{c}}$ closed ideals.