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}^M_\xi$, 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.