Higher order Tsirelson spaces and their modified versions are isomorphic

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-isomorphic 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.