Seminar on Banach and Metric Space Geometry

Date: February 1, 2024

Time: 10:00AM - 11:00AM

Location: BLOC 302

Speaker: Hung Viet Chu, Texas A&M University

Title: Higher Order Tsirelson Spaces and their Modified Version are Isomorphic

Abstract: In this talk, we sketch the proof that the Tsirelson space $T_\xi$ is naturally isomorphic to its modified version $T^M_\xi$ for each countable ordinal $\xi$. We begin by recalling the definition of Schreier families $\mathcal{S}_\xi$, where $\xi$ is a countable ordinal, and their modified version $\mathcal{S}^M_\xi$. These families are defined by transfinite induction on $\xi$. In the case of a successor ordinal $\xi$, i.e., $\xi = \gamma+1$, the family $\mathcal{S}_\xi$ is the collection of unions of sets $E_1< E_2 < \cdots < E_d$ in $\mathcal{S}_\gamma$ with $\min E_1\geqslant d$, where $E_i < E_j$ means that $\max E_i < \min E_j$. On the other hand, the modified version $\mathcal{S}^M_\xi$ only requires the sets $E_i$ to be disjoint instead of being consecutive. From these definitions, we know that $\mathcal{S}_\xi \subset \mathcal{S}^M_\xi$. Our first result shows that $\mathcal{S}_\xi$ is actually equal to $\mathcal{S}^M_\xi$ for each countable ordinal $\xi$, thus answering a question by Argyros and Tolias. This result together with certain tree analysis of the norming sets of $T_\xi$ and $^M_\xi$ give us their isomorphism. As an application, we show that the algebra of linear bounded operators on $T_\xi$ has $2^{\mathfrak c}$ closed ideals. The speaker is thankful to Dr. Schlumprecht for his excellent guidance in this joint work.