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Higher Order Tsirelson Spaces and their Modified Versions are Isomorphic

Hung Viet Chu, Thomas Schlumprecht

Abstract

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

Higher Order Tsirelson Spaces and their Modified Versions are Isomorphic

Abstract

We prove that for every countable ordinal , the Tsirelson's space of order , is naturally, i.e., via the identity, -isomorphc to its modified version. For the first step, we prove that the Schreier family is the same as its modified version , thus answering a question by Argyros and Tolias. As an application, we show that the algebra of linear bounded operators on has closed ideals.
Paper Structure (5 sections, 19 theorems, 98 equations)

This paper contains 5 sections, 19 theorems, 98 equations.

Table of Contents

  1. Introduction
  2. Schreier families of higher order
  3. Equality of the families $\mathcal{S}_\xi$ and $\mathcal{S}^M_\xi$, $\xi<\omega_1$
  4. Modified Tsirelson spaces of higher order
  5. Application: the cardinality of the closed ideals of $\mathcal{L}(T[\mathcal{S}_\xi,\theta])$

Key Result

Proposition 2.1

If $0 < \xi = \gamma + 1 < \omega_1$ and $A\in\text{\rm MAX}(\mathcal{S}_\xi)$, then there are unique elements $A_i\in \text{\rm MAX}(\mathcal{S}_\gamma), i = 1, 2, \ldots, \min A$, so that

Theorems & Definitions (36)

  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Definition 3.1: The $\xi$-analysis of maximal $\mathcal{S}_\xi$ sets
  • Remark 3.2
  • Lemma 3.3: The replacement lemma
  • proof
  • Theorem 3.4
  • Remark 3.5
  • ...and 26 more