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Embedding $\ell_2$ and $J$ into subspaces of $JT$ and $JT^*$

Spiros A. Argyros, Manuel Gonzalez, Pavlos Motakis

Abstract

In the first part of the paper we show that every closed subspace of $JT$ or $JT^*$ contains $\ell_2$ complemented in $JT$ or $JT^*$ respectively, and $JT$ contains uncomplemented copies of $\ell_2$. As a result, the predual $\B$ of $JT$, as well as the spaces $JT$ and $JT^*$, are subprojective and superprojective. In the second part, we prove that every weakly Cauchy sequence that is not weakly convergent in $JT$ has a subsequence equivalent to the basis of $J$. Hence, every non-reflexive subspace of $JT$ contains an isomorphic copy of $J$, and every Schauder basic sequence in $JT$ has a subsequence which is equivalent either to the basis of $\ell_2$ or to the basis of $J$. Moreover these subspaces may be selected to be complemented in $JT$.

Embedding $\ell_2$ and $J$ into subspaces of $JT$ and $JT^*$

Abstract

In the first part of the paper we show that every closed subspace of or contains complemented in or respectively, and contains uncomplemented copies of . As a result, the predual of , as well as the spaces and , are subprojective and superprojective. In the second part, we prove that every weakly Cauchy sequence that is not weakly convergent in has a subsequence equivalent to the basis of . Hence, every non-reflexive subspace of contains an isomorphic copy of , and every Schauder basic sequence in has a subsequence which is equivalent either to the basis of or to the basis of . Moreover these subspaces may be selected to be complemented in .
Paper Structure (14 sections, 52 theorems, 64 equations)

This paper contains 14 sections, 52 theorems, 64 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The dyadic tree
  4. James Tree and its basic properties
  5. The norm of $JT$
  6. The spaces $\mathcal{B}$, $JT^*$
  7. The Space $JT^{**}$
  8. Complemented copies of $\ell_2$ and projectivity
  9. The spaces $\mathcal{B}$, $JT$ and $JT^*$ are subprojective
  10. The spaces $\mathcal{B}$, $JT$ and $JT^*$ are superprojective
  11. Uncomplemented copies of $\ell_2$
  12. Non-Reflexive Subspaces of $JT$
  13. Non-Trivial Weakly Cauchy Sequences in $J$
  14. Non-Trivial Weakly Cauchy Sequences in $JT$

Key Result

Theorem 1.1

Let $X$ be a separable Banach space containing no copies of $\ell_1$ with $X^*$ non-separable. Then there exists a tree-basis $(e_\alpha)_{\alpha\in \mathcal{T}}$ in $X$ satisfying the following properties:

Theorems & Definitions (96)

  • Theorem 1.1: Argyros, Dodos & Kanellopoulos ADK
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Proposition 1.5
  • Proposition 1.6
  • Proposition 3.1
  • Proposition 3.2
  • proof
  • Theorem 3.3
  • ...and 86 more