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A Study of the Extreme Points in the Unit Ball of $JT$

Spiros A. Argyros

Abstract

In this note, we investigate the extreme points of the unit ball of the James Tree space ($JT$). We relate the geometric structure of $JT$ to the classical James space $J$ and provide partial characterizations of extremality based on the concept of separated vectors. We provide a complete characterization for positive vectors and establish the equal sums property for positive extreme points.

A Study of the Extreme Points in the Unit Ball of $JT$

Abstract

In this note, we investigate the extreme points of the unit ball of the James Tree space (). We relate the geometric structure of to the classical James space and provide partial characterizations of extremality based on the concept of separated vectors. We provide a complete characterization for positive vectors and establish the equal sums property for positive extreme points.
Paper Structure (24 sections, 24 theorems, 54 equations)

This paper contains 24 sections, 24 theorems, 54 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Dyadic Tree Structure
  4. The Space $JT$
  5. Extreme Points and Separated Vectors
  6. Necessary Conditions for Extremality
  7. Sufficient Conditions for Extremality
  8. $\ell_2$-Behavior and Extremality
  9. Connection to the James Space $J$
  10. Separated Vectors on Special Supports
  11. Local Isolation Implies $\ell_2$ Norm
  12. Positive Extreme Points
  13. Notation and Basic Properties
  14. The Greedy Algorithm and Norming Partitions
  15. The General Case: Infinite Support
  16. ...and 9 more sections

Key Result

Lemma 2.5

Let $\mathcal{B}$ be a complete subtree of $\mathcal{T}$. Then for every $x\in JT$ we have

Theorems & Definitions (58)

  • Definition 2.1: Segments
  • Definition 2.2: Complete Subtree
  • Definition 2.3: Support and Range
  • Definition 2.4: The Norm of $JT$
  • Lemma 2.5
  • proof
  • Proposition 2.6: Existence of Norming Partitions
  • Definition 2.7: Extreme Point
  • Definition 2.8: Separated Vector
  • Proposition 2.9: Characterization of Extreme Points
  • ...and 48 more