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Stability results of octahedrality in tensor product spaces

Abraham Rueda Zoca

Abstract

We prove that there exists a finite-dimensional Banach space $X$ such that $L_1^\mathbb C([0,1])\widehat{\otimes}_\varepsilon X$ fails the strong diameter two property and $L_\infty^\mathbb C([0,1])\widehat{\otimes}_πX^*$ fails to have octahedral norm. This proves that the octahedrality of the norm (respectively the strong diameter two property) is not automatically inherited from one factor by taking projective tensor product (respectively injective tensor product), which answers [16,Question 4.4].

Stability results of octahedrality in tensor product spaces

Abstract

We prove that there exists a finite-dimensional Banach space such that fails the strong diameter two property and fails to have octahedral norm. This proves that the octahedrality of the norm (respectively the strong diameter two property) is not automatically inherited from one factor by taking projective tensor product (respectively injective tensor product), which answers [16,Question 4.4].

Paper Structure

This paper contains 3 sections, 2 theorems, 9 equations.

Table of Contents

  1. Introduction
  2. Notation and preliminary results
  3. Main result

Key Result

Theorem \oldthetheorem

There exists a two dimensional complex Banach space $X$ such that $L_1^\mathbb C([0,1])\widehat{\otimes}_\varepsilon X$ fails the strong diameter two property, where $L_1^\mathbb C([0,1])$ stands for the space of complex-valued $L_1$-functions.

Theorems & Definitions (5)

  • Theorem \oldthetheorem
  • proof
  • Theorem \oldthetheorem
  • proof
  • Remark \oldthetheorem