Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Transfinite (almost isometric) ideals in Banach spaces

Esteban Martínez Vañó, Abraham Rueda Zoca

TL;DR

The paper develops transfinite analogues of ideals and almost isometric ideals in Banach spaces, linking κ-ideals to κ-injective spaces and κ-(A)UD spaces to (A)UD_<κ spaces. It provides structural results showing how these transfinite notions interact with tensor products and Lipschitz-free spaces, and explores ultrapower behavior, showing sharp preservation results for the injective setting and limitations for the projective case. It also revisits classical results like local reflexivity and Sims-Yost/Abrahamsen in the transfinite setting, highlighting substantial consistency and limitation phenomena under ZFC. Overall, the work unifies extension properties, geometric/norm features, and transfinite density constraints to give a framework for transfinite ideals in Banach spaces with both constructive results and open questions for future study.

Abstract

We present and study some transfinite versions of (almost isometric) ideals in Banach spaces. As these notions are closely related with Lindenstrauss and Gurariĭ spaces respectively, we will present a similar characterization for transfinite injective spaces and spaces of (almost) universal disposition in terms of these transfinite ideals. Furthermore, we construct several examples outside these type of Banach spaces and make a revision of some classical results for transfinite ideals.

Transfinite (almost isometric) ideals in Banach spaces

TL;DR

The paper develops transfinite analogues of ideals and almost isometric ideals in Banach spaces, linking κ-ideals to κ-injective spaces and κ-(A)UD spaces to (A)UD_<κ spaces. It provides structural results showing how these transfinite notions interact with tensor products and Lipschitz-free spaces, and explores ultrapower behavior, showing sharp preservation results for the injective setting and limitations for the projective case. It also revisits classical results like local reflexivity and Sims-Yost/Abrahamsen in the transfinite setting, highlighting substantial consistency and limitation phenomena under ZFC. Overall, the work unifies extension properties, geometric/norm features, and transfinite density constraints to give a framework for transfinite ideals in Banach spaces with both constructive results and open questions for future study.

Abstract

We present and study some transfinite versions of (almost isometric) ideals in Banach spaces. As these notions are closely related with Lindenstrauss and Gurariĭ spaces respectively, we will present a similar characterization for transfinite injective spaces and spaces of (almost) universal disposition in terms of these transfinite ideals. Furthermore, we construct several examples outside these type of Banach spaces and make a revision of some classical results for transfinite ideals.
Paper Structure (14 sections, 29 theorems, 99 equations)

This paper contains 14 sections, 29 theorems, 99 equations.

Table of Contents

  1. Introduction
  2. Notation and preliminary results
  3. kappa injective spaces and spaces of universal disposition
  4. Almost square and octahedral Banach spaces
  5. Tensor product notation
  6. Lipschitz-free spaces
  7. Ultrapowers of Banach spaces
  8. Transfinite ideals
  9. Transfinite (almost) isometric ideals
  10. A revision of classical results
  11. The Principle of Local Reflexivity
  12. Abrahamsen and Sims-Yost theorem
  13. Almost isometric ideals in tensor product and ultrapower spaces
  14. Concluding remarks and open questions

Key Result

Theorem 2.1

Let $X$ be a Banach space. Then, for every finite dimensional subspace $E$ of $X^{**}$, every finite dimensional $F\subseteq X^*$ and every $\varepsilon>0$ there exists a linear operator $T:E\longrightarrow X$ such that

Theorems & Definitions (66)

  • Definition 1.1
  • Theorem 2.1: Principle of local reflexivity
  • Theorem 2.2: Sims-Yost, simyos
  • Theorem 2.3: Abrahamsen, abrahamsen15
  • Example 2.4
  • Remark 2.5
  • Definition 3.1
  • Remark 3.2
  • Theorem 3.3
  • proof
  • ...and 56 more