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Accessible operators on ultraproducts of Banach spaces

Félix Cabello Sánchez

TL;DR

This work builds a bridge between representing linear operators on ultraproducts of Banach spaces via nonlinear maps and the theory of twisted sums through quasilinear maps. A central result is that an accessible operator on an ultraproduct is not the ultraproduct of linear operators precisely when a corresponding short exact sequence fails to split, tying operator accessibility to the three-space problem. The paper then provides explicit, concrete constructions on sequence spaces such as $\ell_p$ (via Ribe and Kalton–Peck maps) to produce accessible functionals and endomorphisms, and develops general procedures to generate accessible operators from a single quasilinear map, including restriction/truncation techniques and considerations of ultrasummands. The discussion culminates with a detailed account of when accessible operators must be ultraproducts, links to $\mathscr{K}$-spaces, and a survey of examples in nonlocally convex and exotic spaces, offering both methodological tools and concrete instances for further exploration in ultraproduct and twisted-sum theory.

Abstract

We address a question by Henry Towsner about the possibility of representing linear operators between ultraproducts of Banach spaces by means of ultraproducts of nonlinear maps. We provide a bridge between these "accessible" operators and the theory of twisted sums through the so-called quasilinear maps. Thus, for many pairs of Banach spaces $X$ and $Y$, there is an "accessible" operator $X_U\to Y_U$ that is not the ultraproduct of a family of operators $X\to Y$ if and only if there is a short exact sequence of quasi-Banach spaces and operators $0\to Y\to Z\to X\to 0$ that does not split. We then adapt classical work by Ribe and Kalton--Peck to exhibit pretty concrete examples of accessible functionals and endomorphisms for the sequence spaces $\ell_p$. The paper is organized so that the main ideas are accessible to readers working on ultraproducts and requires only a rustic knowledge of Banach space theory.

Accessible operators on ultraproducts of Banach spaces

TL;DR

This work builds a bridge between representing linear operators on ultraproducts of Banach spaces via nonlinear maps and the theory of twisted sums through quasilinear maps. A central result is that an accessible operator on an ultraproduct is not the ultraproduct of linear operators precisely when a corresponding short exact sequence fails to split, tying operator accessibility to the three-space problem. The paper then provides explicit, concrete constructions on sequence spaces such as (via Ribe and Kalton–Peck maps) to produce accessible functionals and endomorphisms, and develops general procedures to generate accessible operators from a single quasilinear map, including restriction/truncation techniques and considerations of ultrasummands. The discussion culminates with a detailed account of when accessible operators must be ultraproducts, links to -spaces, and a survey of examples in nonlocally convex and exotic spaces, offering both methodological tools and concrete instances for further exploration in ultraproduct and twisted-sum theory.

Abstract

We address a question by Henry Towsner about the possibility of representing linear operators between ultraproducts of Banach spaces by means of ultraproducts of nonlinear maps. We provide a bridge between these "accessible" operators and the theory of twisted sums through the so-called quasilinear maps. Thus, for many pairs of Banach spaces and , there is an "accessible" operator that is not the ultraproduct of a family of operators if and only if there is a short exact sequence of quasi-Banach spaces and operators that does not split. We then adapt classical work by Ribe and Kalton--Peck to exhibit pretty concrete examples of accessible functionals and endomorphisms for the sequence spaces . The paper is organized so that the main ideas are accessible to readers working on ultraproducts and requires only a rustic knowledge of Banach space theory.
Paper Structure (14 sections, 10 theorems, 60 equations, 1 figure)

This paper contains 14 sections, 10 theorems, 60 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Purpose
  3. Preliminaires
  4. Accessible operators and quasilinear maps
  5. Admissible families and accessible maps
  6. Quasilinear maps and accessible operators
  7. The "three-space" problem
  8. Applications
  9. Examples of accessible operators and functionals on $\ell_p$
  10. Accessible operators arising by restriction or "truncation"
  11. Accessible functionals on Banach spaces
  12. Accessible functionals on nonlocally convex spaces
  13. The derivation out of space
  14. Concluding remarks

Key Result

Lemma 2

Assume that $f_i: X_i\to Y_i$ form an admissible family. If $(f_i)_\mathscr{U}$ is homogeneous, then: Hence every accessible homogeneous map is induced by a uniformly bounded family of homogeneous maps.

Figures (1)

  • Figure :

Theorems & Definitions (33)

  • Definition 1
  • Lemma 2
  • proof
  • Definition 3
  • Theorem 4
  • proof
  • Definition 5
  • Corollary 6
  • Corollary 7
  • Lemma 8: Ribe
  • ...and 23 more