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Subspaces of $\ell_1$ satisfying Grothendieck's theorem

Jesús Suárez

TL;DR

The paper characterizes which subspaces of $\ell_1$ satisfy Grothendieck's theorem (GT-spaces) by linking the GT-property to the extendability of nonnegative quadratic forms. Using push-out diagrams and the theory of $p$-summing operators, it proves that for any quotient $\varphi: \ell_1 \to Z$, the following are equivalent: (i) every short exact sequence $0\to \ell_2 \to Y \to Z\to 0$ splits; (ii) $\ker \varphi$ is GT; (iii) every nonnegative quadratic form on $\ker \varphi$ extends to a nonnegative quadratic form on $\ell_1$; (iv) the extension can be taken delta-semidefinite. The results connect operator-summing theory, extension of quadratic forms, and exact-sequence splitting to provide a precise criterion for GT-spaces among $\ell_1$ subspaces and offer a framework that extends to other GT-spaces. This yields structural insights into the interplay between Grothendieck-type properties and extension phenomena in Banach-space theory.

Abstract

We characterize the subspaces $X$ of $\ell_1$ satisfying Grothendieck's theorem in terms of extension of nonnegative quadratic forms $q:X \longrightarrow \mathbb R$ to the whole $\ell_1$.

Subspaces of $\ell_1$ satisfying Grothendieck's theorem

TL;DR

The paper characterizes which subspaces of satisfy Grothendieck's theorem (GT-spaces) by linking the GT-property to the extendability of nonnegative quadratic forms. Using push-out diagrams and the theory of -summing operators, it proves that for any quotient , the following are equivalent: (i) every short exact sequence splits; (ii) is GT; (iii) every nonnegative quadratic form on extends to a nonnegative quadratic form on ; (iv) the extension can be taken delta-semidefinite. The results connect operator-summing theory, extension of quadratic forms, and exact-sequence splitting to provide a precise criterion for GT-spaces among subspaces and offer a framework that extends to other GT-spaces. This yields structural insights into the interplay between Grothendieck-type properties and extension phenomena in Banach-space theory.

Abstract

We characterize the subspaces of satisfying Grothendieck's theorem in terms of extension of nonnegative quadratic forms to the whole .
Paper Structure (8 sections, 6 theorems, 15 equations)

This paper contains 8 sections, 6 theorems, 15 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Quadratic forms
  4. The $p$-summing operators
  5. Short exact sequences
  6. Push-out space, sequence and diagram
  7. GT-spaces and nonnegative quadratic forms
  8. Remarks

Key Result

Theorem 1

Let $q$ be a quadratic form on a Banach space $X$ and $T:X\longrightarrow X^*$ the symmetric operator that generates $q$. The following conditions are equivalent:

Theorems & Definitions (9)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Proposition 1
  • proof
  • Theorem 4
  • proof
  • Proposition 2
  • proof