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On Subspaces of Indecomposable Banach Spaces

Piotr Koszmider, Zdeněk Silber

TL;DR

The paper investigates which Banach spaces can embed into indecomposable spaces in the nonseparable range, achieving a substantial partial answer for spaces of density at most $\mathfrak{c}$ that do not have $\ell_\infty$ as a quotient. Building on a framework of $C(K)$ spaces with few$^*$ operators, the authors construct indecomposable $C(K)$ spaces above a given $\mathcal X$ by embedding $\mathcal X$ into $C(B_{\mathcal X^*})$ and then into an ambient $C(M)$ with $M$ a Čech–Stone remainder, using Cantor systems, submorphisms, and abundant families to enforce indecomposability and controlled density. A key innovation is a strengthened Haydon-type lemma and a robust connectedness analysis of Gelfand spaces, which together yield indecomposable spaces of density $\le \mathfrak c$ containing a wide class of $\mathcal X$, including Asplund and weakly Lindelöf determined spaces. The work also addresses the presence of $\ell_\infty$ quotients, showing that indecomposable spaces can have $\ell_\infty$ as a quotient under certain set-theoretic assumptions, while highlighting open questions about whether every space not containing $\ell_\infty$ embeds into an indecomposable space.

Abstract

We address the following question: what is the class of Banach spaces isomorphic to subspaces of indecomposable Banach spaces? We show that this class includes all Banach spaces of density not bigger than the continuum which do not admit $\ell_\infty$ as a quotient (equivalently do not admit a subspace isomorphic to $\ell_1(\cc)$). This includes all Asplund spaces and all weakly Lindelöf determined Banach spaces of density not bigger than the continuum. However, we also show that this class includes some Banach spaces admitting $\ell_\infty$ as a quotient. This sheds some light on the question asked in [S. Argyros, R. Haydon, \emph{Bourgain-Delbaen $L^\infty$-spaces, the scalar-plus-compact property and related problems}, Proceedings of the International Congress of Mathematicians (ICM 2018), Vol. III, 1477--1510. Page 1502] whether all Banach spaces not containing $\ell_\infty$ embed in some indecomposable Banach spaces. Our method of constructing indecomposable Banach spaces above a given Banach space is a considerable modification of the method of constructing Banach spaces of continuous functions with few$^*$ operators developed before by the first-named author.

On Subspaces of Indecomposable Banach Spaces

TL;DR

The paper investigates which Banach spaces can embed into indecomposable spaces in the nonseparable range, achieving a substantial partial answer for spaces of density at most that do not have as a quotient. Building on a framework of spaces with few operators, the authors construct indecomposable spaces above a given by embedding into and then into an ambient with a Čech–Stone remainder, using Cantor systems, submorphisms, and abundant families to enforce indecomposability and controlled density. A key innovation is a strengthened Haydon-type lemma and a robust connectedness analysis of Gelfand spaces, which together yield indecomposable spaces of density containing a wide class of , including Asplund and weakly Lindelöf determined spaces. The work also addresses the presence of quotients, showing that indecomposable spaces can have as a quotient under certain set-theoretic assumptions, while highlighting open questions about whether every space not containing embeds into an indecomposable space.

Abstract

We address the following question: what is the class of Banach spaces isomorphic to subspaces of indecomposable Banach spaces? We show that this class includes all Banach spaces of density not bigger than the continuum which do not admit as a quotient (equivalently do not admit a subspace isomorphic to ). This includes all Asplund spaces and all weakly Lindelöf determined Banach spaces of density not bigger than the continuum. However, we also show that this class includes some Banach spaces admitting as a quotient. This sheds some light on the question asked in [S. Argyros, R. Haydon, \emph{Bourgain-Delbaen -spaces, the scalar-plus-compact property and related problems}, Proceedings of the International Congress of Mathematicians (ICM 2018), Vol. III, 1477--1510. Page 1502] whether all Banach spaces not containing embed in some indecomposable Banach spaces. Our method of constructing indecomposable Banach spaces above a given Banach space is a considerable modification of the method of constructing Banach spaces of continuous functions with few operators developed before by the first-named author.
Paper Structure (17 sections, 44 theorems, 65 equations)

This paper contains 17 sections, 44 theorems, 65 equations.

Table of Contents

  1. Introduction
  2. Notation and terminology
  3. Algebras of functions and their Gelfand spaces
  4. Gelfand spaces and maximal sets
  5. Separation of sets by algebras of functions
  6. Copies of $\beta\mathbb{N}$ in Gelfand spaces
  7. Čech-Stone reminders and submorphisms
  8. Čech-Stone reminders
  9. Algebras of functions on almost $P$-spaces
  10. Cantor systems of functions and abundant families
  11. Submorphisms
  12. A strengthening of Haydon's lemma
  13. Algebras with few$^*$ operators
  14. Connectedness of Gelfand spaces
  15. The construction
  16. ...and 2 more sections

Key Result

Theorem 2

If $\mathcal{X}$ is a Banach space of density not bigger than the continuum and it does not admit a quotient isomorphic to $\ell_\infty$, then there is an indecomposable Banach space of density not bigger than the continuum of the form $C(K)$ for a compact Hausdorff $K$ and an isometric embedding of

Theorems & Definitions (95)

  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Remark 6
  • Proposition 7
  • Definition 8
  • Proposition 9
  • proof
  • Lemma 10
  • ...and 85 more