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A negative solution to the complemented subspace problem in Banach lattices

D. de Hevia, G. Martínez-Cervantes, A. Salguero-Alarcón, P. Tradacete

TL;DR

The paper delivers a negative solution to the Complemented Subspace Problem for Banach lattices by constructing the Plebanek-Salguero space PS_2, a 1-complemented subspace of a $C(K)$-space that is not linearly isomorphic to any Banach lattice. The argument hinges on restricting norming free sets in the dual ball via a delicate inductive construction of almost disjoint cylinder families, together with a separation-of-measures framework that rules out $C(K)$ representations. The authors further show PS_2 is a $C_{\\sigma}(K)$-space but not an AM-space, highlighting its place outside well-behaved lattice classes, and extend the construction to complex Banach lattices through a complex variant, proving a negative CSP in the complex setting as well. Collectively, the results clarify that linear and lattice structures can diverge more broadly than previously believed, even for complemented subspaces of classical lattice spaces. The work connects to local unconditional structure, Johnson-Lindenstrauss type constructions, and the duality between $L_1$- and $C(K)$-space representations, with implications for the broader landscape of $\\mathcal{L}_\\infty$-space theory.

Abstract

Building on a recent construction of G. Plebanek and the third named author, it is shown that a complemented subspace of a Banach lattice need not be linearly isomorphic to a Banach lattice. This solves a long-standing open question in Banach lattice theory.

A negative solution to the complemented subspace problem in Banach lattices

TL;DR

The paper delivers a negative solution to the Complemented Subspace Problem for Banach lattices by constructing the Plebanek-Salguero space PS_2, a 1-complemented subspace of a -space that is not linearly isomorphic to any Banach lattice. The argument hinges on restricting norming free sets in the dual ball via a delicate inductive construction of almost disjoint cylinder families, together with a separation-of-measures framework that rules out representations. The authors further show PS_2 is a -space but not an AM-space, highlighting its place outside well-behaved lattice classes, and extend the construction to complex Banach lattices through a complex variant, proving a negative CSP in the complex setting as well. Collectively, the results clarify that linear and lattice structures can diverge more broadly than previously believed, even for complemented subspaces of classical lattice spaces. The work connects to local unconditional structure, Johnson-Lindenstrauss type constructions, and the duality between - and -space representations, with implications for the broader landscape of -space theory.

Abstract

Building on a recent construction of G. Plebanek and the third named author, it is shown that a complemented subspace of a Banach lattice need not be linearly isomorphic to a Banach lattice. This solves a long-standing open question in Banach lattice theory.
Paper Structure (10 sections, 12 theorems, 65 equations)

This paper contains 10 sections, 12 theorems, 65 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and auxiliary results on Banach lattices
  3. The Plebanek-Salguero space
  4. General facts
  5. Norming and free subsets
  6. Separation of measures
  7. The heart of the construction of ${\normalfont\textsf{PS}}_\textsf{2}$
  8. The Plebanek-Salguero space is not a Banach lattice
  9. Relation to other classes of ${\mathcal{L}}_\infty$-spaces
  10. The CSP for complex Banach lattices

Key Result

Proposition 2.1

Let $X$ be a Banach lattice which is an $\mathcal{L}_1$-space. Then $X$ is lattice isomorphic to an $L_1$-space.

Theorems & Definitions (34)

  • Proposition 2.1
  • proof
  • Corollary 2.2
  • proof
  • Corollary 2.3
  • proof
  • Corollary 2.4
  • proof
  • Remark 2.5
  • Remark 2.6
  • ...and 24 more