A negative solution to the complemented subspace problem in Banach lattices

TL;DR

This paper demonstrates that in Banach lattice theory, a complemented subspace need not be isomorphic to a Banach lattice, resolving a long-standing open problem.

Contribution

It provides a negative example showing complemented subspaces in Banach lattices are not necessarily Banach lattices themselves, answering a major open question.

Findings

Complemented subspaces may not be Banach lattices.

Provides a counterexample to a longstanding conjecture.

Advances understanding of structure in Banach lattices.

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.