Order denseness in free Banach lattices

TL;DR

This paper establishes that the free vector lattice over a Banach space is order dense in certain free p-convex Banach lattices only if the space is finite-dimensional, correcting a previous claim and addressing an open question.

Contribution

It proves the precise conditions under which order denseness holds in free Banach lattices, correcting prior misconceptions and advancing understanding in the field.

Findings

Order denseness holds iff the Banach space is finite-dimensional.

Identifies a gap in previous proof claiming universal order denseness.

Provides partial answers to an open question in Banach lattice theory.

Abstract

We prove a fundamental property: the free vector lattice $FVL[E]$ over a Banach space E is order dense in the free p-convex Banach lattice $FBL^{(p)}[E],~~1 ^leq p \leq \infty,$ if and only if E is finite-dimensional. In a recent work, Oikhberg, Tradacete, Taylor, and Troitsky claimed that order denseness holds for all Banach spaces. We point out a gap in their proof, and consequently, any conclusions relying on this claim require reexamination -- a task we also undertake in this present paper. A key tool in our approach, which also leads to a partial answer to an open question recently posed by these authors.