Weakly compactly generated Banach lattices

TL;DR

This paper investigates how weakly compact sets can generate Banach lattices, showing that in order continuous Banach lattices, such generation implies the lattice is weakly compactly generated (WCG).

Contribution

It establishes that in order continuous Banach lattices, the presence of a weakly compact generating set ensures the lattice is WCG, linking generation properties to weak compactness.

Findings

Weakly compact generating sets imply WCG in order continuous Banach lattices.

Existence of a weakly compact set generating the entire lattice is sufficient for WCG.

The study clarifies the relationship between weak compactness and lattice generation properties.

Abstract

We study the different ways in which a weakly compact set can generate a Banach lattice. Among other things, it is shown that in an order continuous Banach lattice $X$, the existence of a weakly compact set $K \subset X$ such that $X$ coincides with the band generated by $K$, implies that $X$ is WCG.