Completeness of Unbounded Convergences

TL;DR

This paper investigates the conditions under which Banach lattices are complete with respect to unbounded order convergence, establishing an equivalence with monotone completeness and exploring properties of minimal topologies.

Contribution

It proves that a Banach lattice is boundedly uo-complete if and only if it is monotonically complete and analyzes the completeness properties of minimal topologies.

Findings

Banach lattice is boundedly uo-complete iff it is monotonically complete

Minimal topologies are exactly the Hausdorff locally solid topologies where uo-convergence implies topological convergence

Established new connections between unbounded convergence and lattice completeness

Abstract

As a generalization of almost everywhere convergence to vector lattices, unbounded order convergence has garnered much attention. The concept of boundedly uo-complete Banach lattices was introduced by N. Gao and F. Xanthos, and has been studied in recent papers by D. Leung, V.G. Troitsky, and the aforementioned authors. We will prove that a Banach lattice is boundedly uo-complete iff it is monotonically complete. Afterwards, we study completeness-type properties of minimal topologies; minimal topologies are exactly the Hausdorff locally solid topologies in which uo-convergence implies topological convergence.