Some loose ends on unbounded order convergence

TL;DR

This paper explores new properties of unbounded order convergence in vector lattices, establishing conditions under which nets are uo-Cauchy and possess uo-limits in order continuous Banach lattices.

Contribution

It provides new results linking uo-Cauchy nets and uo-limits, clarifying aspects of unbounded order convergence in order continuous Banach lattices.

Findings

Norm bounded positive increasing nets are uo-Cauchy

Every uo-Cauchy net has a uo-limit in the universal completion

Advances understanding of unbounded order convergence in Banach lattices

Abstract

The notion of almost everywhere convergence has been generalized to vector lattices as unbounded order convergence, which proves a very useful tool in the theory of vector and Banach lattices. In this short note, we establish some new results on unbounded order convergence that tie up some loose ends. In particular, we show that every norm bounded positive increasing net in an order continuous Banach lattice is uo-Cauchy and that every uo-Cauchy net in an order continuous Banach lattice has a uo-limit in the universal completion.