Bibounded uo-convergence and b-property in vector lattices

TL;DR

This paper introduces bidual uo-convergence in vector lattices and explores its relationship with the b-property, establishing conditions under which order convergence in the space aligns with that in its bidual.

Contribution

It defines bidual uo-convergence and characterizes the b-property via the equivalence of order convergence in a space and its bidual in regular Riesz dual systems.

Findings

Order convergence in X equals order convergence in X^{**} iff X has b-property.

Bidual uo-convergence is a new concept in vector lattices.

The b-property is characterized by convergence equivalences in dual systems.

Abstract

We define bidual bounded $uo$-convergence in vector lattices and investigate relations between this convergence and $b$-property. We prove that for a regular Riesz dual system $\langle X,X^{\sim}\rangle$, $X$ has $b$-property if and only if the order convergence in $X$ agrees with the order convergence in $X^{\sim\sim}$.