Some loose ends on unbounded order convergence

Abstract

Abstract. The notion of almost everywhere convergence has been generalized to vector lattices as unbounded order convergence, which proves to be 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.

Completeness for vector lattices

“…We will prove now that X is laterally complete. This fact is proved in [8] as noted above and we provide here an alternative proof which involves the sup-completion X s . Let (x α ) α∈A be a family of mutually disjoint positive vectors in X and define y F = sup α∈F x α for every finite subset F of A.…”

“…Recall that a Banach lattice is said to satisfy the weak Fatou property if there is a real r > 0 such that for every increasing net (x α ) with the supremum x ∈ X it follows that x ≤ r sup α x α . Since there is a Banach lattice X with weak Fatou property such that X n = {0} the next theorem improves [8,Theorem 2.3]. Theorem 15 will be used again to get a short proof.…”

“…One can ask whether or not there is an analogous characterization of universal completeness. In the recent paper [8], the authors proved that under some extra condition a vector lattice is universal complete if and only if every uo-Cauchy net is uo-convergent. The main result of the present paper (Theorem 15) states that this equivalence is still true without any further assumption.…”

Some results on unbounded absolute weak Dunford–Pettis operators

“…The notion of unbounded order convergence (uo-convergence, for short) was firstly introduced by Nakano in [14], then it was used and systematically investigated in [8,9,10,12,17]. After that, A. Bahramnezhad et al proposed the definition of unbounded order continuous operators in [3].…”

The countable sup property for lattices of continuous functions

“…Although uo-convergence is very exciting on its own, its value shows through its applications in Mathematical finance. For general results on uo-convergence and its unbounded norm version we refer the reader to [GX14,Gao14,KMT17,GTX,LC]. For applications of uoconvergence and its techniques to Mathematical finance we refer the reader to [GLX,GLX2].…”