Unbounded order convergence in dual spaces

Abstract

A net (x α ) in a vector lattice X is said to be unbounded order convergent (or uo-convergent, for short) to x ∈ X if the net (|x α − x| ∧ y) converges to 0 in order for all y ∈ X + . In this paper, we study unbounded order convergence in dual spaces of Banach lattices. Let X be a Banach lattice. We prove that every norm bounded uoconvergent net in X * is w * -convergent iff X has order continuous norm, and that every w * -convergent net in X * is uo-convergent iff X is atomic with order continuous norm. We also characterize among σ-order complete Banach lattices the spaces in whose dual space every simultaneously uo-and w * -convergent sequence converges weakly/in norm.

On the C‐property and ‐representations of risk measures

“…Finally, we remark that most of our results hold in the general framework of Banach lattices; here we need to replace a.e. convergence with the notion of unbounded order convergence, which was recently developed in Gao (2014), Gao and Xanthos (2014), and Gao, Troitsky, and Xanthos (2017).…”

Completeness for vector lattices

“…We show in this section that a vector lattice X is universally complete if and only if it is uo-complete. This answers an open problem in [8] and offers another reason that the notion of unboundedly order convergence deserves the particular attention that it recieved recently by many authors (see [10,9,11,8] and the refenrences cited there). Let first recall some definitions.…”

Unbounded order continuous operators on Riesz spaces

“…The concept of unbounded order convergence or uo-convergence was introduced in [8] and is proposed firstly in [3]. It has recently been intensively studied in several papers [4,5,6]. Recall that a net (x α ) α∈A in a Riesz space E is order convergent (or, o-convergent for short) to x ∈ E, denoted by x α o − → x whenever there exists another net (y β ) β∈B in E such that y β ↓ 0 and that for every β ∈ B, there exists α 0 ∈ A such that |x α − x| ≤ y β for all α ≥ α 0 .…”