Applications of order convergence in Banach lattices

该建议是在巴拿赫理论和向量格。这是泛函分析的一个领域，主要研究巴拿赫空间中的偏序结构。该提案由几个部分组成。1. o-收敛（无界阶收敛）是阶收敛的导数。在最近的一系列论文中，我和我的合作者建立了一些性质，使它成为连接巴拿赫格和函数空间的一个很好的工具，它的重要性变得清晰起来。这导致了序闭凸集、巴拿赫格的前集和风险度量的应用。我将进一步探讨双收敛性的某些性质和应用。J.Grobler和C.Labuschagne最近开发了几种基于向量格的普遍补全的新技术，并使用它们将随机分析的某些结果扩展到向量格设置。我将探索这些技术之间的关系，二收敛，二对偶，二完备，并将其应用于无测度随机理论。我还想把这些技巧与D.Fremlin关于布尔代数上的可测函数空间的普遍完备空间的表述联系起来。3. 二代的序列。基本序列在巴拿赫空间理论中占有重要地位。在与M.Taylor正在进行的一个合作项目中，我们一直在研究双基序列，双基序列是Banach格中的基本序列，其基展开不仅在范数上收敛，而且在序上收敛。我们已经建立了这类序列的许多令人兴奋的性质。在分析中，我们证明了大多数经典的基本序列是双基的。我建议进一步研究双基序列，以及双基序列。特别地，我想确定Banach格的每一个闭子格是否包含双基或双基序列，以及序序完备Banach格中的每一个序基序列是否都是Schauder基。4. 自由巴拿赫格。自由Banach格FBL(A)和FBL[E]是由B.de Pagter， A.Wickstead, a.a aviles等人构造的。他们还发现了一个明确的FBL规范公式[E]。我在[T3]中找到了另一种构建FBL(A)和FBL[E]的方法。在与M.Taylor， P.Tradacete等人正在进行的项目中，我们使用了[T3]的方法来构造自由的p-凸Banach格；我们还为其规范找到了一个公式。我建议研究几个与联邦调查局有关的开放性问题[E]；其中，序列（|xk|）在FBL中是否基本[E]，当（xk）在E中是基本时，我建议使用p多项理论来寻找具有上p估计的自由Banach格的范数的显式公式。我对构造自由巴拿赫格代数也很感兴趣。5. 我要写一本关于向量格和巴拿赫格的书。

The proposal is in the theory of Banach and vector lattices. This is an area of Functional Analysis that focuses on partial order structures in Banach spaces. The proposal consists of several parts. 1. Uo-convergence (unbounded order convergence) is a derivative of order convergence. Its importance became clear after a recent series of papers where my collaborators and I established some properties that make uo-convergence an excellent tool for connecting Banach lattices with function spaces. This has led to applications to order closed convex sets, preduals of Banach lattices, and risk measures. I am going to further explore certain properties and applications of uo-convergence. J.Grobler and C.Labuschagne have recently developed several new techniques based on universal completions of vector lattices and used them to extend certain results of stochastic analysis to a vector lattice setting. I am going to explore the relationship between these techniques, uo--convergence, uo-dualss, and uo-completeness, and apply this to measure-free stochastic theory. I would also like to connect these techniques with D.Fremlin's representation of universally complete spaces as spaces of measurable functions on Boolean algebras. 3. Bibasic sequences. Basic sequences play a major role in the theory of Banach spaces. In an ongoing joint project with M.Taylor, we have been studying bibasic sequences, which are basic sequences in Banach lattices whose basis expansions converge not only in norm but also in order. We have established many exciting properties of such sequences. We proved that most classical basic sequences in Analysis are bibasic. I propose to further study bibasic sequences, as well as uo-bibasic sequences. In particular, I want to determine whether every closed sublattice of a Banach lattice contains a bibasic or a uo-bibasic sequence, and whether every order basic sequence in a sequentially order complete Banach lattice is (Schauder) basic. 4. Free Banach lattices. Free Banach lattices FBL(A) and FBL[E] have recently been constructed by B.de Pagter, A.Wickstead, A.Aviles, et al. They also found an explicit formula for the norm of FBL[E]. I found an alternative way of constructing FBL(A) and FBL[E] in [T3]. In an ongoing project with M.Taylor, P.Tradacete et al, we have used the approach of [T3] to construct free p-convex Banach lattices; we have also found a formula for its norm. I propose to work on several open questions related to FBL[E]; among others, whether the sequence (|xk|) is basic in FBL[E] whenever (xk) is basic in E. I propose to use the theory of p-multinorms to find an explicit formula for the norm of the free Banach lattice with the upper p-estimate. I am also interested in constructing free Banach lattice algebras. 5. I am going to complete writing a book about vector and Banach lattices.