Applications of order convergence in Banach lattices

该建议是在理论的巴拿赫和向量格。这是一个领域的功能分析，重点是在Banach空间的偏序结构。该提案由几个部分组成。1. Uo-收敛（无界阶收敛）是阶收敛的导数。它的重要性在最近的一系列论文中变得清晰起来，在这些论文中，我和我的合作者建立了一些性质，这些性质使得uo收敛成为连接Banach格与函数空间的优秀工具。这导致了应用程序，以封闭的凸集，prefetching的巴拿赫格，和风险措施。我将进一步探讨uo收敛的某些性质和应用。J.Grobler和C.Labuschagne最近发展了几种基于向量格的泛完备化的新技术，并利用它们将随机分析的某些结果推广到向量格环境。我将探讨这些技术之间的关系，uo-收敛，uo-对偶，uo-完备性，并将其应用于无测度随机理论。我也想把这些技巧与D.Fremlin的表示普遍完整的空间作为空间的可测功能的布尔代数。3.双碱基序列。基本序列在Banach空间理论中起着重要作用。在与M.Taylor的一个正在进行的联合项目中，我们一直在研究双基序列，这是Banach格中的基本序列，其基展开式不仅在范数上收敛，而且在序上收敛。我们已经建立了许多令人兴奋的性质，这样的序列。我们证明了分析中的大多数经典基本序列是二元的。我建议进一步研究二元序列，以及uo-二元序列。特别地，我想确定Banach格的每一个闭子格是否包含一个二元序列或一个uo-二元序列，以及序完备Banach格的每一个序基序列是否是（Schauder）基序列. 4.自由Banach格Pagter，A.Wickstead，A.阿维莱斯等人最近构造了自由Banach格FBL（A）和FBL[E]，B.de我在[T3]中找到了另一种构造FBL（A）和FBL[E]的方法。在与M.Taylor，P.Tradacete等人正在进行的一个项目中，我们使用[T3]的方法构造了自由p-凸Banach格，并找到了它的范数公式。我建议研究几个与FBL相关的开放性问题[E];其中，序列是否（|XK|）在FBL[E]中是基本的，只要（xk）在E中是基本的。我建议使用的p-多项式的理论找到一个明确的公式的范数的自由Banach格与上p-估计。我也有兴趣在建设自由巴拿赫格代数。5.我将完成写一本关于向量和巴拿赫格的书。

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.