Applications of order convergence in Banach lattices

阶收敛在 Banach 格中的应用

• 批准号: RGPIN-2020-04855
• 负责人: Troitsky, Vladimir
• 金额: $ 1.31万
• 依托单位: University of Alberta
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2021
• 资助国家: 加拿大
• 起止时间: 2021-01-01 至 2022-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=740262
• 关键词: Applications; order; convergence; Banach; lattices

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.

该提议基于 Banach 和矢量格理论。这是泛函分析的一个领域，重点关注 Banach 空间中的偏序结构。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。在最近的一系列论文中，我和我的合作者建立了一些属性，这些属性使 uo 收敛性成为连接 Banach 格子和函数空间的绝佳工具，它的重要性变得清晰起来。这导致了排序闭凸集、Banach 格的 predual 和风险度量的应用。我将进一步探索 uo 收敛的某些属性和应用。 J.Grobler 和 C.Labuschagne 最近开发了几种基于向量格通用完成的新技术，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、uo-收敛、uo-对偶和 uo-完备性之间的关系，并将其应用于无测度随机理论。我还想将这些技术与 D.Fremlin 将普遍完备空间表示为布尔代数上可测量函数的空间联系起来。 3. 双基本序列。基本序列在巴纳赫空间理论中起着重要作用。在与 M.Taylor 正在进行的联合项目中，我们一直在研究双基序列，它们是 Banach 格中的基本序列，其基展开不仅在范数上收敛，而且按顺序收敛。我们已经确定了此类序列的许多令人兴奋的特性。我们证明了分析中的大多数经典基本序列都是双基的。我建议进一步研究双碱基序列以及双碱基序列。特别是，我想确定 Banach 格子的每个闭合子格子是否包含双基序列或 uo-bibasic 序列，以及顺序有序完整 Banach 格子中的每个基本序列是否是（Schauder）基本序列。 4.自由Banach格子。 B.de Pagter、A.Wickstead、A.Aviles 等人最近构造了自由 Banach 格子 FBL(A) 和 FBL[E]。他们还找到了 FBL[E] 范数的明确公式。我在 [T3] 中找到了构建 FBL(A) 和 FBL[E] 的替代方法。在与 M.Taylor、P.Tradacete 等人正在进行的项目中，我们使用 [T3] 的方法来构造自由 p 凸 Banach 格子；我们还找到了其范数的公式。我建议研究几个与 FBL[E] 相关的开放性问题；其中，只要 (xk) 在 E 中是基本的，序列 (|xk|) 在 FBL[E] 中是否是基本的。我建议使用 p-多范数理论来找到自由 Banach 格子范数的显式公式与较高的 p 估计值。我也对构造免费的巴纳赫格代数感兴趣。 5. 我将完成一本关于矢量和巴纳赫格的书。