Lp-Approximation Properties, Multipliers, and Quantized Calculus

Lp 近似属性、乘子和量化微积分

• 批准号: 2247123
• 负责人: Tao Mei
• 金额: $ 37.9万
• 依托单位: Baylor University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-07-15 至 2026-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2247123&HistoricalAwards=false
• 关键词: Lp; Approximation; Properties; Multipliers; Quantized

Approximation is a fundamental concept in mathematics, and a powerful tool in science and engineering, in which a core notion is that one may study an object of interest by modeling it with simpler objects and using a limiting process. The various approximation properties in mathematical analysis refer to the capacity of an object, such as an infinite-dimensional vector space, to be approximated in useful ways, for instance, by finite-dimensional subspaces. Such properties are central to a number of important fields, such as quantum probability theory, noncommutative geometry, and quantized calculus. This project will focus on Lp-approximation properties of von Neumann algebras and their applications in operator algebras, noncommutative analysis, and beyond. The project incorporates research opportunities for undergraduate and graduate students, as well as training for postdoctoral researchers. In this study, the principal investigator will explore Lp-approximation properties of group von Neumann algebras and their connections to the von Neumann rigidity property and Connes's quantized calculus. The major challenges of the proposed research include the absence of geometric/metric structure and a commutative product in the abstract setting. The proposed research necessitates a reinvention of classical theory on Fourier multipliers in a context with much less structure. Success in this proposed project will strengthen the existing connection between functional analysis and harmonic analysis and will enhance our understanding of the rigidity of discrete groups, the theory of noncommutative geometry, and its applications to quantum information theory.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

近似是数学中的基本概念，也是科学和工程中的强大工具，其中的核心概念是人们可以通过用更简单的对象建模并使用限制过程来研究感兴趣的对象。 数学分析中的各种近似属性是指以有用的方式（例如通过有限维子空间）近似对象（例如无限维向量空间）的能力。 这些性质是许多重要领域的核心，例如量子概率论、非交换几何和量子微积分。 该项目将重点关注冯诺依曼代数的 Lp 近似性质及其在算子代数、非交换分析等方面的应用。 该项目包括为本科生和研究生提供研究机会，以及为博士后研究人员提供培训。 在本研究中，主要研究者将探索冯诺依曼代数群的 Lp 近似性质及其与冯诺依曼刚性性质和 Connes 量化微积分的联系。所提出的研究的主要挑战包括缺乏几何/度量结构和抽象环境中的交换积。拟议的研究需要在结构少得多的背景下重新发明傅里叶乘数的经典理论。该项目的成功将加强泛函分析和调和分析之间的现有联系，并将增强我们对离散群刚性、非交换几何理论及其在量子信息论中的应用的理解。该奖项反映了 NSF 的法定使命，并具有通过使用基金会的智力优点和更广泛的影响审查标准进行评估，被认为值得支持。