Lp-Approximation Properties, Multipliers, and Quantized Calculus

近似是数学中的一个基本概念，也是科学和工程中的一个强大工具，其中的一个核心概念是，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。数学分析中的各种逼近性质指的是对象(例如无限维向量空间)以有用的方式(例如有限维子空间)逼近的能力。这些性质是许多重要领域的核心，例如量子概率论、非对易几何和量子化微积分。本课题将主要研究von Neumann代数的Lp逼近性质及其在算子代数、非对易分析等方面的应用。该项目包括本科生和研究生的研究机会，以及对博士后研究人员的培训。在这项研究中，主要研究者将探讨群von Neumann代数的Lp逼近性质及其与von Neumann刚性性质和Connes量子化微积分的关系。提出的研究的主要挑战包括缺乏几何/度量结构和抽象背景下的交换积。这项拟议的研究需要在结构少得多的情况下对傅里叶乘子的经典理论进行再创造。这个拟议项目的成功将加强泛函分析和调和分析之间的现有联系，并将加强我们对离散群的刚性、非对易几何理论及其在量子信息理论中的应用的理解。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.