Almost Periodic von Neumann Algebras

近周期冯诺依曼代数

• 批准号: 2247047
• 负责人: Brent Nelson
• 金额: $ 37.67万
• 依托单位: Michigan State 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: 2023-05-15 至 2026-04-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2247047&HistoricalAwards=false
• 关键词: Almost; Periodic; von; Neumann; Algebras

Von Neumann algebras are mathematical objects that offer a rigorous framework for the study of quantum physics and can be thought of as infinite-dimensional generalizations of matrix algebras. The theory was initiated by Francis J. Murray and John von Neumann in the 1930s, and since then researchers have discovered a vast number of applications to mathematics as well as biology, physics, and engineering. The von Neumann algebras that occur naturally in physics (e.g. quantum statistical mechanics or relativistic quantum field theory) are typically what are known as non-semifinite von Neumann algebras. This makes them more difficult to study, and in particular they lie outside the scope of the majority of techniques developed for so-called semifinite von Neumann algebras over the past few decades. This project seeks to adapt some of these semifinite techniques to almost periodic von Neumann algebras, which straddle the boundary between semifinite and non-semifinite von Neumann algebras. The project will also involve training and professional development for graduate students and postdocs. The research goal of this project is to study von Neumann algebras that admit almost periodic weights, with a focus on the non-semifinite case. Specifically, the PI will develop a notion of von Neumann dimension for almost periodic von Neumann algebras and use this to generalize free Stein dimension and l^2-Betti numbers to the almost periodic case. These invariants would shed light on the structural properties of such von Neumann algebras, and an extended notion of l^2-Betti numbers has potential applications to non-unimodular groups and non-measure preserving equivalence relations. Additionally, the PI proposes to extend the notions of rigid and co-rigid inclusions from finite von Neumann algebras to von Neumann algebras equipped with almost periodic states.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.

冯·诺伊曼（Von Neumann）代数是数学对象，为量子物理学的研究提供了严格的框架，可以被视为基质代数的无限维概括。该理论是由弗朗西斯·J·默里（Francis J. Murray）和约翰·冯·诺伊曼（John von Neumann）在1930年代发起的，从那时起，研究人员发现了许多用于数学以及生物学，物理和工程的应用。物理学（例如量子统计力学或相对论量子场理论）自然发生的von Neumann代数通常是被称为非米线属的von Neumann代数。这使他们更加难以学习，特别是它们不在过去几十年来为所谓的半决赛von Neumann代数开发的大多数技术的范围。该项目旨在将其中一些半决赛技术适应几乎是周期性的von Neumann代数，后者跨越了半决赛和非米线派和非米线派冯·诺伊曼代数之间的边界。 该项目还将涉及研究生和博士后的培训和专业发展。 该项目的研究目标是研究几乎定期重量的von Neumann代数，重点是非固醇。具体而言，PI将针对几乎周期性的von Neumann代数形成von Neumann维度的概念，并将其概括为几乎是周期性的情况。这些不变的人会阐明这种von Neumann代数的结构特性，而L^2-betti数字的扩展概念具有潜在的应用于非降低基团的应用，并保留了非估量的等价关系。此外，PI提议将刚性和辅助物包含物的概念从有限的von Neumann代数到配备了几乎周期性状态的von Neumann代数到Von Neumann代数。这一奖项反映了NSF的法定任务，并被认为是通过基金会的智力功能和广泛影响的评估来评估CRETERIA的评估。