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·诺依曼代数是为量子物理研究提供严格框架的数学对象，可以被视为矩阵代数的无限维推广。该理论由 Francis J. Murray 和 John von Neumann 在 20 世纪 30 年代提出，从那时起，研究人员发现了数学、生物学、物理学和工程学的大量应用。物理学中自然出现的冯诺依曼代数（例如量子统计力学或相对论量子场论）通常被称为非半有限冯诺依曼代数。这使得它们更难以研究，特别是它们超出了过去几十年为所谓的半有限冯诺依曼代数开发的大多数技术的范围。该项目旨在将其中一些半有限技术应用于几乎周期性的冯诺依曼代数，这些代数跨越了半有限和非半有限冯诺依曼代数之间的界限。 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承认几乎周期性权重的冯诺依曼代数，重点是非半有限情况。具体来说，PI 将为几乎周期的冯诺依曼代数开发冯诺依曼维数的概念，并使用它来将自由斯坦因维数和 l^2-Betti 数推广到几乎周期的情况。这些不变量将揭示此类冯诺依曼代数的结构特性，并且 l^2-Betti 数的扩展概念对于非幺模群和非测度保留等价关系具有潜在的应用。此外，PI 提议将刚性和共刚性包含的概念从有限冯诺依曼代数扩展到具有几乎周期性状态的冯诺依曼代数。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力评估进行评估，被认为值得支持。优点和更广泛的影响审查标准。