Almost Periodic von Neumann Algebras

冯·诺伊曼代数是一种数学对象，它为量子物理的研究提供了一个严格的框架，可以被认为是矩阵代数的无限维推广。该理论是由弗朗西斯·j·默里和约翰·冯·诺伊曼在20世纪30年代提出的，从那时起，研究人员在数学、生物学、物理学和工程学上发现了大量的应用。冯·诺伊曼代数在物理学（如量子统计力学或相对论量子场论）中自然出现，通常被称为非半有限冯·诺伊曼代数。这使得它们更加难以研究，特别是它们超出了过去几十年来为所谓的半有限冯诺依曼代数开发的大多数技术的范围。该项目旨在将这些半有限技术中的一些适应于几乎周期的冯·诺伊曼代数，它跨越了半有限和非半有限冯·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。本课题的研究目标是研究具有几乎周期权值的von Neumann代数，重点研究其非半有限情况。具体来说，PI将发展冯·诺伊曼维数的概念用于几乎周期的冯·诺伊曼代数并利用它将自由斯坦维数和l^2-贝蒂数推广到几乎周期的情况。这些不变量将揭示这些von Neumann代数的结构性质，并且l^2-Betti数的扩展概念在非单模群和非保测度等价关系中具有潜在的应用。此外，PI还提出将刚性和共刚性包涵的概念从有限的冯·诺依曼代数扩展到具有几乎周期态的冯·诺依曼代数。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.