Almost Periodic von Neumann Algebras

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，可以被认为是矩阵代数的无限维推广。这一理论是由弗朗西斯·J·默里和约翰·冯·诺伊曼在20世纪30年代提出的，从那时起，研究人员发现了大量在数学以及生物学、物理学和工程学中的应用。在物理学(例如量子统计力学或相对论量子场论)中自然出现的von Neumann代数通常是所谓的非半有限von Neumann代数。这使得对它们的研究更加困难，尤其是它们超出了过去几十年来为所谓的半有限von Neumann代数开发的大多数技术的范围。这个项目试图将这些半有限技巧中的一部分应用于几乎周期von Neumann代数，它跨越了半有限和非半有限von Neumann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概周期权的von Neumann代数，重点研究非半正定情形。具体地说，PI将发展概周期von Neumann代数的von Neumann维的概念，并利用它将自由Stein维和L^2-Betti数推广到概周期情形。这些不变量将揭示这类von Neumann代数的结构性质，而推广的L^2-Betti数的概念在非么模群和非保测等价关系中具有潜在的应用。此外，PI建议将刚性和余刚性包含的概念从有限von Neumann代数扩展到配备概周期态的von Neumann代数。这一奖项反映了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。

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.