Rigidity for von Neumann Algebras and Applications

冯·诺依曼代数是某些复数的无限矩阵的集合，称为希尔伯特空间算子。它们是在20世纪30年代引入的，目的是为量子力学提供数学基础。从那时起，冯·诺依曼代数理论蓬勃发展成为一个独立的领域，与数学和科学的几个领域建立了卓有成效的联系，包括量子物理。冯·诺依曼代数自然而然地产生于各种数学结构，如对称组和作用组，它们存在于数学的许多领域。他们的研究与群论和遍历理论中的主题密切相关。这个项目将通过研究这些区域和算子代数界面上的一些公开问题来加深它们之间的联系。该项目还将为研究生的培训和专业发展提供机会。该项目的目的是扩大von Neumann代数中刚性的范围，并发现在群论、遍历理论和C*-代数中的新应用。该项目的第一个目标是研究群von Neumann代数的刚性。在Connes著名的1980年刚性猜想的基础上，PI建议寻找在von Neumann代数的背景下是刚性的新的性质(T)群族并度量等价。该项目的第二个目标是研究群的两个相关的刚性性质：完全群C*-代数的Hilbert-Schmidt稳定性和局部提升性质。PI建议研究关于几乎交换矩阵和局部提升性质的两个长期存在的问题。该项目的第三个目标是研究von Neumann代数的渐近问题。这项拟议的研究有望在算子代数、(几何和可测量的)群论和遍历理论之间产生新的互动。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 collections of certain infinite matrices of complex numbers, called Hilbert space operators. They were introduced in the 1930s in order to provide a mathematical foundation for quantum mechanics. Since then, the theory of von Neumann algebras has flourished into an independent area with fruitful connections to several areas of mathematics and science, including quantum physics. Von Neumann algebras arise naturally from a variety of mathematical structures, such as groups of symmetries and actions, which are present in many areas of mathematics. Their study is closely connected to topics in group theory and ergodic theory. This project will deepen the connections between these areas and operator algebras, by investigating a number of open problems at their interface. The project will also provide opportunities for the training and professional development of graduate students.The aim of this project is to expand the scope of rigidity in von Neumann algebras and uncover new applications to group theory, ergodic theory and C*-algebras. The first objective of the project is to investigate rigidity for group von Neumann algebras. Building on recent work on Connes' famous 1980 rigidity conjecture, the PI proposes to find new families of property (T) groups which are rigid in the context of von Neumann algebras and measure equivalence. The second objective of the project is to investigate two related rigidity properties for groups: Hilbert-Schmidt stability and the local lifting property for full group C*-algebras. The PI proposes to investigate two longstanding problems concerning almost commuting matrices and the local lifting property. The third objective of the project is to study asymptotic problems of von Neumann algebras. The proposed research is expected to lead to new interactions between operator algebras, (both geometric and measured) group theory and ergodic 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.