Approximation properties in von Neumann algebras

冯·诺依曼代数在1930年的S和40年的S被引入，以研究群的表示论，并用作发展量子物理的数学基础的工具。自那以后，它们已经发展成为一个完整的研究领域，成为度量理论的一个自然的非对易概念。不久之后，拓扑学(C*-代数)的非对易环境出现了，这两个学科在历史上一直紧密联系在一起。这个项目探索这些联系，以发展新的想法，接触到广泛的数学社区，并为该领域的新学生提供参与和支持。研究人员正在积极参与对学生和博士后进行冯·诺依曼代数的培训，该项目的研究将直接影响到这些学生和博士后。这位项目研究者正在研究von Neumann代数和C*-代数的逼近性质，特别是与群von Neumann代数和群测度空间结构有关的性质。这在历史上一直是算子代数分类的一个重要研究领域，顺从性/内射性在von Neumann代数的发展中起着重要的作用，而核性在C*-代数的理论中起着重要的相应作用。Popa的形变/刚性理论的出现使von Neumann代数的分类在顺从性环境之外取得了许多突破，而逼近性质，如小泽的双正合群的概念，为研究von Neumann代数的逼近性质创造了新的机会。本项目研究这些逼近性质，在C*和von Neumann代数之间建立了新的联系。这允许在von Neumann代数的设置中使用新的C*-代数工具，导致群和群测量空间von Neumann代数的新的结构结果，并使人们对算子代数、遍历理论和几何群论之间的相互作用有了更深入的了解。该奖项反映了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 were introduced in the 1930's and 40's to study representation theory of groups, and to use as a tool for developing a mathematical foundation for quantum physics. They have since developed into a full area of study as a natural noncommutative notion of measure theory. The noncommutative setting of topology (C*-algebras) emerged shortly after, and the two subjects have historically been closely connected. This project explores these connections to develop new ideas, to reach a broad mathematical community and providing engagement and support for new students in the field. The investigator is actively participating in the training of students and postdocs in von Neumann algebras and the research from this project will directly impact these students and postdocs. The project investigator is studying approximation properties (or the lack thereof) in von Neumann algebras and C*-algebras, especially relating to group von Neumann algebras and group measure space constructions. This has historically been a significant area of study in the classification of operator algebras, with amenability/injectivity playing a major role in the development of von Neumann algebras, and nuclearity playing a major corresponding role in the theory of C*-algebras. The emergence of Popa's deformation/rigidity theory has led to numerous breakthroughs in the classification of von Neumann algebras beyond the amenability setting, and approximation properties, such as Ozawa's notion of a biexact group, have created new opportunities to study approximation properties in the setting of von Neumann algebras. The research developed in this project investigates these approximation properties, creating new connections between C* and von Neumann algebras. This allows new C*-algebraic tools to be used in the setting of von Neumann algebras, leading to new structural results for group and group measure space von Neumann algebras, and giving a deeper insight into interactions between operator algebras, ergodic theory, and geometric group 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.