Approximation properties in von Neumann algebras

冯·诺依曼代数在20世纪30年代和40年代被引入，用于研究群的表示理论，并用作开发量子物理数学基础的工具。 他们已经发展成为一个完整的研究领域作为一个自然的非交换概念的措施理论。拓扑的非交换设置（C*-代数）出现后不久，这两个主题在历史上一直密切相关。该项目探索这些联系，以发展新的想法，达到广泛的数学社区，并为该领域的新学生提供参与和支持。 研究人员积极参与冯诺依曼代数的学生和博士后的培训，该项目的研究将直接影响这些学生和博士后。该项目研究人员正在研究冯诺依曼代数和C*-代数中的近似性质（或缺乏近似性质），特别是与群冯诺依曼代数和群测度空间构造有关的性质。这在历史上一直是算子代数分类中的一个重要研究领域，顺从性/内射性在冯诺依曼代数的发展中发挥了重要作用，而核性在C*-代数理论中发挥了重要的相应作用。波帕的变形/刚性理论的出现导致了冯诺依曼代数分类的许多突破，超越了顺从性设置，近似性质，如小泽的双精确群的概念，创造了新的机会来研究冯诺依曼代数设置中的近似性质。该项目中开发的研究调查了这些近似性质，在C* 和冯诺依曼代数之间建立了新的联系。这使得新的C*-代数工具可以用于冯诺依曼代数的设置，导致群和群测度空间冯诺依曼代数的新结构结果，并对算子代数，遍历理论，该奖项反映了NSF的法定使命，并已被认为是值得通过评估使用基金会的智力价值和更广泛的支持。影响审查标准。

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.