Approximation properties in von Neumann algebras

冯·诺依曼代数中的近似性质

• 批准号: 2400040
• 负责人: Jesse Peterson
• 金额: $ 29.16万
• 依托单位: Vanderbilt University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2024
• 资助国家: 美国
• 起止时间: 2024-06-01 至 2027-05-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2400040&HistoricalAwards=false
• 关键词: Approximation; properties; von; Neumann; algebras

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.

在1930年代和40年代引入了冯·诺伊曼（Von Neumann）代数，以研究群体的表示理论，并用作为量子物理学发展数学基础的工具。 从那以后，他们已将整个研究领域发展为量度理论的自然非共同概念。不久之后出现了拓扑（C* - 代数）的非交通设置（C*-Algebras），并且这两个受试者历来密切相关。该项目探讨了这些联系，以发展新的想法，建立一个广泛的数学社区，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加冯·诺伊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。该项目研究者正在研究von Neumann代数和C* - 代数中的近似特性（或缺乏），尤其是与von Neumann代数和组测量空间结构有关的近似值。从历史上看，这一直是操作员代数分类的重要研究领域，并且在von Neumann代数的发展中具有不良性/注入性在C* - 代数理论中起着相应的相应作用。 POPA的变形/刚性理论的出现导致了冯·诺伊曼代数的分类，超出了不可理性的设置，而近似属性（例如Ozawa对Biexact群体的概念）为VONNEUMANN ALGEBRAS设置中研究近似属性创造了新的机会。该项目开发的研究研究了这些近似属性，从而在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，从而为小组和小组测量太空von Neumann代数带来新的结构性结果，并更深入地深入了解操作员代数，善良的理论和几何学理论之间的依据，以反映了NSF的构建奖，并反映了NSF的构建奖。更广泛的影响审查标准。