Operator Algebras Associated to Groups and Noncommutative Convexity

与群和非交换凸性相关的算子代数

• 批准号: RGPIN-2018-05191
• 负责人: Kennedy, Matthew
• 金额: $ 2.55万
• 依托单位: University of Waterloo
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=685740
• 关键词: Operator; Algebras; Associated; Groups; Noncommutative

Beginning with the work of von Neumann on the mathematical foundations of quantum physics, mathematicians have found it profitable to view various structures arising in the theory of operator algebras as noncommutative counterparts of classical mathematical objects. This philosophy underlies my work, which spans a wide range of topics within the field of operator algebras. My most significant recent work is concentrated in two areas: the structure of operator algebras associated to groups and the structure of noncommutative convex sets.***The first major component of my proposal concerns the structure of operator algebras associated to groups and dynamical systems. On the one hand, group-theoretic constructions of operator algebras provide a valuable source of examples and questions. On the other hand, it is becoming increasingly clear that many problems about the analytic structure of groups and dynamical systems are most naturally studied within an operator-algebraic framework. The fundamental problem considered in my research is how to relate properties of the group or dynamical system to properties of the corresponding operator algebra.***The second major component of my proposal concerns the theory of noncommutative convexity. In 1969, motivated by groundbreaking work on the structure of convex sets in infinite dimensional spaces, Arveson proposed a theory of noncommutative convexity as a framework for the study of objects arising from noncommutative mathematics. Despite the enormous potential of these ideas, they went undeveloped for many years until recent breakthroughs, the first by Arveson himself and the second by current author in joint work with K.R. Davidson. I plan to continue developing these ideas, which have already been applied with great success to problems in fields like quantum information theory, group theory and semidefinite optimization.

从冯·诺依曼在量子物理学的数学基础上的工作开始，数学家发现将算子代数理论中出现的各种结构视为经典数学对象的非交换对应物是有利的。这一哲学是我的工作的基础，它涵盖了算子代数领域的广泛主题。我最近最重要的工作集中在两个领域：与群相关的算子代数的结构和非交换凸集的结构。***我提案的第一个主要组成部分涉及与群和动力系统相关的算子代数的结构。一方面，算子代数的群论构造提供了有价值的示例和问题来源。另一方面，越来越清楚的是，关于群和动力系统的分析结构的许多问题是在算子代数框架内最自然地研究的。我的研究考虑的基本问题是如何将群或动力系统的性质与相应算子代数的性质联系起来。***我的建议的第二个主要组成部分涉及非交换凸性理论。 1969 年，受无限维空间中凸集结构的开创性工作的推动，Arveson 提出了非交换凸性理论，作为研究由非交换数学产生的对象的框架。尽管这些想法潜力巨大，但它们多年来一直没有得到发展，直到最近取得突破，第一个突破是由阿维森本人实现的，第二个突破是由当前作者与 K.R. 共同完成的。戴维森。我计划继续发展这些想法，这些想法已经成功地应用于量子信息论、群论和半定优化等领域的问题。