Operator Algebras Associated to Groups and Noncommutative Convexity

与群和非交换凸性相关的算子代数

基本信息

批准号：
RGPIN-2018-05191
负责人：
Kennedy, Matthew
金额：
$ 5.1万
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：
2022
资助国家：
加拿大
起止时间：
2022-01-01 至 2023-12-31
项目状态：
已结题

来源：
https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=764412
关键词：
Operator Algebras Associated Groups Noncommutative

项目摘要

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·诺伊曼（von Neumann）在量子物理学的数学基础上的工作开始，数学家发现，将在操作者代数理论中引起的各种结构视为经典数学对象的非共同对应物是有利可图的。这种哲学是我的作品的基础，该作品涵盖了操作员代数领域中的广泛主题。我最近最重要的工作集中在两个领域：与群体相关的运算符代数的结构和非交通式凸组的结构。我的提案的第一个主要组成部分涉及与组和动态系统相关的操作员代数的结构。一方面，操作员代数的群体理论构造提供了一个宝贵的示例和问题来源。另一方面，越来越清楚的是，在操作员 - 代数框架中最自然地研究了组和动态系统的分析结构的许多问题。我的研究中考虑的基本问题是如何将组或动力系统的属性与相应运算符代数的属性联系起来。我的提案的第二个主要组成部分涉及非共同凸性的理论。 1969年，阿尔维森（Arveson）以无限尺寸空间的凸集结构进行了突破性的工作，提出了一种非交通性凸度的理论，作为研究非交通性数学引起的对象的框架。尽管这些想法具有巨大的潜力，但它们一直没有开发多年，直到最近突破，第一次是阿尔维森本人，第二位是与K.R.共同合作的现任作者。戴维森。我计划继续开发这些思想，这些想法已经在量子信息理论，群体理论和半决赛优化等领域的问题上取得了巨大成功。