C*-algebras of Groups and Quantum Groups: Rigidity and Structure Theory

算子代数理论是在20世纪30年代作为量子力学的数学基础而发展起来的。这个理论也是一个自然的数学框架，其中物理系统随着时间的推移（称为动力系统）可以被表示和研究；群是这张图中表示时间的代数结构。群也为物理系统和数学结构的对称性建模，自从它们在19世纪被正式引入以来，它们的应用已经深入到几乎所有主要的数学领域。群的表示理论提供了将算子代数分配给给定群的规范方法，因此在两个理论之间建立了一座桥梁，允许一个理论将思想和概念从一个理论引入另一个理论。自冯·诺伊曼时代以来，对群及其相关算子代数之间的联系的研究一直是算子代数理论的核心部分，它对这两个理论以及其他几个相关领域如遍历理论、动力系统和表示理论产生了巨大的影响。本项目旨在通过发现群的结构性质和它们的算子代数之间的新联系，在这个方向上取得进一步的进展。本项目为研究生提供研究训练机会。项目的第一部分是针对离散群的C*-代数的新算子-代数刚性现象的几个猜想和问题。这是基于将高阶群遍历理论的重要经典结果推广到算子代数集的一种新方法。感兴趣的主要概念是由酉表示生成的C*-代数的不变子代数，它们被视为正规子群的非交换推广。群体的边界理论（在弗斯滕伯格的意义上）在首席研究员的策略中起着核心作用，以解决所提出的问题。在过去的几十年里，Furstenberg的边界理论一直是高阶格遍历理论和证明其刚性性质的主要工具。最近，它也被用于算子-代数刚性问题，在这一领域取得了重大进展。该项目旨在开发利用边界作用的新技术，以证明由离散群的酉表示生成的C*-代数的不变子代数的刚性结果。本课题的另一部分涉及离散量子群C*-代数的简单性和唯一迹性的几个开放性问题；目标是进一步发展量子群的边界理论，并将其应用于这些问题。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

The theory of operator algebras was developed in the 1930s as the mathematical foundation of quantum mechanics. This theory is also a natural mathematical framework in which physical systems as they evolve in time (called dynamical systems) can be represented and studied; groups are the algebraic structures that represent the time in this picture. Groups also model symmetries of physical systems and mathematical structures, and since their formal introduction in the 19th century, their applications have expanded deep into almost every major area of mathematics. Representation theory of groups provides canonical ways to assign operator algebras to a given group, hence establishes a bridge between the two theories which allows one to import ideas and concepts from one theory to the other. The study of the connections between groups and their associated operator algebras has been a central part of operator algebra theory since the time of von Neumann, and it has had tremendous impact on both theories, as well as in several other related areas such as ergodic theory, dynamical systems, and representation theory. This project aims to make further progress in this direction by discovering new connections between structural properties of groups and their operator algebras. The project provides research training opportunities for graduate students. The first part of the project is aimed at several conjectures and problems related to new operator-algebraic rigidity phenomena for discrete groups in terms of their C*-algebras. This is based on a new approach to extending important classical results from ergodic theory of higher rank groups to the operator-algebraic setting. The main concept of interest is that of invariant subalgebras of C*-algebras generated by unitary representations, which are viewed as non-commutative generalizations of normal subgroups. The boundary theory of groups (in the sense of Furstenberg) plays a central role in the Principal Investigator’s strategies to approach the proposed problems. Furstenberg's boundary theory has been, in the past few decades, a major tool in ergodic theory of higher rank lattices and in proving their rigidity properties. More recently, it has also been used in operator-algebraic rigidity problems, resulting in significant progress in this area. The project aims to develop new techniques in utilizing boundary actions in order to prove the proposed rigidity results for invariant subalgebras of C*-algebras generated by unitary representations of discrete groups. Another part of this project concerns several open problems regarding simplicity and unique trace property of C*-algebras of discrete quantum groups; the goal is to further develop the boundary theory of quantum groups and apply them in these problems.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.