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Boundary Actions and Applications in Operator Algebras

算子代数中的边界作用和应用

基本信息

批准号：
1700259
负责人：
Mehrdad Kalantar
金额：
$ 12万
依托单位：
University of Houston
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2017
资助国家：
美国
起止时间：
2017-07-01 至 2020-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1700259&HistoricalAwards=false
关键词：
Boundary Actions Applications Operator Algebras

项目摘要

The theory of Operator Algebras was developed in the 1930s as the mathematical foundation of quantum mechanics. This theory also provides the 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 dynamical systems. Today, each of the theories of Operator Algebras, Groups, and Dynamical Systems are independently of each other among most important and well-established parts of modern mathematics and mathematical physics. But the framework of operator algebras still allows deep interactions between these mathematical concepts. The aim of this project is to exploit the existing, and develop new bridges between these different theories. This allows one to utilize the current advanced mathematical technology of each area to help overcome some open problems in the others.This project is concerned with various aspects of interaction between analytic group theory (and its quantum version) and operator algebras, particularly via the theory of boundaries. Measure-theoretical boundaries (e.g. the Poisson boundary), and topological boundaries (e.g. the Furstenberg boundary) of groups were introduced and developed in the 1960s and 1970s in the seminal work of Furstenberg. These concepts were used as a tool to prove certain rigidity results for lattices in Lie groups. The former type of boundaries have since been vastly investigated and used as the main tool in some of the most substantial results in the past few decades in both ergodic theory of groups and rigidity theory of von Neumann algebras. However, topological boundaries are much less understood, and surely have not been fully exploited as a similar powerful tool in continuous ergodic theory or C*-algebraic rigidity problems. These boundaries are defined abstractly in general for all discrete (and also non-discrete) groups, and they compromise all natural notions of boundaries such as Gromov's boundaries of hyperbolic groups or Furstenberg boundaries of semisimple Lie groups. The project aims to further develop the understanding of such boundaries. The goal is to apply them to generalize various results in operator algebra theory that have been proven by using special cases of such boundary actions, for example, the existing results concerning maximal injective von Neumann subalgebras and character-rigidity. In another direction, the project aims to develop and study the boundary theory of quantum groups and investigate their possible applications to problems such as C*-simplicity in the quantum setting.

算子代数理论是在20世纪30年代发展起来的，作为量子力学的数学基础。这个理论还提供了自然的数学框架，在这个框架中，随着时间的推移，物理系统（称为动力系统）可以被表示和研究;群是表示动力系统中时间的代数结构。今天，算子代数、群和动力系统的每一个理论都是现代数学和数学物理中最重要和最成熟的部分。但算子代数的框架仍然允许这些数学概念之间的深层相互作用。这个项目的目的是利用现有的，并开发这些不同的理论之间的新的桥梁。这使得一个利用目前先进的数学技术的每一个领域，以帮助克服一些开放的问题，在其他。这个项目是关注的各个方面之间的相互作用的分析群论（及其量子版本）和算子代数，特别是通过理论的边界。群的测度-理论边界（如泊松边界）和拓扑边界（如弗斯滕伯格边界）是在1960年代和1970年代由弗斯滕伯格的开创性工作引入和发展的。这些概念被用来证明李群中格的某些刚性结果。前一种类型的边界已被广泛调查和使用的主要工具，在一些最实质性的成果，在过去几十年中遍历理论的群体和刚性理论的冯诺依曼代数。然而，拓扑边界的理解要少得多，肯定还没有充分利用连续遍历理论或C*-代数刚性问题作为一个类似的强大工具。这些边界是抽象地定义在所有离散（以及非离散）群中的，并且它们妥协了所有自然的边界概念，例如双曲群的格罗莫夫边界或半单李群的弗斯滕伯格边界。该项目旨在进一步加深对这种界限的理解。我们的目标是应用它们来推广算子代数理论中的各种结果，这些结果已经通过使用这种边界作用的特殊情况得到了证明，例如，关于极大内射von Neumann子代数和特征刚性的现有结果。在另一个方向上，该项目旨在发展和研究量子群的边界理论，并研究它们在量子环境中的C*-简单性等问题中的可能应用。