Boundary Actions and Applications in Operator Algebras

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。这个理论也提供了自然的数学框架，在这个框架中，随时间演变的物理系统（称为动力系统）可以被表示和研究；群是动力系统中表示时间的代数结构。今天，算子代数、群和动力系统的每一个理论都是相互独立的，是现代数学和数学物理中最重要和最完善的部分。但是算子代数的框架仍然允许这些数学概念之间的深度交互。这个项目的目的是利用现有的，并在这些不同的理论之间建立新的桥梁。这允许人们利用当前每个领域的先进数学技术来帮助克服其他领域的一些开放问题。这个项目关注分析群论（及其量子版本）和算子代数之间相互作用的各个方面，特别是通过边界理论。群的测量理论边界（如泊松边界）和拓扑边界（如Furstenberg边界）在20世纪60年代和70年代在Furstenberg的开创性工作中被引入和发展。这些概念被用作证明李群中晶格的某些刚性结果的工具。在过去的几十年里，在群的遍历理论和冯·诺伊曼代数的刚性理论中，前一种类型的边界被广泛地研究，并作为一些最重要的结果的主要工具使用。然而，拓扑边界的理解要少得多，并且肯定没有被充分利用作为一个类似的强大工具在连续遍历理论或C*-代数刚性问题。一般来说，这些边界是抽象定义的，适用于所有离散（和非离散）群，它们妥协了所有自然的边界概念，如双曲群的Gromov边界或半单李群的Furstenberg边界。该项目旨在进一步发展对这些边界的理解。目的是应用它们来推广算子代数理论中已经用这种边界作用的特殊情况证明的各种结果，例如关于极大内射von Neumann子代数和特征刚性的现有结果。另一方面，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C*-简单性等问题上的可能应用。

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.