Classification, STructure, Amenability and Regularity

分类、结构、顺应性和规律性

• 批准号: EP/X026647/1
• 负责人: Stuart White
• 金额: $ 249.34万
• 依托单位: University of Oxford
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2023
• 资助国家: 英国
• 起止时间: 2023 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FX026647%2F1
• 关键词: Classification; STructure; Amenability; Regularity

Operator algebras arise as families of bounded operators on a Hilbert space, closed under algebraic operations, the Hilbert space adjoint, and under taking limits. There are two main types of operator algebra - von Neumann algebras and C*-algebras - which are closed under pointwise and uniform convergence respectively. Both through the structure of abelian algebras, and also the tools used to work with them, von Neumann algebras have the flavour of measure theory, while C*-algebras are topological in nature. Examples arise wherever mathematics touches Hilbert spaces, so operator algebras occur naturally from group representations, dynamics, mathematical physics, to name but a few.The central theme of this proposal is the structure and classification of amenable C*-algebras, with an emphasis on examples coming from dynamics. This aims for the full C*-algebra analogue of Connes' groundbreaking advances in the structure theory of von Neumann algebras from the 1970's which led to the complete classification of amenable von Neumann algebras (the Connes-Haagerup classification of injective factors) and has remained critical ever since, powering dramatic subsequent developments in measurable dynamics, subfactors and rigidity problems. The proposal seeks to obtain definitive classification theorems for amenable morphisms between C*-algebras together with powerful structure theorems which identify classifiable algebras and morphisms both abstractly and in important families of examples. This will be used to initiate a deep study of quantum symmetries of amenable C*-algebras through a classification of actions of tensor categories, aiming for the profound impact seen in the von Neumann algebraic framework through Jones theory of subfactors.A driving theme throughout the project is the explicit transfer of ideas and techniques from the von Neumann algebraic framework to C*-algebras: the use of von Neumann techniques in C*-algebras.

算子代数是希尔伯特空间上的有界算子族，在代数运算、希尔伯特空间伴随和取极限下是封闭的。算子代数主要有两种类型：von Neumann代数和C*-代数，它们分别在点态收敛和一致收敛下是闭的。通过交换代数的结构，以及使用它们的工具，冯·诺依曼代数具有测度论的味道，而C*-代数本质上是拓扑的。例子出现在数学接触希尔伯特空间的地方，所以算子代数自然地出现在群表示，动力学，数学物理中，仅举几例。这个提议的中心主题是顺从C*-代数的结构和分类，重点是来自动力学的例子。这旨在充分C*-代数模拟康纳斯的突破性进展的结构理论冯诺依曼代数从20世纪70年代，导致了完整的分类顺从冯诺依曼代数（康纳斯-Haagerup分类的内射因子），并一直保持关键至今，供电戏剧性的后续发展可测量的动态，子因子和刚性问题。该建议旨在获得明确的分类定理，顺应态射之间的C*-代数与强大的结构定理，确定可分类的代数和态射的抽象和重要的家庭的例子。这将用于通过张量范畴的作用分类来启动对顺从C*-代数的量子对称性的深入研究，旨在通过子因子的琼斯理论在冯诺依曼代数框架中看到的深刻影响。整个项目的驱动主题是从冯诺依曼代数框架到C*-代数的思想和技术的明确转移：冯·诺依曼技巧在C*-代数中的应用