Classification, STructure, Amenability and Regularity

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闭合。有两种主要类型的算子代数——von Neumann代数和C*-代数，它们分别在点向和一致收敛下闭合。无论是通过阿贝尔代数的结构，还是用来处理它们的工具，冯·诺伊曼代数都有测量理论的味道，而C*代数本质上是拓扑的。只要数学涉及到希尔伯特空间，例子就会出现，所以算子代数自然地出现在群表示、动力学、数学物理等领域。本提案的中心主题是可服从C*-代数的结构和分类，重点是来自动力学的例子。这是一个完整的C*-代数模拟，从20世纪70年代开始，Connes在冯诺依曼代数的结构理论中取得了突破性的进展，这导致了可修正的冯诺依曼代数的完整分类（Connes- haagerup内射因子分类），并且从那时起一直至关重要，推动了可测量动力学，子因子和刚性问题的戏剧性后续发展。该提案旨在获得C*-代数之间的可适应态射的确定分类定理，以及在抽象和重要的示例族中识别可分类代数和态射的强大结构定理。这将用于通过张量范畴的作用分类来启动可调节C*-代数的量子对称性的深入研究，旨在通过琼斯子因子理论在冯·诺伊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·诺伊曼代数框架到C*-代数的思想和技术的明确转移：在C*-代数中使用冯·诺伊曼技术。

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.