Classification, STructure, Amenability and Regularity

算子代数是作为Hilbert空间上的有界算子族而产生的，在代数运算、Hilbert空间伴随和取值极限下闭合。算子代数有两种主要类型--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。Von Neumann代数具有测度论的味道，而C*-代数本质上是拓扑的，无论是通过交换代数的结构，还是通过使用它们所使用的工具。例子出现在数学接触希尔伯特空间的任何地方，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，仅举几个例子。这个建议的中心主题是服从C*-代数的结构和分类，重点是来自动力学的例子。这是Cones自1970年S以来在von Neumann代数结构理论方面取得的开创性进展的完全C*-代数模拟，该进展导致了服从von Neumann代数的完全分类(Connes-Haagerup内射因子分类)，并从那时起一直是关键，推动了可测动力学、子因子和刚性问题的巨大发展。该方案试图得到C*-代数之间可归类态射的确定性分类定理，以及识别抽象的和重要的例族中的可归类代数和态射的强有力的结构定理。这将被用来通过张量范畴的作用的分类来启动对服从C*-代数的量子对称性的深入研究，旨在通过琼斯的子因子理论在冯·诺伊曼代数框架中看到深刻的影响。整个项目的一个驱动力主题是从冯·诺伊曼代数框架到C*-代数的思想和技术的明确转移：在C*-代数中冯·诺伊曼技巧的使用。

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.