Classifying *-Homomorphisms

本课题涉及可调算子代数的分类和结构。算符代数是数学的一个领域，始于默里和冯·诺伊曼在20世纪30年代的工作，部分原因是海森堡用无限矩阵来研究量子力学的方法。冯·诺伊曼在希尔伯特空间上对算子的发展为海森堡的思想奠定了严谨的基础。在该理论中，可观测数据由Hilbert空间上的某些算子（即无限矩阵）表示。量子力学中最著名的概念之一是海森堡的测不准原理，即粒子的速度和位置不能同时已知。这个陈述的数学严谨版本是，测量位置和动量的算子P和Q不交换；即PQ和QP是不相等的（然而，有一个关于P和Q的精确公式）。算子代数是研究算子集合之间代数关系的学科。可顺从算子代数是一类特别重要的算子代数。近年来，对简单可服从算子代数进行了大量的分类研究。从某种意义上说，简单的例子是理论的基本组成部分。这个项目的目标是理解简单的可调节算子代数之间的关系（例如一个可以嵌入到另一个代数中的方式）和这些代数的对称性，并利用这些想法来揭示关于算子代数的整体结构信息。该项目还将通过培养研究生为美国劳动力的教育做出贡献。从技术上讲，最近Elliott程序的进展表明，UCT类中的可分离的、简单的、核的、正则的C*-代数通过它们的算子k理论群、它们的迹单纯形和它们之间的配对被分类到同构。这应该被看作是cones - haagerup对可分离作用的注入因子的分类的直接类比，就其类型和权重流动而言。cones - haagerup分类已成为现代冯·诺依曼代数理论的基石，而C*-代数类比也有望在C*-代数理论中发挥同等重要的作用。在最近与Carrion， Gabe， Tikuisis和White的联合工作中，PI证明了“可分类”C*-代数的嵌入可以由k理论数据确定为近似幺正等价。该项目的目的是完善和扩展这些技术，着眼于等变分类和非简单分类，以及正则理论的应用，例如计算一类新的C*-代数的核维数。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This project relates to the classification and structure of amenable operator algebras. Operator algebras is an area of mathematics which began with the work of Murray and von Neumann in the 1930's motivated in part by Heisenberg's approach to quantum mechanics in terms of infinite matrices. Von Neumann's development of operators on a Hilbert space put Heisenberg's ideas on rigorous foundations. In this theory, the observable data is represented by certain operators (i.e. infinite matrices) on a Hilbert space. One of the most famous concepts of quantum mechanics is Heisenberg's uncertainty principle that the speed and position of a particle cannot be known simultaneously. The mathematically rigorous version of this statement is that the operators P and Q which measure position and momentum do not commute; i.e. PQ and QP are not equal (there is, however, a precise formula relating P and Q). Operator algebras is the study of algebraic relations between collections of operators. Amenable operators algebras form a particularly important class of operators algebras. In recent years, there has been a substantial work in classifying simple amenable operator algebras. In a certain sense, the simple examples are the basic building blocks of the theory. The goal of this project is to understand the relations between the simple amenable operator algebras (such as ways one can be embedded into the other) and the symmetries of such algebras and to exploit these ideas to uncover structural information about operator algebras as a whole. This project also will be contributing to the education of the US workforce through the training of graduate students.More technically, recent progress in Elliott's Program shows separable, simple, nuclear, regular C*-algebras in the UCT class are classified up to isomorphism via their operator K-theory groups, their trace simplex, and the pairing between them. This should be viewed as a direct analogue to the Connes-Haagerup classification of separably acting injective factors in terms of their type and flow of weights. The Connes-Haagerup classification has become a corner stone of modern von Neumann algebra theory, and the C*-algebraic analogue may be expected to have an equally important role in C*-algebra theory. In recent joint work with Carrion, Gabe, Tikuisis, and White, the PI showed that embeddings of "classifiable" C*-algebras are determined up to approximate unitary equivalence by K-theoretic data. The aim of this project is to refine and expand on these techniques with an eye towards equivariant classification and non-simple classification, as well as applications to the regularity theory, such as computing the nuclear dimension of a new classes of C*-algebras.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.