The Structure of Simple Amenable C*-Algebras and their Homomorphisms.

在这个项目中，主要的研究者将研究单位的，单的，服从的C*-代数的结构以及从一个这样的C*-代数到另一个这样的C*-代数的同态。根据Gelfand的一个定理，我们知道一个单位交换C*-代数同构于紧Hausdorrf空间上的连续函数代数。因此，交换C*-代数的结构完全由基础空间或空间的拓扑结构决定。从一个交换C*-代数到另一个交换C*-代数的同态是由一个从一个基础空间到另一个基础空间的连续映射诱导和确定的。因此，在非对易环境下，本课题归结为对非对易拓扑的研究。该项目的中心目标是(1)利用与K-理论相关的数据来分类可分的、单的、可服从的C*-代数，(2)确定同态的近似酉等价类，以及(3)在非对易拓扑和拓扑动力系统的研究中找到应用。其次最简单的C*-代数是复数矩阵系统。一般而言，C*-代数是算符系统(可以被认为是矩阵的推广)。例如，微分和积分是某些函数空间上的运算符。例如，运算符也可以用来作为微观物理世界的可观测对象的模型。运算符系统具有加法和乘法的结构，就像数字系统一样。与二乘三等于三乘二的数字系统不同，在一般的C*-代数中，乘积A乘B可能不等于B乘A。这种非对易反映了量子物理的现实，并与著名的海森伯格测不准原理相对应。C*-代数出现在科学和工程的许多不同领域，量子力学只是其中一个重要的例子。为了应用的目的，以及出于理论上的原因，了解C*-代数的结构，或者从不同的应用形成的算子系统的结构是重要的。这个项目的目的是找到决定C*-代数的结构和C*-代数之间存在的关系的最简单的基本数据，以便应用成为可能。为了有用，这些数据应该很容易获得，并且相对容易计算。此外，如果两个C*-代数产生相同的数据集，则对于所有应用而言，这两个代数应该是相同的。主要调查者必须搜索这些数据并发明工具来提供证据，证明这些数据确实可以用来完全确定相应C*-代数的结构。预期的直接应用将是动力学系统的研究。然而，数学的许多其他领域(例如，线性代数、算子论、群表示法、非对易拓扑学、非对易几何)都应该感受到长期的影响。在过去的几年里，一些相关的研究涉及到几个博士生的培养。该项目还将包括研究生培训和博士后研究人员的指导。

In this project the principal investigator will study the structure of unital, simple, amenable C*-algebras and homomorphisms from one such C*-algebra to another. From a theorem of Gelfand we know that a unital commutative C*-algebra is isomorphic to the algebra of continuous functions on some compact Hausdorrf space. The structure of a commutative C*-algebra is thus completely determined by the underlying space, or the topological structure of the space. A homomorphism from one commutative C*-algebra to another is induced and determined by a continuous map from one underlying space to another. Thus, in the noncommutative setting, this project comes down to a study of noncommutative topology. The central goals of the project are (1) to use K-theory-related data to classify separable, simple, amenable C*-algebras, (2) to determine approximate unitary equivalence classes of homomorphisms, and (3) to find applications to the study of noncommutative topology and topological dynamical systems.The simplest C*-algebra is the system of all complex numbers. The next most simple C*-algebras are systems of matrices of complex numbers. In general, C*-algebras are systems of operators (which can be thought of as generalizations of matrices). For example, differentiation and integration are operators on certain function spaces. Operators can also be used, for example, as models for observables for the microscopic physical world. A system of operators has the structure of addition and multiplication, just like the system of numbers. Unlike the system of numbers, where two times three is the same as three times two, in a general C*-algebra the product A times B may not be the same as B times A. This noncommutativity reflects the reality of quantum physics and corresponds to the famous Heisenberg uncertainty principle. C*-algebras arise in many diverse areas of science and engineering, of which quantum mechanics is just one important example. For purposes of application, as well as for theoretical reasons, it is important to understand the structure of C*-algebras, or the structure of systems of operators formed from different applications. The aim of this project is to find the simplest essential data that determine both the structure of a C*-algebra and the relations that exist between C*-algebras so that applications become possible. To be useful, the data should be easy to obtain and relatively easy to compute. Furthermore, if two C*-algebras give rise to the same set of data, then the algebras should be identical for purposes of all applications. The prinicipal investigator has to search these data and invent tools to provide a proof that such data can indeed be used to determine completely the structure of the corresponding C*-algebras. The expected immediate applications will be to the study of dynamical systems. However, a long-term impact should be felt in many other areas of mathematics (e.g., linear algebra, operator theory, group representations, noncommutative topology, noncommutative geometry). In the last few years, some related research involved the training of several Ph.D. students. This project will also include both graduate student training and the mentoring of postdoctoral researchers.