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

在这个项目中，首席研究员将研究单一的、简单的、可服从的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。由Gelfand的一个定理可知，单位交换C*代数与紧化Hausdorrf空间上的连续函数代数是同构的。因此，交换C*-代数的结构完全由底层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*-代数的同态是由从一个底层空间到另一个底层空间的连续映射导出并确定的。因此，在非交换的情况下，这个项目可以归结为对非交换拓扑的研究。该项目的中心目标是(1)使用k理论相关数据对可分离的，简单的，可修改的C*-代数进行分类，(2)确定同态的近似幺正等价类，以及(3)寻找非交换拓扑和拓扑动力系统研究的应用。最简单的C*代数是所有复数的系统。下一个最简单的C*代数是复数矩阵的系统。一般来说，C*-代数是算子系统（可以认为是矩阵的推广）。例如，微分和积分是某些函数空间上的算子。例如，操作符还可以用作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。与数字系统不同，2乘以3等于3乘以2，在一般的C*-代数中，a乘以B的乘积可能与B乘以a不同。这种非交换性反映了量子物理的现实，并与著名的海森堡测不准原理相对应。C*代数出现在许多不同的科学和工程领域，其中量子力学只是一个重要的例子。为了应用的目的，以及理论上的原因，理解C*-代数的结构，或由不同应用形成的算子系统的结构是很重要的。这个项目的目的是找到最简单的基本数据来确定C*代数的结构和C*代数之间存在的关系，从而使应用成为可能。为了有用，数据应该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，那么在所有应用中，这些代数应该是相同的。首席研究员必须搜索这些数据并发明工具来证明这些数据确实可以用来完全确定相应的C*-代数的结构。预期的直接应用将是对动力系统的研究。然而，在数学的许多其他领域（例如，线性代数、算子理论、群表示、非交换拓扑、非交换几何）应该会感受到长期的影响。在过去的几年里，一些相关的研究涉及到几个博士生的培训。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.