Topological and Related Aspects of the Structure of C* - Algebras

C* 结构的拓扑和相关方面 - 代数

• 批准号: 9706850
• 负责人: Norman Phillips
• 金额: $ 8.34万
• 依托单位: University of Oregon Eugene
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1997
• 资助国家: 美国
• 起止时间: 1997-06-01 至 2000-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9706850&HistoricalAwards=false
• 关键词: Topological; Related; Aspects; Structure; Algebras

Abstract Phillips The Principal Investigator, N. Christopher Phillips, proposes three lines of research: (1) work on the classification and structure of simple C*-algebras; (2) a functional analytic characterization of the algebra of smooth functions on a smooth manifold; and (3) a continuation of his previous work on exponential rank. In (1), he proposes to work with Qing Lin on understanding the structure of transformation group C*-algebras of minimal diffeomorphisms. The eventual goal is to show that they are direct limits of sub-homogeneous C*-algebras, which would put them close to the classifiable class of stably finite simple C*-algebras. Several already interesting intermediate results, such as cancellation results in the K-theory of these algebras, are closer to realization. He also proposes to follow up recent work on the purely infinite case of the classification in several ways: a possibility of interesting invariants in the non-nuclear case, a long shot possibility for proving that nuclearity implies the Universal Coefficient Theorem, and a very plausible approach to the real case of the known classification theorem. In (2), he proposes to try to prove a functional analytic characterization of the algebra of smooth functions on a smooth manifold, in an effort to gain a better understanding of what a noncommutative manifold should be. In (3), he proposes to search for a simple C*-algebra with large exponential rank, and to try to understand better the exponential rank of stable and homogeneous C*-algebras. The purpose of this project is to improve the understanding of he "simple" C*-algebras. A C*-algebra is a kind of algebraic system (somewhat like the set of real numbers, with its operations of addition, subtraction, multiplication, and division, but somewhat more complicated). It has additional structure which, roughly speaking, describes when something is "large" or "small" (again, somewhat like the set of real numbers). C*-algebras turn out to be one of the more impor tant kinds of structures in mathematics. They have significant applications to other parts of mathematics which at first sight seem rather unrelated (geometry, for example), and they are also one of the kinds of structure that is important in quantum mechanics, the (rather counterintuitive) physical theory needed to deal properly with atoms and other very small objects. The simple C*-algebras are those that cannot be broken into smaller pieces, and in some sense all C*-algebras are built out of them. The project has two main goals. One is to advance the understanding of the "internal structure" of simple C*-algebras, more or less to know about each one all that it is possible to know. This is relevant when they are used in other subjects. The other is to improve the knowledge of the classification of C*-algebras: one wants a complete list of all of them, with a recipe for deciding when two simple C*-algebras, obtained in different ways, are actually the same.

抽象的菲利普斯 首席研究员，N。克里斯托弗菲利普斯，提出了三条线的研究：（1）工作的分类和结构的简单C*-代数;（2）一个功能分析表征代数的光滑函数光滑流形;和（3）继续他以前的工作指数秩。在（1）中，他建议与 林青关于极小同态的变换群C*-代数的结构的认识。最终的目标是证明它们是次齐次C*-代数的直接极限，这将使它们接近稳定有限单C*-代数的可分类类。几个已经有趣的中间结果，如取消结果在K-理论的这些代数，更接近实现。他还建议跟进最近的工作在纯粹无限的情况下的分类在几个方面：一个可能性有趣的不变量在非核的情况下，一个长远的可能性证明，核意味着普遍系数定理，和一个非常合理的办法，真实的情况下，已知的分类定理。在（2）中，他提出试图证明光滑流形上光滑函数代数的泛函分析特征，以更好地理解非交换流形应该是什么。在（3）中，他提出寻找具有大指数秩的简单C*-代数，并试图更好地理解稳定和齐次C*-代数的指数秩。 这个项目的目的是提高对“简单”C*-代数的理解。C*-代数是一种代数系统（有点像真实的数的集合，有加、减、乘、除的运算，但更复杂）。它有额外的结构，粗略地说，描述了当 某个东西是“大”或“小”的（同样，有点像真实的数的集合）。C ~*-代数是数学中一种重要的结构。它们在数学的其他领域也有重要的应用，这些领域乍看上去似乎是不相关的（例如几何学），它们也是量子力学中重要的结构之一，量子力学是正确处理原子和其他非常小的物体所需要的（相当违反直觉的）物理理论。简单的C*-代数是那些不能被分解成更小的块的代数，在某种意义上，所有的C*-代数都是由它们构成的。该项目有两个主要目标。一个是推进对简单C*-代数的“内部结构”的理解，或多或少地了解每个C *-代数的所有可能的知识。当它们用于其他科目时，这是相关的。另一个是提高C*-代数分类的知识：人们想要一个所有C *-代数的完整列表，并有一个方法来决定两个简单的C*-代数，以不同的方式获得，实际上是相同的。