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.

首席研究员N. Christopher Phillips提出了三个研究方向：(1)研究简单C*-代数的分类和结构；(2)光滑流形上光滑函数代数的泛函解析刻划；(3)继续他之前关于指数秩的研究。在(1)中，他提议与林青一起研究极小微分同态的变换群C*-代数的结构。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简单C*-代数的可分类类。一些已经很有趣的中间结果，如这些代数的k理论中的消去结果，更接近于实现。他还建议从几个方面对最近关于分类的纯无限情况的工作进行后续研究：在非核情况下有趣的不变量的可能性，证明核意味著普适系数定理的可能性，以及对已知分类定理的真实情况的一种非常合理的方法。在(2)中，他提出尝试证明光滑流形上光滑函数代数的泛函解析表征，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（有点像实数的集合，具有加、减、乘、除的运算，但更复杂一些）。它有额外的结构，粗略地说，描述什么时候是“大”或“小”（再一次，有点像实数的集合）。C*代数是数学中最重要的结构之一。它们在数学的其他部分有重要的应用，这些部分乍一看似乎不相关（例如几何），它们也是量子力学中重要的一种结构，量子力学是一种（相当违反直觉的）物理理论，需要适当地处理原子和其他非常小的物体。简单的C*代数是那些不能被分解成小块的代数，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“内部结构”的理解，或多或少地了解每一个代数，所有可能知道的。当它们用于其他科目时，这是相关的。另一个目的是提高对C*-代数分类的认识：人们想要一个C*-代数的完整列表，并有一个公式来判断两个简单的C*-代数，以不同的方式获得，实际上是相同的。