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.

摘要菲利普斯（Phillips）主要研究员N. Christopher Phillips提出了三条研究：（1）简单C*-Algebras的分类和结构； （2）平滑歧管上平滑函数代数的功能分析表征； （3）他以前在指数级上的工作的延续。在（1）中，他提议与清林合作了解最小差异性的转化组C*代数的结构。最终的目标是表明它们是亚均匀c*-ergebras的直接限制，这将使它们接近可分类的稳定有限的简单C*-Algebras类。这些代数的K理论中的一些已经有趣的中间结果（例如取消结果）更接近实现。他还建议以几种方式跟进有关纯粹的分类案例的最新工作：在非核案例中有一种有趣的不变性的可能性，证明核性的长期可能性意味着通用系数定理，并且是对已知分类定理的真实情况的非常合理的方法。在（2）中，他提出试图证明平滑函数代数的功能分析表征，以便更好地了解非交通歧管的内容。在（3）中，他提议搜索具有指数级较高的简单C*-Algebra，并试图更好地理解稳定和同质C*-Algebras的指数等级。 该项目的目的是提高对HE“简单” C* - 代数的理解。 c* - 代数是一种代数系统（有点像一组实数，其加法，减法，乘法和除法的操作，但更复杂）。它具有附加的结构，该结构粗略地描述了何时“大”或“小”（同样，有点像实数集）。 C* - 代数被证明是数学中最重要的结构之一。它们对数学的其他部分具有重要的应用，乍一看似乎是无关的（例如几何形状），并且它们也是在量子力学中很重要的结构之一，即正确处理原子和其他非常小的物体所需的（相当反直觉的）物理理论。简单的c* - 代数是那些不能分成较小的碎片的代数，从某种意义上说，所有c*-ergebras都是由它们构建的。该项目有两个主要目标。一种是促进对简单C* - 代数的“内部结构”的理解，或多或少地了解每个人都有可能知道的一切。当它们用于其他主题时，这很重要。另一个是提高对C*-Algebras的分类知识：一个人想要一个完整的列表，并确定何时以不同方式获得的两个简单的C* - 代数何时实际上是相同的。