Mathematical Sciences: Noncommutative Homotopy Theory and Proper Actions on C*-Algebras

数学科学：非交换同伦理论和 C*-代数的适当作用

• 批准号: 8905812
• 负责人: Norman Phillips
• 金额: $ 3.56万
• 依托单位: University of Georgia Research Foundation Inc
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1989
• 资助国家: 美国
• 起止时间: 1989-06-15 至 1991-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8905812&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Noncommutative; Homotopy; Theory

Professor Phillips will continue his ongoing study of the homotopy theory of inverse limits of operator algebras. This includes a noncommutative version of the Bott periodicity theorem describing the iterated loop spaces of the infinite unitary group up to homotopy, and the search for noncommutative classifying algebras for K-theory and other cohomology theories. Another facet of his project involves proper actions of noncompact groups on noncommutative operator algebras, where it is hoped that the theory will develop along the lines of the topological case of groups acting on spaces. The mathematical research for this project falls under the heading of noncommutative algebraic topology. Topology is the study of shape, the overall conformation of objects (called topological spaces when one looks at them from this point of view). Topology becomes algebraic when one systematically computes invariants (numbers, say, or slightly more complicated mathematical objects) that are defined for classes of topological spaces. For instance, algebraic topology assigns invariants that count numbers of holes of various dimensions in topological spaces, useful when the space is itself too high-dimensional to permit a pictorial rendering. One popular topological procedure is to take a standard space, like a circle or a sphere, and look at how it maps into the space under examination, two such mappings being considered identical if one can be continuously deformed into the other; this is homotopy theory. Topology becomes noncommutative when you replace the space by an appropriate algebra of complex functions on it, and then forget about the space and allow the algebra to be more or less arbitrary. One result of this is to be able to count holes, so to speak, in objects entirely outside the traditional purview of topology, objects such as operator algebras that have heretofore been of interest mainly to functional analysts. The more general framework also yields some insights on the older subject matter by providing more elbow room for constructions. Professor Phillips' project is concerned with certain facets of this work of extending the machinery of topology into new domains.

菲利普斯教授将继续他正在进行的研究， 算子代数逆极限同伦理论这 包括一个非交换版本的博特周期性定理 描述无限酉群的迭代圈空间 直到同伦，以及寻找非交换分类 代数的K-理论和其他上同调理论。另一 他的项目的一个方面涉及非紧凑群体的适当行动 在非交换算子代数上，希望 理论将沿着拓扑情况的路线发展， 作用于空间的群。 该项目的数学研究属于福尔斯 非交换代数拓扑学的标题。拓扑是 研究形状，物体的整体构造（称为 当我们从这一点来看拓扑空间时， 视图）。当一个人系统地 计算不变量（比如数字，或者稍微复杂一点的 数学对象）定义的拓扑类 空间.例如，代数拓扑分配不变量， 计算拓扑中各种尺寸的孔的数量 空间，当空间本身维数太高， 允许一种绘画的表现。一个流行的拓扑过程 就是找一个标准的空间，比如一个圆或者一个球，然后看 在它如何映射到被检查的空间，两个这样的 映射被认为是相同的，如果一个映射可以连续 变形成另一个;这就是同伦理论。拓扑 将空格替换为 适当的代数的复杂函数，然后忘记 让代数或多或少 任意.这样做的一个结果是能够计数孔， 说，在对象完全超出了传统的范围， 拓扑，对象，如算子代数，迄今为止， 主要是功能分析师的兴趣。更一般的 框架也产生了一些关于旧主题的见解 为建筑提供更多的活动空间。教授 菲利普斯的项目涉及这项工作的某些方面 将拓扑学的机制扩展到新的领域。