Operator Algebras, K-theory and Groups

算子代数、K 理论和群

• 批准号: 0500693
• 负责人: Marius Dadarlat
• 金额: $ 27.52万
• 依托单位: Purdue University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-03-01 至 2009-02-28
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0500693&HistoricalAwards=false
• 关键词: Operator; Algebras; theory; Groups

Research conducted in the last ten years has revealed unexpected rigidity properties of noncommutative C*-algebras. Whereas the cohomological invariants of a space (commutative C*-algebra) will determine the space at most up to homotopy equivalence, in the class of nuclear simple C*-algebras, the objects are often determined up to isomorphism by their K-theoretical invariants. Elliott's conjecture states that, far from being an accident, this is always the case for the entire class of separable nuclear simple C*-algebras. (Tracial invariants are needed if the real rank is nonzero.) The proposed research aims to uncover and explain rigidity properties of nuclear C*-algebras. The basic idea beyond the classification program is that that the simplicity and the stable or real rank conditions for a nuclear C*-algebra translate to certain internal dynamical properties of the algebra which forces a behavior typical to that of a combinatorial object. The C*-algebra becomes a rigid object built around its K-theory skeleton. The ramifications of the classification theory into the structure theory of C*-algebras will be explored with emphasis on dynamical systems and group C*-algebras. The investigator will analyze the impact of the recent advances around the Baum-Connes conjecture on the classification theory with the long term goal of formulating and exploring a Baum-Connes type conjecture for general nuclear C*-algebras. This is closely tied with the universal coefficient theorem problem in KK-theory and deformation theory of C*-algebras.Geometry was developed in an attempt to describe the ambient physical space. Its history has seen a series of remarkable achievements from the Euclidian geometry to the non-Euclidian geometries which culminated with the Riemannian geometry providing a successful model for large-scale spacetime in general relativity. The noncommutative geometry of Alain Connes is a far reaching generalization of the Riemannian geometry, well adapted for the study of a variety of large and small scale structures. The theory can be viewed as a significant development in the quest of quantizing of mathematics following the successful quantization of physics. As in quantum physics, the coordinates in this theory are no longer ordinary numbers but noncommuting operators acting on infinite dimensional Hilbert spaces. The ordinary spaces are being replaced by algebras of operators. The proposed project aims to contribute to the extensive effort of a community of researchers to extend the mathematics of commutative spaces to operator algebras.

在过去的十年中，研究人员发现了非交换C*-代数的刚性性质。 而空间（交换C*-代数）的上同调不变量将最多在同伦等价的情况下确定空间，在核单C*-代数类中，对象通常由它们的K-理论不变量确定到同构。 艾略特猜想指出，这绝不是一个偶然，对于整个可分核单C*-代数类来说，情况总是如此。（如果真实的秩非零，则需要Tracial不变量。） 该研究旨在揭示和解释核C*-代数的刚性性质。 分类程序之外的基本思想是，核C*-代数的简单性和稳定或真实的秩条件转化为代数的某些内部动力学性质，这些性质迫使组合对象具有典型的行为。C*-代数变成了一个围绕其K-理论骨架构建的刚性对象。 分支的分类理论到结构理论的C*-代数将探讨重点是动力系统和群C*-代数。 研究人员将分析Baum-Connes猜想的最新进展对分类理论的影响，长期目标是制定和探索一般核C*-代数的Baum-Connes型猜想。 这与KK-理论和C*-代数的变形理论中的泛系数定理问题密切相关。几何学的发展是为了试图描述周围的物理空间。从欧几里得几何到非欧几里得几何，黎曼几何在广义相对论中为大尺度时空提供了一个成功的模型。 阿兰·康纳斯的非对易几何是黎曼几何的一个深远的推广，非常适合于研究各种大小尺度的结构。这一理论可以被看作是继物理学量子化之后，数学量子化探索的一个重大发展。 与量子物理中一样，这个理论中的坐标不再是普通的数字，而是作用于无限维希尔伯特空间的非对易算子。 普通的空间正被算子代数所取代。拟议的项目旨在促进社区研究人员的广泛努力，将交换空间的数学扩展到算子代数。