C*-Algebras, K-theory and Groups

C*-代数、K 理论和群

• 批准号: 0200601
• 负责人: Marius Dadarlat
• 金额: $ 24.08万
• 依托单位: Purdue University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-06-01 至 2006-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0200601&HistoricalAwards=false
• 关键词: Algebras; theory; Groups

ABSTRACTDadarlatResearch 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.

在过去的十年中进行的AbstractDadarlatresearch揭示了非交通c* - 代数的意外刚性特性。 尽管空间的共生不变剂（交换性C* - 代数）最多将确定空间最多达到同型等效性，而在核简单的C* - 代数类别中，通常会通过其K理论不变性确定对象的同构。 埃利奥特（Elliott）的猜想指出，对于整个可分离的核简单c* - 代数始终是偶然的，始终是这种情况。 （如果实际等级为非零，则需要奇异的不变性。）拟议的研究旨在揭示和解释核C* - 代数的刚性特性。 除分类计划以外的基本思想是，核C*-Algebra的简单性和稳定或实际等级条件转化为代数的某些内部动力学特性，迫使组合对象的行为典型地行为。 c* - 代数变成了围绕其K理论骨骼建立的刚性对象。 分类理论对C* - 代数的结构理论的影响将探索，重点是动力学系统和C* - 代数组。 研究者将分析鲍姆 - 康涅狄格州猜想对分类理论的最新进展的影响，其长期目标是制定和探索对一般核C*-Algebras的Baum-Connes型猜想。 这与KK理论中的通用系数定理问题与C*-Algebras。几何形式的变形理论紧密相关，以描述环境物理空间。它的历史已经看到了一系列显着的成就，从欧几里得的几何形状到非欧国人的几何形状，这些几何形状以riemannian的几何形状达到顶峰，从而为大规模的时空提供了成功的模型。 Alain Connes的非交通性几何形状是Riemannian几何形状的广泛概括，非常适合研究各种大型和小型结构。该理论可以被视为在成功量化物理学后量化数学的重大发展。 与量子物理学一样，该理论中的坐标不再是普通数字，而是作用于无限尺寸希尔伯特空间的非公认操作员。 普通空间被操作员代数代替。拟议的项目旨在为研究人员社区的广泛努力做出贡献，以将换向空间的数学扩展到操作员代数。