C*-Algebras, K-theory and Groups

摘要近十年来的研究揭示了非交换C*-代数意想不到的刚性性质。然而空间（交换C*-代数）的上同调不变量将决定空间至多达到同伦等价，在核简单C*-代数类中，对象通常由它们的k理论不变量决定到同构。艾略特的猜想指出，这绝不是偶然的，对于整个可分离核简单C*代数类来说，情况总是如此。（如果实秩非零，则需要Tracial不变量。）提出的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实秩条件转化为代数的某些内部动态特性，这些特性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。分类理论在C*-代数结构理论中的分支将被探索，重点是动力系统和群C*-代数。研究人员将分析围绕Baum-Connes猜想的最新进展对分类理论的影响，长期目标是形成和探索一般核C*-代数的Baum-Connes猜想。这与kk理论中的普适系数定理问题和C*-代数的变形理论密切相关。几何学是为了描述周围的物理空间而发展起来的。它的历史见证了从欧几里得几何到非欧几里得几何的一系列显著成就，其中以黎曼几何为顶峰，为广义相对论中的大尺度时空提供了一个成功的模型。Alain cones的非交换几何是对黎曼几何的一个深远的推广，非常适合于各种大小尺度结构的研究。这个理论可以看作是继物理学成功的量子化之后，数学在追求量子化方面的一个重大发展。与量子物理一样，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交换算子。普通空间被算子的代数所取代。提议的项目旨在为研究人员社区的广泛努力做出贡献，将交换空间的数学扩展到算子代数。

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.