Operator Algebras, K-theory and Groups

近十年来的研究揭示了非交换C*-代数的意外刚性性质。虽然空间(交换C*-代数)的上同调不变量至多决定空间的同伦等价，但在核单C*-代数类中，对象通常由它们的K-理论不变量确定为同构。埃利奥特的猜想指出，这远非偶然，而是整个可分核单C*-代数的情形。(如果实数秩非零，则需要轨迹不变量。)本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该代数的某些内部动力学性质，从而迫使组合对象的典型行为。C*-代数成为围绕其K-理论骨架构建的刚性对象。分类理论在C*-代数的结构理论中的分支将被重点放在动力系统和群C*-代数上。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，长期目标是建立和探索一般核C*-代数的Baum-Connes型猜想。这与KK-理论中的泛系数定理问题和C*-代数的形变理论密切相关。几何是为了描述环境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着成就，最终以黎曼几何为广义相对论中的大尺度时空提供了一个成功的模型。Alain Connes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。这一理论可以被视为继物理学量子化之后，在数学量子化方面的一项重大发展。正如在量子物理学中一样，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。普通空间正在被算子的代数所取代。这项拟议的项目旨在促进一个研究团体的广泛努力，将交换空间的数学扩展到算子代数。

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.