Mathematical Sciences: Analytic and Geometric Aspects of Operator Algebras

这项研究是在该地区的运营商代数和 包括三个项目。 第一，是对问题的研究。 极大复同态扩张的唯一性 交换对角子代数到代数的纯态。 第二，研究将涉及指数理论的工作， 任意因子的包含与“智”的研究 有限因子不变。 第三，研究将继续下去 以前的工作包含冯诺依曼代数， 涉及称为“量化群”的类群对象。 本研究涉及算子的一般理论， 空间 物理学中的一个基本概念是 空间上算子的对称群。 最 重要的群作为结构空间的对称性而出现， 例如来自集合的对称、线性和酉群， 线性空间和希尔伯特空间。 在量子 物理学中，空间被一个非交换代数所取代， 希尔伯特空间上的算子 量子群，其中 经典群的变形，已经被证明是重要的， 物理学和数学的几个领域。

This research is in the area of operator algebras and consists of three projects. First, is a study of the problem of uniqueness of extension of complex homomorphisms on maximal abelian diagonal subalgebras to pure states of the algebra. Second, the research will involve work on the Index theory of inclusions of arbitrary factors and the study of the 'chi' invariant for finite factors. Third, the research will continue previous work on the inclusions of von Neumann algebras which involve group-like objects called 'quantized groups'. This research concerns the general theory of operators on a space. One of the fundamental concepts in physics is the notion of the symmetry groups of operators on a space. The most important groups arise as the symmetries of structured spaces, such as the symmetric, linear, and unitary groups from a set, linear space, and Hilbert space, respectively. In quantum physics, the space is replaced by a noncommutative algebra of operators on a Hilbert space. The quantum groups, and among them the deformations of classical groups, have proved important in several areas of physics and mathematics.