Mathematical Sciences: Quasitriangularity in von Neumann Algebras and Other Topics

本课题是发展半有限冯·诺伊曼代数的拟三角形子代数理论。将要研究的一些问题是：准三角形代数中元素的特征、邻近性问题、局部性、有限宽度CSL代数的应用、外部巢和B(H)中的超自反性问题。将与插补理论、谐波分析和控制理论相结合。提出的方法包括联合算子范数/Hilbert-Schmidt范数逼近和密度参数。这个项目的一般数学领域在希尔伯特空间算子的代数理论中有其基础。运算符可以被认为是有限或无限的复数矩阵。特殊类型的运算符通常放在一个代数中，自然地称为运算符代数。这些看似抽象的对象有着各种各样的应用。例如，它们在结理论中起着关键作用，而结理论目前正被用于研究DNA的结构。

The project is to develop a theory of quasitriangular subalgebras of semifinite von Neumann algebras. Some of the problems that will be studied are: characterizations of elements in a quasitriangular algebra, proximinality questions, locality, applications of finite width CSL algebras, external nests, and hyperreflexivity questions in B(H). Connection with interpolation theory, harmonic analysis and control theory will be pursued. The proposed methods include joint operator norm/Hilbert-Schmidt norm approximations, and density arguments. The general area of mathematics of this project has its basis in the theory of algebras of Hilbert space operators. Operators can be thought of as finite or infinite matrices of complex numbers. Special types of operators are often put together in an algebra, naturally called an operator algebra. These seemingly abstract objects have a surprising variety of applications. For example, they play a key role in knot theory, which in turn is currently being used to study the structure of DNA.