Mathematical Sciences: Quasitriangularity in von Neumann Algebras and Other Topics

本课题的目的是发展半有限von Neumann代数的拟三角子代数理论。将研究的问题包括：拟三角代数中元素的刻画、邻近性问题、局部性、有限宽CSL代数的应用、外套以及B(H)中的超自反性问题。与插补理论、调和分析和控制理论相结合。所提出的方法包括联合算子范数/Hilbert-Schmidt范数逼近和密度变元。这个项目的一般数学领域以希尔伯特空间算子的代数理论为基础。运算符可以被认为是复数的有限或无限矩阵。特殊类型的运算符通常放在一个代数中，自然称为运算符代数。这些看似抽象的对象有着令人惊讶的各种应用。例如，它们在纽结理论中扮演着关键角色，而纽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.