Mathematical Sciences: Operator Algebras

这个项目的中心主题是诺姆德塔 矩阵空间，位于某些赋范线性 空间，例如有界代数的子空间 希尔伯特空间上的算子完全有界映射与张量 产品将在此背景下进行研究。的附加方面 该项目包括函数上希尔伯特模的工作 代数，以及算子不变子空间的研究 由遍历变换和乘法组成。 这里要进行的数学研究旨在 算子代数理论中的基本问题。 运算符可以被认为是丰富的数，服从相同的 算术定律作为两个显著的普通数 例外：运算符的乘法取决于 其中因子被取，并且不是每个非零运算符都有 一个逆。在有限维的情况下，算子是 只是数字的方阵，线性的基本对象， 代数，甚至在这个阶段，更丰富的结构变得 很明显从无限维的角度来看，大小和距离的概念 作为衡量什么是所谓的规范的运营商，成为 至关重要的.这些规范在受到各种各样的 建筑业是该项目的重点之一。另一个重点 是对某些算子代数的研究， 高维复杂几何

The central theme of this project is the tower of normed matrix spaces that sits over certain kinds of normed linear spaces, for instance subspaces of the algebra of bounded operators on Hilbert space. Completely bounded maps and tensor products will be studied in this setting. Additional aspects of the project include work on Hilbert modules over function algebras, and investigation of invariant subspaces for operators composed of an ergodic transformation and a multiplication. The mathematical research to be pursued here is aimed at fundamental questions in the theory of operator algebras. Operators may be thought of as enriched numbers, obeying the same laws of arithmetic as ordinary numbers with two notable exceptions: multiplication of operators depends upon the order in which the factors are taken, and not every non-zero operator has an inverse. In the finite-dimensional situation, operators are just square matrices of numbers, the basic objects of linear algebra, and even at this stage the richer structure becomes apparent. Infinite-dimensionally, notions of size and distance, as measured by what is called the norm of an operator, become crucial. How these norms behave when subjected to various constructions is one of the foci of this project. Another focus is the study of certain algebras of operators that come from higher-dimensional complex geometry.