Mathematical Sciences: Operator Spaces, Operator Algebras and Completely Bounded Maps

阮教授的项目将利用简单但 希尔伯特空间上的算子矩阵是 同样是一个运算符，因此，以规范和自然的方式， norm.他和合著者最近的特点单位Banach 希尔伯特空间上的算子代数， 矩阵范数他在当前项目中的目标之一是 以获得该结果的共轭空间模拟。他还将 算子代数张量积上交叉范数研究 从矩阵的角度来看，并将研究一个多变量 局部紧群的傅立叶代数的一个版本。 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.阮教授研究的精髓在于发现， 通过考虑规范的行为，什么样的数学 对象可以表示为运算符。

Professor Ruan's project will exploit the simple but consequential fact that a matrix of operators on Hilbert space is again an operator and so, in a canonical and natural way, has a norm. He and coauthors have recently characterized unital Banach algebras of operators on Hilbert space in terms of these matricial norms. One of his objectives in the current project is to obtain a conjugate-space analogue of this result. He will also investigate cross-norms on tensor products of operator algebras from a matricial point of view, and will study a multivariable version of the Fourier algebra of a locally compact group. 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. The essence of Professor Ruan's research is to discover, by considering the behavior of norms, what sorts of mathematical objects can be represented as operators.