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

阮教授的项目将利用一个简单但重要的事实，即希尔伯特空间上的算子矩阵又是一个算子，因此，以规范和自然的方式，有一个范数。他和他的合著者最近用这些矩阵范数刻画了Hilbert空间上算子的单位Banach代数。他在当前项目中的目标之一是获得这一结果的共轭空间模拟。他还将从矩阵的角度研究算子代数张量积的交叉范数，并将研究局部紧群的多变量形式的傅立叶代数。这里要进行的数学研究是针对算子代数理论中的基本问题的。运算符可以被认为是丰富的数，遵守与普通数相同的算术法则，但有两个明显的例外：运算符的乘法取决于因子的取值顺序，并且并不是每个非零运算符都有逆运算。在有限维的情况下，运算符只是数字的平方矩阵，是线性代数的基本对象，即使在这个阶段，更丰富的结构也变得明显。在无限维度上，大小和距离的概念变得至关重要，这些概念是通过所谓的算子范数来衡量的。阮教授研究的实质是，通过考虑规范的行为，发现什么样的数学对象可以用算子来表示。

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.