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

数学科学：算子空间、算子代数和全有界图

• 批准号: 8902467
• 负责人: Zhong-Jin Ruan
• 金额: $ 3.27万
• 依托单位: University of Illinois at Urbana-Champaign
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1989
• 资助国家: 美国
• 起止时间: 1989-05-15 至 1991-10-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8902467&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Operator; Spaces; Algebras

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.

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具有范数。他和合著者最近根据这些矩阵范数描述了希尔伯特空间上算子的酉巴拿赫代数的特征。他当前项目的目标之一是获得该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傅立叶代数的多变量版本。 这里要进行的数学研究针对的是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字，遵循与普通数字相同的算术定律，但有两个值得注意的例外：运算符的乘法取决于因子的取值顺序，并且并非每个非零运算符都有逆元。在有限维情况下，算子只是数字的方阵，是线性代数的基本对象，甚至在这个阶段，更丰富的结构也变得显而易见。在无限维上，尺寸和距离的概念（通过所谓的算子范数来测量）变得至关重要。阮教授研究的本质是通过考虑规范的行为来发现什么样的数学对象可以表示为算子。