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.

鲁恩教授的项目将利用一个简单但因此的事实，即希尔伯特·太空的运营商矩阵再次成为操作员，因此，以规范和自然的方式具有规范。他和合着者最近在希尔伯特空间上的Unital Banach代数来表征了这些母系规范。他在当前项目中的目标之一是获得该结果的共轭空间类似物。他还将从临床角度研究操作员代数的张量产品的跨字符，并将研究本地紧凑型组的傅立叶代数的多变量版本。 这里要进行的数学研究旨在涉及操作者代数理论中的基本问题。 可以将操作员视为富集的数字，遵守与普通数字相同的算法定律，有两个值得注意的例外：操作员的乘法取决于接收因素的顺序，而不是每个非零操作员都具有反向。在有限维的情况下，操作员只是数字的正方形矩阵，是线性代数的基本对象，甚至在此阶段，较富裕的结构变得显而易见。在无限的情况下，由所谓的操作员的规范衡量的大小和距离的概念变得至关重要。 Ruan教授的研究的本质是通过考虑规范的行为来发现哪种数学对象可以表示为操作员。