Mathematical Sciences: Operator Algebras and Their Applications

数学科学：算子代数及其应用

• 批准号: 8908281
• 负责人: Masamichi Takesaki
• 金额: $ 36.69万
• 依托单位: University of California-Los Angeles
• 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: 1989-07-01 至 1993-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8908281&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Operator; Algebras; Their

Professors Effros, Popa, and Takesaki will continue their investigations into operator algebra theory and its applications. Effros will focus on matricial norms and orderings. This is expected to lead to new approaches to Fourier analysis on non- abelian groups and to the structure theory of non-injective von Neumann algebras. Popa will work on the structure of factors, index theory for subfactors, amenability, and rigidity. Takesaki will study group actions on injective factors, and explore the relationship between modular theory for von Neumann algebras and Connes' cyclic cohomology. The Hilbert space operators that lie at the heart of this project may be thought of as a species of enriched numbers. They obey the same laws of arithmetic as numbers with two notable exceptions, namely that the result of multiplication depends on the order in which the factors are taken, and not every nonzero operator has an inverse. Mathematical analysis knows only two number systems, the real and complex, but there is an immense variety of operator algebras. Situations unsuitable for numerical representation can often be represented in some fashion by operator algebras, thanks to the greater degree of flexibility available with the latter. The price for this is that understanding the structure and classification of operator algebras requires considerable effort, as for instance the work to be undertaken in this project.

Effros，Popa和Takeaki教授将继续对操作员代数理论及其应用进行调查。 EFFRO将专注于母系规范和顺序。预计这将导致对非阿贝尔群体的傅立叶分析的新方法，以及非注射性von Neumann代数的结构理论。 POPA将致力于因素的结构，子因子的索引理论，舒适性和僵化。 Takeaki将研究对注射因素的小组行动，并探讨von Neumann代数的模块化理论与Connes Cyclicslogology之间的关系。 位于该项目核心的希尔伯特太空运营商可能被认为是一种丰富的数字。他们遵守与数字相同的算法定律，并且有两个值得注意的例外，即乘法的结果取决于接收因素的顺序，而不是每个非零运算符都具有反向。数学分析仅知道两个数字系统，即真实和复杂，但是有多种操作员代数。由于后者提供的灵活性更大，通常可以通过运算符代数来代表数值表示不适合数值表示的情况。这样做的价格是了解操作员代数的结构和分类需要大量的努力，例如在本项目中要进行的工作。