Mathematical Sciences: Operator Algebras and Their Applications

Effros、Popa和Takesaki教授将继续研究算子代数理论及其应用。Effros将专注于婚姻规范和秩序。这有望为非交换群上的傅里叶分析和非内射von Neumann代数的结构理论提供新的途径。Popa将致力于要素的结构、子要素的指数理论、适应性和刚性。Takesaki将研究内射因子上的群作用，并探索von Neumann代数的模理论与Connes的循环上同调之间的关系。位于该项目核心的希尔伯特空间运算符可以被认为是一种丰富的数字。它们遵守与数字相同的算术定律，但有两个明显的例外，即乘法的结果取决于因子的取值顺序，并且并不是每个非零运算符都有一个逆运算符。数学分析只知道两个数字系统，实数和复数，但运算符代数种类繁多。不适合用数字表示的情况通常可以用算符代数以某种方式表示，这要归功于后者提供的更大程度的灵活性。这样做的代价是，理解算子代数的结构和分类需要相当大的努力，例如这个项目中要进行的工作。

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.