Mathematical Sciences: Studies in Automorphism Groups and Operator Algebras

Arveson和Vocerescu教授将研究非对易分析的几个领域，涉及von Neumann代数、量子化指数理论、C*-代数、Fredholm模和非对易随机变量。Arveson的项目部分涉及因子自同态半群的指数理论和分类理论，以及与这些半群相关的C*-代数的结构。Vocerescu将继续他关于赋范理想的拟中心近似单位及其与Fredholm模的关系的研究，以及自由积的非对易概率理论。他将进一步研究核C*-代数与算子逼近理论之间的关系。这项数学研究项目涉及希尔伯特空间上涉及算子的各种构造和技巧。运算符可以被认为是一种丰富的数字。它们遵循与数字相同的算术规则，只是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。事实证明，在理论物理和纯数学中，看看在给定的上下文中用运算符替换数字时会发生什么，这是一个非常有成效的练习。例如，Arveson教授正在研究依赖时间的数字被依赖于时间的运算符取代时出现的系统。沃库列斯库教授正在进行的一项研究涉及概率论的非对易版本，在该理论中，随机变量被视为算符取值而不是数值取值。

Professors Arveson and Voiculescu will investigate several areas of noncommutative analysis involving von Neumann algebras, quantized index theory, C* - algebras, Fredholm modules, and noncommutative random variables. Arveson's part of the project involves the index theory and classification theory of semigroups of endomorphisms of factors, and the structure of the C* - algebras associated to such semigroups. Voiculescu will continue his research on quasicentral approximate units relative to normed ideals and their relationship to Fredholm modules, and on the noncommutative probability theory of free products. He will look further into the relation between nuclear C* - algebras and the approximation theory of operators. This mathematical research project is concerned with various constructions and techniques involving operators on Hilbert space. Operators may be thought of as a species of enriched numbers. They obey the same rules of arithmetic as numbers except that the result of multiplication depends upon the order in which the factors are taken, and not every nonzero operator has an inverse. It has proven to be a very fruitful exercise, both in theoretical physics and in pure mathematics, to see what happens when numbers are replaced by operators in a given context. Professor Arveson, for instance, is studying systems that arise when time-dependent numbers are replaced by time-dependent operators. One of Professor Voiculescu's ongoing investigations involves a noncommutative version of probability theory in which random variables are treated as if they were operator-valued rather than number-valued.