Mathematical Sciences: Investigation of the Structure of C*-Algebras

9400904菲利普斯 该项目将涉及四个方面的研究。(1)C*-代数中指数秩概念的继续研究。特别是，将试图获得更好的理解的指数秩的齐次C*-代数和稳定的C*-代数。(2)研究与乘法算子有关的工作。(3)继续研究光滑交叉积的K-理论。(4)自由群的约化C ~*-代数的元素谱的研究。 算子是矩阵的无限维推广。有一个自然的乘法和加法运算符，因此，一些运算符的集合可以被认为是数学结构称为算子代数。 算子上的对合很像矩阵上的转置对合。在这个对合作用下闭的算子代数称为星星代数。一个C*-代数是一个具有额外拓扑性质的星星算子代数。在许多交换代数中，具有乘法逆的元素具有幂，或者等价地，可以写为指数。这在非交换C*-代数中不是这样。这个项目的一部分旨在确定将可逆元素写为指数乘积所需的指数因子的数量。 ***

9400904 Phillips The project will involve four lines of research. (1) A continuing investigation of the notion of exponential rank in C*- algebras. In particular, an attempt will be made to obtain a better understanding of the exponential rank of homogeneous C*- algebras and stable C*-algebras. (2) Investigation of related work involving multiplicative commutators. (3) Continuation of work on the K-theory of smooth crossed products. (4) A study of spectra of elements of the reduced C*-algebras of free groups. Operators are infinite dimensional generalizations of matrices. There is a natural multiplication and addition of operators and, consequently, some collections of operators can be considered as mathematical structures called operator algebras. There is an involution on operators much like the transposition involution on matrices. An operator algebra that is closed under the action of this involution is called a star algebra. A C*- algebra is a star operator algebra that has additional topological properties. In many commutative algebras, elements that have multiplicative inverses have logarithms or, equivalently, can be written as an exponential. This is not the case in non-commutative C*-algebras. A portion of this project is aimed at determining the number of exponential factors required to write invertible elements as products of exponential. ***