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

9400904 Phillips这个项目将包括四条研究线。(1)继续研究C*-代数中指数秩的概念。特别地，我们将尝试更好地理解齐次C*-代数和稳定C*-代数的指数秩。(2)对乘法换易子相关工作的考察。(3)光滑交叉积k理论的继续研究。(4)自由群的约化C*代数元素谱的研究。算子是矩阵的无限维推广。运算符的乘法和加法是自然存在的，因此，一些运算符的集合可以被认为是称为运算符代数的数学结构。算子的对合与矩阵的转置对合非常相似。在这种对合作用下闭合的算子代数称为星形代数。C*-代数是具有附加拓扑性质的星型算子代数。在许多交换代数中，具有乘法逆的元素具有对数，或者等价地，可以写成指数。这在非交换的C*代数中是不成立的。这个项目的一部分旨在确定将可逆元素写成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. ***