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

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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. ***