Certain C*-Algebras of Toeplitz Operators and Singular Integral Operators

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9400600 Xia The investigation will examine the structure of C*-algebras generated by Toeplitz operators and singular integral operators associated within-dimensional flows. This class of C*-algebras includes many classical examples. The investigation is particularly directed to the case where n is greater than 1. In this case each algebra contains a chain of ideals that will be analyzed carefully. These ideals will be used to compute K- groups, which are important invariants for the classification of C*-algebras. The k-groups will also be used to study the invertibility of systems of Toeplitz operators. In this connection an attempt will be made to determine the stable rank of certain commutator ideals. An analysis will be made of certain automorphisms of the Toeplitz algebra on the unit circle which are induced by homeomorphisms of the circle. This project investigates certain algebras generated by classical operators that arise from the study of singular integral equations. These equations can be traced back to applications in engineering and manufacturing such as the equation of the airfoil and the stamping of metal plates. The abstract problem here is the classification of these algebras using invariants. The contributions will be to many fields of pure mathematics including ergodic theory, analysis and geometry. ***