Simple C*-Algebras and Dynamical Systems: Classification and Applications

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************Abstract*****************************************The principal investigators, Guihua Gong and Liangqing Li, propose to continue their research on the classification of separable, amenable C*-algebras. They also plan to study the cross product C*-algebras which arise naturally in the study of dynamical systems. The methods to be employed include KK-theory, E-theory, homotopy path with controlled length, and spectrum cutting and absorption.The passage from a finite to an infinite number of degrees of freedom in quantum physics led to the mathematical theory of certain infinite dimensional algebras, called C*-algebras. A C*-algebra is an algebraic system, similar to that of numbers, with its operations of addition, subtraction, multiplication, and division. But unlike the multiplication for numbers, the multiplication in a C*-algebra is not commutative ---- that is, in general, X times Y is not as same as Y times X. This important feature corresponds to Heisenberg uncertainty principle in Quantum Mechanics. The investigators propose to work on a complete enumeration (or classification) of a large class of amenable C*-algebras. They expect that progress on the proposed research will result in important contributions to several mathematical fields including operator algebras, differential topology, and also to the understanding of the infinite dimensional world of quantum physics.