The Structure of Nuclear C*-Algebras

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Abstract Lin The project is to study nuclear C*-algebras by studying their homomorphisms, isomorphic types and their invariants, and by studying extensions of C*-algebras. Continuous maps between topological spaces are of fundamental importance in topology. As their noncommutative counterparts, homomorphisms are of fundamental importance in the theory of C*-algebras. One part of the project is to determine when two homomorphisms from one unital separable nuclear C*-algebra to another C*-algebra are (stably) approximately unitarily equivalent. It is always in the center of the theory of C*-algebras to determine when two C*-algebras are isomorphic and to describe their invariants. Another part of the project is to classify (simple) nuclear separable C*-algebras via their K-theory. These two parts are closely related and both involve the general theory of C*-algebras, KK-theory, the theory of automorphisms, the C*-algebra extension theory, the group representations as well as algebraic topology. Many important and significant problems in engineering, the physical sciences, and social sciences require the knowledge of linear algebras (or matrix algebras) to solve them. Many problems in linear algebras, in particular asymptotic and approximation problems in linear algebras are best described in infinite dimensional linear algebras. The study of C*-algebras is the study of infinite dimensional linear algebras. However the applications of C*-algebras range from linear algebras, operator theory, group representations, dynamic systems, to model quantum field theory and quantum statistical mechanics. For the development of the theory of C*-algebras as well as for the purpose of applications to other areas, it is primarily important to have a better understanding of the structure of C*-algebras. It is the ambitious goal of this project to determine the structure of C*-algebras with fewest possible data.