Simple Amenable C*-algebras and K-theory

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In quantum mechanics, or in the microscopical physical world, an observable may be modeled by a self-adjoint operator on a Hilbert space. A system of such operators forms a C*-algebra. In the finite dimensional cases, these operators are just matrices. In more general cases, these operators may be represented by infinite matrices, or matrices on infinite dimensional Hilbert space. In the mathematical formulation of quantum mechanics, a pair of self-adjoint operators representing observables are typically non-commutative which corresponds to the Heisenberg uncertainty principle. The study of C*-algebra is often viewed as the study of non-commutative analogues of the algebra of continuous functions on a topological space. There are many C*-algebras that come from different fields of sciences and the study of C*-algebras has variety of applications. It is known that C*-algebras with different appearance from different applications can be the same. On the other hand, C*-algebras from the same source may look the same but have essentially different structure. For application purposes as well as the theoretical point of view, it is extremely important to determine the structure of C*-algebras from their appearance. This project is an attempt to use a small number of computable data to completely determine C*-algebras in certain classes. In other words one may study the computable data to understand the structure of C*-algebras. This project is a study of classification of amenable C*-algebras using K-theory related invariants and applications of the classification. The main part of this project is to search for a broad classification of non-unital simple amenable C*-algebras. Part of the strategy is to create new technical tools to deal with non-commutative version of non-compact spaces. The goal is to establishe a new type of existence theorem as well as a K-theoretic characterization of asymptotic unitary equivalence for non-unital simple C*-algebras. A closely related topic is to verify certain C*-algebras satisfy the Universal Coefficient Theorem. The research of this project will also provide a new passage to greatly simplify the way in which the theory of C*-algebras is applied to various other related research areas, notably, to non-commutative homotopy theory and the study of dynamical systems. For example, it is proposed, using this broad classification theory, to study a new relation among minimal homeomorphisms on a compact metric space which is called asymptotic conjugacy, and ultimately characterize this relation via K-theoretical data.