This proposal is to study when two C*-algebras are isomorphic by comparing their K-theoretic data. In particular, it attempts to use the K-theoretic data to determine whether two unital separable simple amenable C*-algebras which are approximately divisible (or Z-stable) and satisfies the Universal Coefficient Theorem are isomorphic. It also proposes to study a closely related problem whether K-theoretic data of a minimal dynamic system could determine the structure of the minimal dynamic system by studying the associated transformation C*-algebra together with other K-theoretic data of the system. Viewing C*-algebras as non-commutative topological spaces, it also proposes to study (approximate) homotopy theory in C*-algebras.
In the micro-scopical physical world, an observable may be modeled by a self-adjoint operator on a Hilbert space, according to Dirac and von Neumann. A system of such operators forms a C*-algebra. Such a system has the structure of addition and multiplication like the system of complex numbers. However, in a C*-algebra, multiplication may not be commutative which corresponds to the Heisenberg uncertainty principle. Let X be a compact metric space and F be a transformation from X to X which is assumed to be invertible and both F and its inverse are continuous. The pair (X, F) forms the associated transformation C*-algebra. To study the dynamical structure of (X, F), one may start with the associated C*-algebra. The study of the structure of the associated C*-algebra provides the information of the original dynamical system. One of such examples is the special case that X is the unit circle and F is an irrational rotation on the circle. The associated C*-algebra is a unital separable simple amenable C*-algebra. This C*-algebra can also be formed by a typical non-commutative relation of two unitary operators. It is also known as non-commutative torus. There are many C*-algebras come from different fields of sciences and the study of C*-algebras has variety of applications. For example, C*-algebras may be formed by operators on some Hilbert spaces, by classical dynamic systems, by non-commutative geometry, by group representations, or, by many other studies such as quantization. To classify a class of C*-algebras is to use a few computable data to completely determine C*-algebras in the class and their structure, in the process, one may also understand the related operators on Hilbert spaces, dynamical systems, non-commutative geometry, group representations, and, in turn, these may further provide applications to other parts of the scientific world.
