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.

Agency
National Science Foundation (NSF)
Institute
Division of Mathematical Sciences (DMS)
Type
Standard Grant (Standard)
Application #
0754813
Program Officer
Bruce P. Palka
Project Start
Project End
Budget Start
2008-06-01
Budget End
2011-05-31
Support Year
Fiscal Year
2007
Total Cost
$143,976
Indirect Cost
Name
University of Oregon Eugene
Department
Type
DUNS #
City
Eugene
State
OR
Country
United States
Zip Code
97403