The Structure of Simple Amenable C*-Algebras and their Homomorphisms.

University Of Oregon Eugene, Eugene OR

Investigators

Abstract

In this project the principal investigator will study the structure of unital, simple, amenable C*-algebras and homomorphisms from one such C*-algebra to another. From a theorem of Gelfand we know that a unital commutative C*-algebra is isomorphic to the algebra of continuous functions on some compact Hausdorrf space. The structure of a commutative C*-algebra is thus completely determined by the underlying space, or the topological structure of the space. A homomorphism from one commutative C*-algebra to another is induced and determined by a continuous map from one underlying space to another. Thus, in the noncommutative setting, this project comes down to a study of noncommutative topology. The central goals of the project are (1) to use K-theory-related data to classify separable, simple, amenable C*-algebras, (2) to determine approximate unitary equivalence classes of homomorphisms, and (3) to find applications to the study of noncommutative topology and topological dynamical systems. The simplest C*-algebra is the system of all complex numbers. The next most simple C*-algebras are systems of matrices of complex numbers. In general, C*-algebras are systems of operators (which can be thought of as generalizations of matrices). For example, differentiation and integration are operators on certain function spaces. Operators can also be used, for example, as models for observables for the microscopic physical world. A system of operators has the structure of addition and multiplication, just like the system of numbers. Unlike the system of numbers, where two times three is the same as three times two, in a general C*-algebra the product A times B may not be the same as B times A. This noncommutativity reflects the reality of quantum physics and corresponds to the famous Heisenberg uncertainty principle. C*-algebras arise in many diverse areas of science and engineering, of which quantum mechanics is just one important example. For purposes of application, as well as for theoretical reasons, it is important to understand the structure of C*-algebras, or the structure of systems of operators formed from different applications. The aim of this project is to find the simplest essential data that determine both the structure of a C*-algebra and the relations that exist between C*-algebras so that applications become possible. To be useful, the data should be easy to obtain and relatively easy to compute. Furthermore, if two C*-algebras give rise to the same set of data, then the algebras should be identical for purposes of all applications. The prinicipal investigator has to search these data and invent tools to provide a proof that such data can indeed be used to determine completely the structure of the corresponding C*-algebras. The expected immediate applications will be to the study of dynamical systems. However, a long-term impact should be felt in many other areas of mathematics (e.g., linear algebra, operator theory, group representations, noncommutative topology, noncommutative geometry). In the last few years, some related research involved the training of several Ph.D. students. This project will also include both graduate student training and the mentoring of postdoctoral researchers.