C*-algebra theory, Classification and its applications

University Of Oregon Eugene, Eugene OR

Investigators

Abstract

This project is a study of the classification of C*-algebras and the development of its applications. The study of C*-algebras is a study of some special algebraic systems. These algebraic systems are systems of square matrices, possibly of infinite size. C*-algebras have a strong presence in many diverse fields of science and mathematics such as dynamical systems, noncommutative geometry, and quantum mechanics, to name just a few. C*-algebras are very complicated mathematical objects that usually exist in infinitely many dimensions. What "classification" does is to allow one to learn the identity and structure of a C*-algebra from relatively small amounts of fairly simple numerical data, analogous to the way in which fingerprints are used to identify an individual. In applications, C*-algebras arise in many areas and often appear in disguise, so it is of great advantage to use limited numerical data to determine their structure. The present project aims to classify the largest class of C*-algebras considered thus far, including algebras that are particularly important in many applications. The main goal of this project is to obtain a broad classification result for amenable C*-algebras. Part of the strategy is to create new technical tools to facilitate the interaction between the C*-algebra and its invariants. One of these tools is an enhancement of the Basic Homotopy Lemma and another is a K-theoretic characterization of asymptotic unitary equivalence. Another important objective is to verify that certain commonly appearing C*-algebras actually do belong to our classifiable class. This research will therefore greatly simplify the way in which the theory of C*-algebras is applied. It is worth noting one application, namely, to noncommutative homotopy theory and the study of dynamical systems. The principal investigator will use this broad classification theory to study a new relation among minimal homeomorphisms on a compact metric space, a relation known as asymptotic conjugacy, and to characterize it by means of K-theoretical data.