Operator Algebras and Noncommutative Topology

Board Of Regents, Nshe, Obo University Of Nevada, Reno, Reno NV

Investigators

Abstract

ABSTRACT Blackadar Blackadar will study a range of questions concerning the structure of various classes of operator algebras and their relation to noncommutative topology. Topics will include generalized inductive limits of finite-dimensional C*-algebras and the structure of nuclear C*-algebras, semiprojectivity, nonstable K-theory and the Universal Coefficient Theorem, and bivariant cohomology theories on C*-algebras and their dense subalgebras. An idea which has revolutionized the study of operator algebras in recent years, and which has led to some spectacular applications throughout mathematics and mathematical physics, is to view operator algebras as generalizations of topological spaces, or more precisely the set of continuous functions on a topological space; the principal new feature is that the multiplication of "functions" is no longer assumed to be commutative. It has long been recognized that operator algebras provide the "right" framework for the mathematical formulation of quantum mechanics, and it has been increasingly recognized recently that noncommutative "function spaces" (operator algebras) arise naturally in problems in subjects as diverse as knot theory (with applications to the structure of DNA and other molecules, as well as mathematical physics), dynamical systems, and even mathematical logic. It has become apparent that it will be possible to give an explicit description of all operator algebras in large classes, including most algebras arising in applications, far beyond what was thought possible only a few years ago. This project concerns some of the central remaining problems in the classification of one particularly important class, the nuclear C*-algebras. Understanding the mathematical structures that can occur will give great insight into the nature and behavior of the associated applied problems.