Mathematical Sciences: Noncommutative Homotopy Theory and Proper Actions on C*-Algebras

点击翻译按钮获取中文摘要

Professor Phillips will continue his ongoing study of the homotopy theory of inverse limits of operator algebras. This includes a noncommutative version of the Bott periodicity theorem describing the iterated loop spaces of the infinite unitary group up to homotopy, and the search for noncommutative classifying algebras for K-theory and other cohomology theories. Another facet of his project involves proper actions of noncompact groups on noncommutative operator algebras, where it is hoped that the theory will develop along the lines of the topological case of groups acting on spaces. The mathematical research for this project falls under the heading of noncommutative algebraic topology. Topology is the study of shape, the overall conformation of objects (called topological spaces when one looks at them from this point of view). Topology becomes algebraic when one systematically computes invariants (numbers, say, or slightly more complicated mathematical objects) that are defined for classes of topological spaces. For instance, algebraic topology assigns invariants that count numbers of holes of various dimensions in topological spaces, useful when the space is itself too high-dimensional to permit a pictorial rendering. One popular topological procedure is to take a standard space, like a circle or a sphere, and look at how it maps into the space under examination, two such mappings being considered identical if one can be continuously deformed into the other; this is homotopy theory. Topology becomes noncommutative when you replace the space by an appropriate algebra of complex functions on it, and then forget about the space and allow the algebra to be more or less arbitrary. One result of this is to be able to count holes, so to speak, in objects entirely outside the traditional purview of topology, objects such as operator algebras that have heretofore been of interest mainly to functional analysts. The more general framework also yields some insights on the older subject matter by providing more elbow room for constructions. Professor Phillips' project is concerned with certain facets of this work of extending the machinery of topology into new domains.