The Structure of Crossed Product C*-algebras

University Of Oregon Eugene, Eugene OR

Investigators

Abstract

Abstract Phillips The Principal Investigator has recently proved that crossed products of compact manifolds by minimal diffeomorphisms can be represented as direct limits of systems of recursive subhomogeneous C*-algebras with no dimension growth. He proposes to attempt to generalize this result, by relaxing the differentiability assumptions (many interesting minimal homeomorphisms are not smooth, or not even on manifolds), and by considering more general groups (such as the real numbers or the direct sum of several copies of the integers). He also proposes to apply the direct limit decomposition, by attempting to prove a classification theorem for such direct limits, and by using the results it implies about K-theory to investigate the connections between dynamics and C*-algebras for interesting minimal homeomorphisms. The Principal Investigator further proposes to use methods of free probability to study isomorphism questions for C*-algebras related to the reduced C*-algebras of free groups. This project concerns the classification of simple C*-algebras. C*-algebras are a fascinating and beautiful part of mathematics in their own right, and moreover they have surprising applications to other parts of mathematics (such as geometry and topology) and even to parts of physics (such as quantum mechanics and statistical mechanics). For these reasons, and others, one wants to identify and describe all C*-algebras. Without some limitations, this task is presently hopeless, and my project focuses on C*-algebras which are "simple" (that is, the usual way of taking them apart into smaller pieces doesn't work), and also "not too large" in several other technical senses. Under these limitations, some classification theorems have been proved. That is, one can identify, label, and describe the objects in certain classes of C*-algebras, sort of like the periodic table of the elements or like the naming of the species of living things. In one particular case of interest (the "stably finite" case), the classification results have been proved for simple C*-algebras arising from one particular construction ("direct limits"), but the most interesting source of examples is a different construction ("crossed products"). In a sense, the wrong algebras have been classified. The aim of this project is to show that some algebras of the more interesting kind are actually the same as some of the ones already classified, or are at least close enough that the old classification methods from direct limits might still be applied; and to extend those methods far enough to actually classify the algebras of the more interesting type.