Regularity, complexity, and perturbation for C*-algebras

Purdue University, West Lafayette IN

Investigators

Abstract

This research concerns three broad projects, each with several subproblems of varying difficulty. The first project aims to prove the equivalence of three regularity properties for separable simple nuclear C*-algebras: one topological, one homological, and one algebraic. The proof of this fact, which has recently seen rapid progress toward a solution under some restrictions on traces, would represent a deep generalization of Kirchberg's characterization of purely infinite simple nuclear C*-algebras. The second project concerns the interplay between descriptive set theory and C*-algebras, and specifically the use of the notion of Borel reducibility to answer, for various classes of functional analytic objects, the question: How complicated is isomorphism? The objects that the principal investigator will consider include nuclear, exact, and locally reflexive C*-algebras, and operator spaces and systems. The third project examines uniform perturbations of C*-algebras and the degree to which they preserve structure and invariants. Important questions here include whether or not Z-stability and stability are preserved by such perturbations. Many fields of scientific inquiry require analyzing infinite-dimensional systems and the ways in which they can be transformed. Examples include models for quantum physics, signal analysis, and weather patterns. Infinite-dimensional systems are, of course, complicated. Understanding them often proceeds by approximating them with simpler finite-dimensional systems. This project uses this approach in an effort to understand infinite-dimensional systems called C*-algebras. The finite-dimensional approximating objects are square arrays with complex number entries. Our aim is to identify conditions under which one can approximate the infinite-dimensional system arbitrarily closely using only a fixed finite number of overlapping arrays. This last property is known to have powerful consequences for the original system, consequences that reveal a great deal of detail about its structure.