Regularity, complexity, and perturbation for C*-algebras

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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.