Continuous fields of C*-algebras were discovered as natural structures that underline C*-algebras with Hausdorff primitive spectrum. But more importantly, they have become effective tools of noncommutative geometry in a large array of contexts: E-theory, strict deformation quantization, the Novikov and the Baum-Connes conjectures, representation theory and index theory. We will study approximation techniques of continuous fields by continuous fields with controlled complexity. In many instances, the approximating fields can be analyzed by homological methods (sheaf theory, parametrized KK-theory), leading to far-reaching generalizations of the classical work of Dixmier and Douady on fields with fibres the compact operators. We will pursue this direction of research in collaboration with Jim McClure. In a different direction we propose an approach for proving that large classes of C*-algebras absorb tensorially the Jiang-Su algebra. The goal of this research in collaboration with Andrew Toms and Chris Phillips is to give classification results for C*-algebras associated with smooth minimal dynamical systems, based on results of Phillips and Q. Lin and very recent results of Winter and Huaxin Lin.
Quantization arises in the process of relating classic mechanics to quantum mechanics. In this process, the commutative algebra of classical observables is deformed into a noncommutative algebra of quantum observables. The theory of continuous fields provides one possible mathematical context for the study of these deformations. Much of the versatility of continuous fields comes from the fact that while they are bundles of operator algebras in the sense of general topology, they are not necessarily locally trivial and hence they allow for just the right amount of continuity necessary for deformations that capture and propagate interesting topological invariants. The proposed project aims to contribute to the extensive effort of a community of researchers to extend classical ideas of mathematics to noncommutative contexts.
1. Significance of research and description of results The award supported research in Operator Algebras, an area of mathematics that has emerged from the matriceal formulation of Quantum Mechanics and its subsequent refinements. The theory of Operator Algebras is a significant development in the quest of quantizing of mathematics following the successful quantization of physics. As in quantum physics, numerical functions are replaced in this theory by non-commuting infinite matrices (operators) acting on infinite dimensional Hilbert spaces. Often, in this process, the commutative algebra of classical observables is deformed into a non-commutative algebra of quantum observables. The theory of continuous fields studied in our project provides one possible mathematical context for the study of these deformations. We obtained a complete classification of the Cuntz-Pimsner algebras associated to vector bundles, by K-theory invariants. In joint work with R. Meyer, we developed an analogue of E-theory, a deformation theory for C*-algebras over non-separated spaces. This enabled us to give isomorphism results for nuclear continuous fields over infinite dimensional compact Hausdorff spaces that were previously unaccessible. In joint work with A. Toms we studied a theory of rank functions on simple C*-algebras and as a result we were able to verify substantial cases of a central regularity conjecture in the theory of nuclear simple C*-algebras. In joint work with C. Phillips, I. Hirshberg, A. Toms and W. Winter we revealed interesting new phenomena in the algebraic topology of simple operator algebras. We obtained effective criteria for embeddability of continuous fields into almost finite dimensional C*-algebras. The principal mathematical results of this project resulted in 9 publications in scientific mathematical journals. 2. Broader Impacts The Principal investigator advised 3 PhD students and co-mentored one post-doctoral fellow. One of the students completed his PhD thesis in April 2012. The other two are expected to graduate in 2013. Results obtained in our research were presented at several international meetings and in extended series of lectures at Universities in Europe. We have co-organized: the perennial Wabash seminars and mini-conferences, a BIRS (Banff) 5-day Workshop and have served as member of various scientific committees for well-established national and international scientific events in Operator Algebras.
