Operator Algebras and K-theory

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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.