Operator Algebras, Groups, and Topological Invariants

Purdue University, West Lafayette IN

Investigators

Abstract

Three projects concerned with analytical and topological aspects of operator algebras are to be investigated. Operator algebras is an area of mathematics that emerged from the matrix mechanics formulation of quantum mechanics discovered by Heisenberg and further developed by von Neumann. The new (quantum) variables used in this theory are ensembles of (possibly infinite) matrices that satisfy suitable regularity properties. They are organized in various algebraic structures, called operator algebras, that can realized as linear operators acting on Hilbert spaces. The passage from functions to matrices often involves a process of quantization/deformation in which geometric structures are discretized in such a way that important topological spatial relations are distilled in the form of numerical invariants. The principal investigator will explore the theory of such deformations in the larger context of operator algebras. The goal is to exhibit and study new classes of algebraic and geometric structures which admit special deformations into finite matrices that capture subtle topological properties. The invariants that are to be investigated are intimately related to those that arise in the physics of novel materials such as topological insulators with crystalline symmetry. In more technical terms, the purpose of the first project is to explore the class of connective C*-algebras. Connectivity is a homotopy invariant property with significant permanence properties. It has important consequences pertaining to absence of projections, finite dimensional approximations such as quasidiagonality, and geometric realizations of K-homology. The second project is devoted to embeddability into AF-algebras. The principal investigator will investigate K-theory invariants associated to quasi-representations and topological obstructions to quasidiagonality. For the third project, the principal investigator will explore characteristic classes of continuous fields of strongly self-absorbing C*-algebras in the framework of the generalized Dixmier-Douady theory that were developed with Ulrich Pennig.