Approximation Properties of C*-algebras

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This project will address approximation properties of C*-algebras and their connection to noncommutative dynamics, group theory, and operator spaces. Perron-Frobenius operators are a natural and useful way to study the time-evolution of densities. The principal investigator will study the dynamics of classical Perron-Frobenius operators as well as noncommutative Perron-Frobenius operators, with a special focus on topological entropy. In another direction, he will investigate how the completely bounded approximation property behaves with respect to reduced free products, focusing on the analogies with free products and weak amenability for groups. Finally, the project will study how the local linear structure (from an operator space point-of-view) of a C*-algebra controls the algebraic, topological, and representation theoretic properties of a C*-algebra.Approximation theory is an extremely useful concept in mathematics. The basic idea is as follows: take an infinite (and therefore difficult for the human mind to comprehend) mathematical object that one seeks to understand; then approximate this object by similar, yet finite (and therefore theoretically easier for the human mind to comprehend) mathematical objects; finally, fit all of these approximations together to compose a clear understanding of the infinite mathematical object. The mathematical objects in this proposal bear the technical name C*-algebras. They were first studied in the context of quantum mechanics but have since realized numerous connections to nearly all areas of mathematics, for example, to dynamical systems and to group theory. This project aims to treat three seemingly separate areas of mathematics (dynamical systems, group theory, and operator spaces) with the same tool of approximation theory, all in the context of C*-algebras.