C*-algebras Associated to Minimal and Hyperbolic Dynamical Systems

Mathematicians use invariants to study and classify highly abstract objects. The key properties of a useful invariant are that it is computable and that it distinguishes many different objects. For dynamical systems, an important class of mathematical objects, the construction of invariants is very difficult. This project studies invariants of dynamical systems using abstract operator algebras (in particular C*-algebras). The specific invariant is the K-theory of the relevant operator algebra, which in turn is an invariant of the relevant dynamical system. The project will involve significant contributions from early-career researchers, graduate students, and undergraduate students, who will benefit from training through research involvement. The principal investigator will study the range of the Elliott invariant, which consists of K-theory and tracial information, for C*-algebras constructed from minimal and hyperbolic dynamical systems. This research represents a continuation of the investigator’s previous work. However, in addition to the study of K-theory as an abstract graded group, the investigator will also study the order structure on K-theory and the traces of the relevant C*-algebras. Furthermore, this research goes beyond the study of uniformly hyperbolic dynamical systems (e.g., Smale spaces) to study general expansive systems. Specific projects include the study of minimal homeomorphisms on odd dimensional spheres, the order structure on K-theory for minimal crossed products, the finer structure of Smale space C*-algebras, and the HK-conjecture for C*-algebras associated to dynamical systems. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.