Structure of C*-algebras arising from groups

Dynamical systems in its simplest form describe how a point in a geometrical space depends on time. At every time the system has a state and one considers the action of time on the state space as a transformation of the space. Therefore, it is natural to generalize the concept of dynamical systems to actions of groups on such spaces, where groups often represent symmetries. The study of C*-algebras has its origin as quantum mechanical models of physical observables, in the first place as algebras of operators on Hilbert spaces. In the modern language, C*-algebras are referred to as noncommutative spaces. Even though the theory of C*-algebras arose from quantum mechanics, the topic is also of great interest from a purely mathematical viewpoint. In particular, it has many striking and deep connections between different branches of mathematics as algebra, analysis, and geometry. An important part of the theory deals with group actions on C*-algebras, giving rise to C*-dynamical systems, thought of as noncommutative dynamical systems. For each such system one associates another C*-algebra, a so-called crossed product, and in this project we study how this process can be reversed, up to various types of equivalences. Finding out when C*-dynamical systems are simple or primitive, and thus representing the smallest building blocks of the theory, is an essential challenge. In the project, the problem is addressed when the C*-algebra is trivial, that is, all the information about the system is contained in the group, but where the product rule in the group is twisted by a scalar. This topic is closely related to the study of projective representations of groups. Invariants are also crucial for understanding the structure of C*-algebras, especially for classification, where K-theory plays a particularly important role. In this project, the K-theory of certain types of C*-dynamical systems arising from number theory is computed.

The theory of C*-dynamical systems has proved enormously important in the study of operator algebras. In addition, it has many striking connections to subjects such as representation theory, ergodic theory, K-theory, number theory, and quantum physics. The project involves several topics within the study of C*-algebras in connection with groups and dynamical systems, and can roughly be divided into three parts: One of the main objectives is to find conditions for uniqueness of trace, simplicity, primitivity, and primeness of twisted group C*-algebras. A challenge that will be pursued is to characterize the simple twisted group C*-algebras with unique trace corresponding to amenable discrete groups. Another goal is to continue the study of Cuntz-Li algebras by a crossed product construction. In the first place, the focus is completing the computation of K-theory of the so-called a-adic algebras, and then on classifying these algebras. Finally, there are duality theories for actions and coactions both for the nondegenerate categories and for the categories of C*-correspondences. The outer category is in some sense between those two, and the project aims to give a categorical duality theory also in this case. Each part of the proposal has the potential to generate deep and important insights into the theory and applications of C*-algebras, and to forge potentially fruitful connections among operator algebras, geometry, topology, and number theory. The majority of the work will be conducted at the University of Oslo and at Arizona State University, where in both places the collaborating research group is highly recognized in the international community. The results will be published in top mathematical journals, and be presented at national and international conferences and workshops.