C*-algebras of Groups and Quantum Groups: Rigidity and Structure Theory

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The theory of operator algebras was developed in the 1930s as the mathematical foundation of quantum mechanics. This theory is also a natural mathematical framework in which physical systems as they evolve in time (called dynamical systems) can be represented and studied; groups are the algebraic structures that represent the time in this picture. Groups also model symmetries of physical systems and mathematical structures, and since their formal introduction in the 19th century, their applications have expanded deep into almost every major area of mathematics. Representation theory of groups provides canonical ways to assign operator algebras to a given group, hence establishes a bridge between the two theories which allows one to import ideas and concepts from one theory to the other. The study of the connections between groups and their associated operator algebras has been a central part of operator algebra theory since the time of von Neumann, and it has had tremendous impact on both theories, as well as in several other related areas such as ergodic theory, dynamical systems, and representation theory. This project aims to make further progress in this direction by discovering new connections between structural properties of groups and their operator algebras. The project provides research training opportunities for graduate students. The first part of the project is aimed at several conjectures and problems related to new operator-algebraic rigidity phenomena for discrete groups in terms of their C*-algebras. This is based on a new approach to extending important classical results from ergodic theory of higher rank groups to the operator-algebraic setting. The main concept of interest is that of invariant subalgebras of C*-algebras generated by unitary representations, which are viewed as non-commutative generalizations of normal subgroups. The boundary theory of groups (in the sense of Furstenberg) plays a central role in the Principal Investigator’s strategies to approach the proposed problems. Furstenberg's boundary theory has been, in the past few decades, a major tool in ergodic theory of higher rank lattices and in proving their rigidity properties. More recently, it has also been used in operator-algebraic rigidity problems, resulting in significant progress in this area. The project aims to develop new techniques in utilizing boundary actions in order to prove the proposed rigidity results for invariant subalgebras of C*-algebras generated by unitary representations of discrete groups. Another part of this project concerns several open problems regarding simplicity and unique trace property of C*-algebras of discrete quantum groups; the goal is to further develop the boundary theory of quantum groups and apply them in these problems.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.