Non-unimodular L2-Invariants

This project concerns the theory of von Neumann algebras, a branch of mathematics that was initiated by Francis J. Murray and John von Neumann in the 1930s in an effort to provide a rigorous framework for quantum physics as well as to study objects from other areas of mathematics such as groups and dynamical systems. Since the inception of theory, researchers have discovered many new applications to mathematics as well as to biology, physics, and engineering. Von Neumann algebras also provide a framework for so-called quantum groups, the central focus of this project, which are profound generalizations of the groups of geometric symmetries exhibited by everyday objects (think of rotating a ball versus a cube). Quantum groups arise in the study of quantum systems and have been essential to advances in quantum communications and quantum computing. This project seeks to use von Neumann algebras to better understand quantum groups through new numerical invariants. This project will also provide numerous training opportunities for early career researchers through summer funding and travel support. The research goal of this project is to extend the theory of L2-Betti numbers to a variety of non-tracial settings including: non-unimodular groups, non-Kac type quantum groups, non-measure preserving equivalence relations, and almost periodic von Neumann algebras. L2-Betti numbers have already proven to be a powerful classification tool in the tracial case of each of these settings. In particular, the first L2-Betti number gives a measure of the number of unbounded 1-cocycles or derivations, and its behavior has been a crucial hypothesis in a number of structural and non-isomorphism results in von Neumann algebras. Most aspects of these results admit natural generalizations to the non-tracial case, with the exception of a notion of L2-Betti numbers. Thus, by providing a canonical definition of L2-Betti numbers in this case, this proposal seeks to expand the scope of these tracial results and obtain new structural and classification results. 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.