Von Neumann Algebras: Classification, Rigidity and Applications

Von Neumann algebras are certain collections of infinite matrices, called Hilbert space operators, which are closed under matrix addition and multiplication. Their study was initiated almost 100 years ago in order to provide a mathematical foundation for quantum mechanics. Over the years, the theory of von Neumann algebras has broadened and developed in remarkable ways. Nowadays, it has deep connections to many fields of mathematics and science, including ergodic theory, group theory, logic, free probability, quantum field theory and quantum information theory. Von Neumann algebras arise naturally from fundamental mathematical structures such as groups, actions and graphs, which are studied throughout mathematics. This makes von Neumann algebras a powerful framework for studying rigidity phenomena. The aim of this proposal is to widen the scope of rigidity for von Neumann algebras and find new applications to ergodic theory, group theory, logic and C*-algebras by investigating several open problems across these areas. The project will also provide opportunities for the training and professional development of graduate students. This project has three main objectives. The first objective of the project is to investigate rigidity aspects of groups, in the context of von Neumann algebras and measured group theory. The second objective of the project is to explore the classification and rigidity of large (that is, non-separable) von Neumann algebras. The third objective of the project is to study trace spaces of groups and C*-algebras, with the goal of finding new classes of groups and C*-algebras whose extreme traces are dense in the space of all traces. 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.