Rigidity for von Neumann Algebras and Applications

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Von Neumann algebras are collections of certain infinite matrices of complex numbers, called Hilbert space operators. They were introduced in the 1930s in order to provide a mathematical foundation for quantum mechanics. Since then, the theory of von Neumann algebras has flourished into an independent area with fruitful connections to several areas of mathematics and science, including quantum physics. Von Neumann algebras arise naturally from a variety of mathematical structures, such as groups of symmetries and actions, which are present in many areas of mathematics. Their study is closely connected to topics in group theory and ergodic theory. This project will deepen the connections between these areas and operator algebras, by investigating a number of open problems at their interface. The project will also provide opportunities for the training and professional development of graduate students.The aim of this project is to expand the scope of rigidity in von Neumann algebras and uncover new applications to group theory, ergodic theory and C*-algebras. The first objective of the project is to investigate rigidity for group von Neumann algebras. Building on recent work on Connes' famous 1980 rigidity conjecture, the PI proposes to find new families of property (T) groups which are rigid in the context of von Neumann algebras and measure equivalence. The second objective of the project is to investigate two related rigidity properties for groups: Hilbert-Schmidt stability and the local lifting property for full group C*-algebras. The PI proposes to investigate two longstanding problems concerning almost commuting matrices and the local lifting property. The third objective of the project is to study asymptotic problems of von Neumann algebras. The proposed research is expected to lead to new interactions between operator algebras, (both geometric and measured) group theory and ergodic theory.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.