Rigidity results for von Neumann algebras and equivalence relations

Von Neumann algebras were introduced to provide a mathematical foundation for the study of quantum mechanics and can be viewed as infinite-dimensional generalizations of matrix algebras. The groundbreaking work of Francis J. Murray and John von Neumann in the 1930s already demonstrated that von Neumann algebras are complex objects with exceptionally rich mathematical structures. Since then, the theory of von Neumann algebras has developed into an independent field, establishing fruitful connections with various branches of mathematics and science. Natural classes of von Neumann algebras arise from a variety of mathematical structures, such as groups of symmetries and their actions. This highlights the close relationship between von Neumann algebras and the mathematical areas of group theory and ergodic theory. The main goal of this project is to advance the connections among these research areas by addressing a range of open problems at their intersection. The project will also incorporate opportunities for the involvement of graduate students. In this research project, the principal investigator (PI) aims to obtain new classification and structural results for von Neumann algebras arising from groups and their actions on probability spaces. As a consequence, he will derive several new structural results for equivalence relations associated with group actions. In general, von Neumann algebras do not retain much information about the groups or actions from which they are constructed. However, Popa’s deformation/rigidity theory has revealed that many structural properties of these groups and actions can, in fact, be detected through their associated von Neumann algebras. Building on this foundation, the PI will develop new techniques to investigate this rigidity phenomenon within von Neumann algebras. The project will focus on several key research directions. First, the PI will expand the understanding of rigidity by identifying new classes of groups and actions that can be completely reconstructed from their von Neumann algebras. Second, he will investigate primeness and unique prime factorization for von Neumann algebras arising from actions of higher-rank lattices. Finally, the PI will pursue new rigidity results within the framework of measure equivalence of von Neumann algebras. 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.