Structure, rigidity and classification of von Neumann algebras via noncommutative boundary techniques

Von Neumann algebras are infinite-dimensional algebras of operators that were introduced in the 1930s as part of the mathematical foundation of quantum mechanics. They now form a central area of modern analysis, with deep connections to group theory, ergodic theory, probability, mathematical physics, noncommutative geometry, and quantum information theory. A central challenge is to classify these algebras and determine which features of the symmetries, dynamical systems, or probabilistic models that produce them are retained by the algebra itself. Boundary methods have played an important role in group theory, and this project develops their von Neumann algebraic counterparts for classification problems. By strengthening connections between operator algebras and neighboring fields, the project advances NSF's mission to promote the progress of science. The investigator's seminars and workshops will also support education and workforce development by creating research opportunities for graduate students, postdoctoral researchers, and early-career mathematicians, and the resulting tutorial materials will be made publicly available. Building on the investigator's recent work on noncommutative analogues of group boundaries, the project studies structure, rigidity, and classification problems for von Neumann algebras in three related directions. First, the investigator applies and develops these methods for q-Araki-Woods factors, a class of von Neumann algebras arising from deformed probability and mathematical physics. Second, the investigator studies von Neumann equivalence, a noncommutative analogue of measure equivalence, with the goal of finding new invariants and rigidity principles connecting group theory, measured dynamics, and operator algebras. Third, the investigator develops foundations for noncommutative Furstenberg boundaries for von Neumann algebras, motivated by applications to Cartan subalgebras and classification problems. These directions expand the role of boundary methods in von Neumann algebras and clarify their use in structure and classification 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. NSF Award ID: 2554483 | Program: 01002627DB NSF RESEARCH & RELATED ACTIVIT | Principal Investigator: Changying Ding | Institution: University of California-Los Angeles, LOS ANGELES, CA | Award Amount: $117,567 View on NSF Award Search: https://www.nsf.gov/awardsearch/show-award/?AWD_ID=2554483 View on Research.gov: https://www.research.gov/awardapi-service/v1/awards/2554483.html