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This 42-minute lecture presents Jesse Peterson of Vanderbilt University discussing "A hierarchy of Haagerup-type approximation properties" at IPAM's Free Entropy Theory and Random Matrices Workshop, recorded on February 27, 2025. Explore the concept of the Haagerup property for groups and von Neumann algebras, a significant approximation property that extends deformability phenomena beyond amenable cases into free group factors. Learn about successive weakenings of the Haagerup property indexed by ordinal numbers, and discover how for each countable ordinal α, the α-Haagerup property serves as an invariant of the group von Neumann algebra and transfers to von Neumann subalgebras. Examine the construction of countable groups possessing the α-Haagerup property but lacking the β-Haagerup property for any β < α, providing a new proof of Ozawa's theorem on the non-existence of a universal separable II1 factor. This presentation covers joint work with Fabian Salinas, hosted by the Institute for Pure & Applied Mathematics (IPAM) at UCLA.
