Probabilistic Operator Algebra Seminar: Non amenable von Neumann algebras embed wildly into ultraproducts

Seminar | November 18 | 3-5 p.m. | 736 Evans Hall

 Kunnawalkam Elayavalli Srivatsav, Vanderbilt University

 Department of Mathematics

K. Jung showed in 2005 that non-amenable tracial von Neumann algebras have at least 2 non unitarily-conjugate embeddings into $R^w$ (the ultraproduct of the hyperfinite $II_1$ factor). N. Brown later studied this space of embeddings (of a non-amenable domain) up to unitary conjugation in $R^w$, with a natural metric topology, and showed that it is uncountable (Ozawa improved this in an appendix to show that this space is not even second countable). In this talk, we will generalize both Jung's and Ozawa's result (thereby answering a question of Popa), where the target space is an arbitrary ultraproduct of $II_1$ factors. Our generalization of Jung's result concerning a more general notion of ucp-conjugation, and uses a technical characterization of semidiscreteness due to Kishimoto. The core of our solution to Popa's question relies on a theorem of Haagerup regarding delta relatedness of finite families of unitaries. This is based on joint work with S. Atkinson. See arXiv:1907.03359 for submitted paper.