Factorial relative commutants and the generalized Jung property for II1 factors

Abstract

The findings reported in this paper aim to garner the interest of both model theorists and operator algebraists alike. Using a novel blend of model theoretic and operator algebraic methods, we show that the family of II1 factors elementarily equivalent to the hyperfinite II1 factor R all admit embeddings into RU with factorial relative commutant. This answers a long standing question of Popa for an uncountable family of II1 factors. We introduce the notion of a generalized Jung factor: a II1 factor M for which any two embeddings of M into its ultrapower MU are equivalent by an automorphism of MU. As an application of the result above, we show that R is the unique RU-embeddable generalized Jung factor. Using the concept of building von Neumann algebras by games and the recent refutation of the Connes embedding problem, we also show that there exists a generalized Jung factor which does not embed into RU. Moreover, we find that there are uncountably many non RU-embeddable generalized Jung type II1 von Neumann algebras. We study the space of embeddings modulo automorphic equivalence of a II1 factor N into an ultrapower II1 factor MU and equip it with a natural topometric structure, yielding cardinality results for this space in certain cases. These investigations are naturally connected to the super McDuff property for II1 factors: the property that the central sequence algebra is a II1 factor. We provide new examples, classification results, and assemble the present landscape of such factors. Finally, we prove a transfer theorem for inducing factorial commutants on embeddings with several applications.