On the generalized Jung property for II\(_1\) factors

Abstract

We introduce the notion of a generalized Jung factor: a II\(_1\) factor \(M\) for which any two embeddings of \(M\) into its ultrapower \(M^{\mathcal U}\) are equivalent by an automorphism of \(M^{\mathcal U}\). We show that \(\mathcal R\) is not the unique generalized Jung factor but is the unique \(\mathcal R^{\mathcal U}\)-embeddable generalized Jung factor. We use model-theoretic techniques to obtain these results. Integral to the techniques used is the result that if \(M\) is elementarily equivalent to \(\mathcal R\), then any elementary embedding of \(M\) into \(\mathcal R^{\mathcal U}\) has factorial relative commutant. This answers a long-standing question of Popa for an uncountable family of II\(_1\) factors. We also provide new examples and results about the notion of super McDuffness, which is a strengthening of the McDuff property for II\(_1\) factors.