Factorial relative commutants and the generalized Jung property for II$_1$ factors
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
