Abstract: A landmark theorem of Jung is that the hyperfinite II_1 factor $\mathcal R$ is the unique separable factor with the property that any two embeddings of it into its ultrapower $\mathcal R^\mathcal{U}$ are conjugate by a unitary. In 2020, Atkinson and Kunnawalkam Elayavalli observed that $\mathcal R$ is the unique separable $R^\{\mathcal{U}}$ embeddable II_1 factor $N$ with the property that any two embeddings of $N$ into $N^\mathcal{U}$ are conjugate by a unitary. In the first half of this talk, I will discuss a recent result (joint with Atkinson and Kunnawalkam Elayavalli) showing that $\mathcal R$ is the unique separable II_1 factor $N$ with the property that any two embeddings of $N$ into $N^\mathcal{U}$ are conjugate by an arbitrary (not necessarily inner) automorphism. The proof is a blend of operator algebraic and model theoretic techniques. Along the way, we show that any separable II_1 factor elementarily equivalent to $\mathcal R$ admits an embedding into $\mathcal R^{\mathcal U}$ with factorial commutant, thus providing continuum many examples of factors satisfying the conclusion of a longstanding open problem of Popa, which we refer to as the Factorial Commutant Embedding Problem (FCEP). In the second half of the talk, I will discuss a recent result showing that there is a separable II_1 factor M for which all property T factors admit an embedding into $M^\mathcal{U}$ with factorial commutant, thus providing a “poor man’s” resolution to the FCEP for property T factors. We will also identify two barriers from extending this result to a full resolution of the FCEP for property T factors. No knowledge of model theory will be assumed.
