On Popa’s factorial commutant embedding problem

Abstract

An open question of Sorin Popa asks whether or not every R U \mathcal {R}^{\mathcal {U}} -embeddable factor admits an embedding into R U \mathcal {R}^{\mathcal {U}} with factorial relative commutant. We show that there is a locally universal McDuff II 1 _1 factor M M such that every property (T) factor admits an embedding into M U M^{\mathcal {U}} with factorial relative commutant. We also discuss how our strategy could be used to settle Popa’s question for property (T) factors if a certain open question in the model theory of operator algebras has a positive solution.