Super McDuff and Uniformly Super McDuff $\rm II_1$ factor

Abstract

In this paper, we prove that every super McDuff $\rm II_1$ factor is uniformly super McDuff. In particular, the classes of super McDuff and uniformly super McDuff $\rm II_1$ factors coincide. This answers \cite[Question D]{GJKEP} affirmatively. For factors with separable predual, super McDuffness is therefore equivalent to the Brown property and is preserved under elementary equivalence, also answering \cite[Question~6.2.1]{AGK22} affirmatively. We further show that every infinite tensor product of non-$Γ$ $\mathrm{II}_1$ factors is uniformly super McDuff. For a finite tensor product of separable $\mathrm{II}_1$ factors that is super McDuff, each tensor factor is either non-$Γ$ or uniformly super McDuff, and at least one is uniformly super McDuff.