Uniformly super McDuff II1 factors

We introduce and study the family of uniformly super McDuff II1 factors. This family is shown to be closed under elementary equivalence and also coincides with the family of II1 factors with the Brown property introduced in Atkinson et al. (Adv. Math. 396, 108107, 2022). We show that a certain family of existentially closed factors, the so-called infinitely generic factors, are uniformly super McDuff, thereby improving a recent result of Chifan et al. (Embedding Universality for II1 Factors with Property (T). arXiv preprint, 2022). We also show that Popa's family of strongly McDuff II1 factors are uniformly super McDuff. Lastly, we investigate when finitely generic II1 factors are uniformly super McDuff.