Embedding universality for II$_1$ factors with property (T)

Abstract

We prove that every separable tracial von Neumann algebra embeds into a II1 factor with property (T) which can be taken to have trivial outer automorphism and fundamental groups. We also establish an analogous result for the trivial extension over a non-atomic probability space of every countable p.m.p. equivalence relation. These results are obtained by using the class of wreath-like product groups introduced recently in [CIOS21].

Uniformly Super McDuff II$_1$ Factors

“…In this section, we use Proposition 4.4 to show that infinitely generic factors are uniformly super McDuff. This result improves the result of Chifan, Drimbe, and Ioana from [CDI22] showing that infinitely generic factors are super McDuff. A main ingredient in the proof of the latter result is the embedding universality of property (T) factors, also proved in [CDI22].…”

“…This result improves the result of Chifan, Drimbe, and Ioana from [CDI22] showing that infinitely generic factors are super McDuff. A main ingredient in the proof of the latter result is the embedding universality of property (T) factors, also proved in [CDI22]. Our proof in this subsection also utilizes this fact (and its proof).…”

“…The first family is a certain subclass of the family of existentially closed factors (see [Far+16]) arising in the model theory of II 1 factors. Our result for this case improves upon the work of [CDI22], where it was shown that infinitely generic II ). However, for our purposes, we will only need to know the following two properties of a finitely generic factor N (besides their existence):…”

“…To prove this, we will use the following Theorem, which is extracted from the discussion at the end of the introduction to [CDI22].…”