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

TL;DR

This paper demonstrates that all separable tracial von Neumann algebras can embed into a II$_1$ factor with property (T), trivial outer automorphism, and trivial fundamental group, using wreath-like product groups.

Contribution

It introduces a universal embedding result for von Neumann algebras into property (T) factors with trivial automorphisms, expanding the understanding of their structure.

Findings

Every separable tracial von Neumann algebra embeds into a property (T) II$_1$ factor.

Constructs II$_1$ factors with trivial outer automorphism and fundamental groups.

Extends results to trivial extensions of countable p.m.p. equivalence relations.

Abstract

We prove that every separable tracial von Neumann algebra embeds into a II$_1$ 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 \cite{CIOS21}.