There Is An Equivalence Relation Whose von Neumann Algebra Is Not Connes Embeddable

TL;DR

This paper demonstrates the existence of a relation whose associated von Neumann algebra cannot be embedded into the Connes framework, using advanced quantum complexity and group theory techniques.

Contribution

It introduces a new notion of hyperlinearity for IRS and simplifies the reduction to non-local games, establishing non-embeddability of certain von Neumann algebras.

Findings

Existence of a non co-hyperlinear IRS on any non-abelian free group

Existence of a relation with a non-Connes embeddable von Neumann algebra

Simplified reduction of Aldous-Lyons to non-local games

Abstract

The landmark quantum complexity result MIP$^*$=RE was used to prove the existence of a non Connes embeddable tracial von Neumann algebra. Recently, similar ideas were used to give a negative solution to the Aldous-Lyons conjecture: there is a non co-sofic IRS on any non-abelian free group. We define a notion of hyperlinearity for an IRS and show that there is a non co-hyperlinear IRS on any non-abelian free group. As a corollary, we prove that there is a relation whose von Neumann algebra is not Connes embeddable. We do this by significantly simplifying the reduction of Aldous-Lyons to non-local games, removing the need for subgroup tests entirely.