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

TL;DR

This work addresses the Connes embedding problem by establishing the existence of a non-Connes embeddable von Neumann algebra arising from an invariant random subgroup (IRS) of a non-abelian free group. It introduces co-hyperlinear IRS via amenable traces and links Connes embeddability of $L(\Gamma/H)$ to amenability of the corresponding IRS trace, bridging IRS theory with finite-dimensional approximations. By translating traces into non-local game strategies and developing a computable NPA-type hierarchy, the authors show there exists a non-local game with a gap $\omega_{IRS}(\mathfrak{G})>\omega^*(\mathfrak{G})$, yielding a non-co-hyperlinear IRS and hence a non-Connes embeddable $L(\Gamma/H)$. As a corollary, they deduce the existence of a relation whose von Neumann algebra is not Connes embeddable, highlighting a concrete obstruction to finite-dimensional approximation in this IRS-quotient framework.

Abstract

MIP$^*$=RE [Ji+22] was used to prove the existence of a non Connes embeddable tracial von Neumann algebra. Recently, similar ideas were used in [Bow+24, BCV24] 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, which reproves the main results of [Bow+24, BCV24]. As a corollary, we prove that there is a relation whose von Neumann algebra is not Connes embeddable.

Theorems & Definitions (43)

• Theorem 1.1
• Theorem 1.2
• Definition 2.1
• Example 2.2
• Example 2.3
• Theorem 2.4
• Definition 2.5
• Proposition 2.6
• Theorem 2.7
• Definition 2.8