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

Aareyan Manzoor

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arXiv · 2502.06697

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

Aareyan Manzoor

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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.

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Aareyan Manzoor. 2025-02-10. There Is An Equivalence Relation Whose von Neumann Algebra Is Not Connes Embeddable. https://arxiv.org/abs/2502.06697

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