There Is An Equivalence Relation Whose von Neumann Algebra Is Not Connes Embeddable
arXiv:2502.06697
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.
22 pages, v2: to appear in Compositio Mathematica, major change in introduction to be more readable for people outside of operator algebras, minor change in body and other exposition