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

2025/02/10 by Aareyan Manzoor, Manzoor, Aareyan · 2 citations
Computer Science · Physics and Astronomy · #46L99 #Computability, Logic, AI Algorithms #FOS: Mathematics #Group Theory (math.GR) #Operator Algebras (math.OA) #Quantum Mechanics and Applications

paper · pdf · doi:10.48550/arxiv.2502.06697

openalex publication_date 2025/02/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Cited by

• Surjunctivity does not characterize cosoficity of invariant random subgroups
• Invariant Random Subgroups, Soficity, and Lück's determinant conjecture

Related

• The Connes Embedding Problem: A guided tour
• von Neumann's problem and extensions of non-amenable equivalence\n relations
• Invariant Random Subgroups, Soficity, and Lück's determinant conjecture
• A Survey on Connes' Embedding Conjecture
• Non-isomorphism of A^*n, 2≤ n ≤ ∞, for a non-separable abelian von Neumann algebra A