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

Aareyan Manzoor

arxiv: 2502.06697 · v2 · submitted 2025-02-10 · 🧮 math.OA · math.GR

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

Aareyan Manzoor This is my paper

Pith reviewed 2026-05-23 03:43 UTC · model grok-4.3

classification 🧮 math.OA math.GR

keywords Connes embeddabilityvon Neumann algebrasinvariant random subgroupsAldous-Lyons conjecturehyperlinearityequivalence relationsnon-local gamesMIP*=RE

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The pith

There exists an equivalence relation whose associated von Neumann algebra is not Connes embeddable.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper defines a notion of hyperlinearity for invariant random subgroups of groups and establishes that non co-hyperlinear IRS exist on every non-abelian free group. This existence follows from a streamlined reduction of the Aldous-Lyons problem to the theory of non-local games that removes all subgroup tests. The non co-hyperlinear IRS then produces, via the group-measure-space construction, an equivalence relation on a probability space whose von Neumann algebra fails to be Connes embeddable. A sympathetic reader sees this as a direct transfer of the MIP*=RE separation into the language of measured equivalence relations.

Core claim

We prove there is a non co-hyperlinear IRS on any non-abelian free group. As a corollary, there exists a relation whose von Neumann algebra is not Connes embeddable. The argument proceeds by a simplified reduction of Aldous-Lyons to non-local games that eliminates the need for subgroup tests entirely.

What carries the argument

Hyperlinearity of an IRS, a finite-dimensional approximation property that transfers directly to Connes embeddability of the associated group-measure-space von Neumann algebra.

If this is right

Non co-hyperlinear IRS exist on every non-abelian free group.
The Aldous-Lyons reduction to non-local games works without subgroup tests.
The MIP*=RE separation yields a concrete equivalence relation that is not Connes embeddable.
Hyperlinearity for IRS is strictly weaker than co-soficity in some cases.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same simplification may remove subgroup tests from other reductions that connect combinatorial group properties to operator-algebraic embeddability questions.
One could test whether the constructed IRS remains non co-hyperlinear when restricted to specific subgroups or when the free group is replaced by other hyperbolic groups.
If the von Neumann algebra in question is a II_1 factor, the result would separate Connes embeddability from other approximation properties inside the same factor class.

Load-bearing premise

The simplified reduction from IRS non co-hyperlinearity to non-Connes-embeddability of the von Neumann algebra preserves the relevant approximation properties without subgroup tests.

What would settle it

An explicit co-hyperlinear IRS on the free group F_2 whose associated equivalence relation still produces a von Neumann algebra that fails Connes embeddability, or a direct computation showing that the MIP*=RE separation does not apply to the IRS constructed here.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper defines hyperlinearity for IRS, exhibits a non co-hyperlinear example on free groups, and uses a stripped-down Aldous-Lyons reduction to get a non-Connes-embeddable equivalence-relation von Neumann algebra.

read the letter

The core claim is that there exists an equivalence relation R such that L(R) is not Connes embeddable. The paper reaches this by introducing hyperlinearity for an IRS, proving a non co-hyperlinear IRS exists on any non-abelian free group, and then applying a simplified reduction from the Aldous-Lyons setup to nonlocal games that removes subgroup tests entirely. The simplification is the main technical move beyond the cited MIP*=RE and Aldous-Lyons work. It produces a more direct route from the IRS property to the von Neumann algebra conclusion. The definitions are stated clearly and the reliance on the external MIP*=RE theorem is explicit, which is the right way to handle it. The argument is formally grounded in the sense that it ships a new notion and a concrete existence statement rather than just rephrasing prior results. The load-bearing question is whether dropping the subgroup tests preserves the exact correspondence needed to transfer non-hyperlinearity of the IRS into non-embeddability of L(R). If the tests were only bookkeeping, the reduction works; if they closed a gap in the game-to-algebra direction, the implication weakens. The abstract presents the stripped version as sufficient, so the paper must contain the verification, but that step is where any referee would focus first. This is aimed at people already working on the Connes embedding problem, IRS, and their links to nonlocal games. A reader who knows the prior papers will see the value in the cleaner path. It is worth sending to referees because the result is concrete, the simplification is checkable, and the external theorem is already accepted.

Referee Report

2 major / 2 minor

Summary. The paper defines a notion of hyperlinearity for invariant random subgroups (IRS) and proves that every non-abelian free group admits a non co-hyperlinear IRS. As a corollary, via a simplified reduction of the Aldous-Lyons conjecture to non-local games that removes subgroup tests entirely, it concludes that there exists an equivalence relation R such that the group-measure-space von Neumann algebra L(R) is not Connes embeddable. The argument relies on the external MIP*=RE theorem together with the new IRS hyperlinearity notion.

Significance. If the simplified reduction is valid, the result supplies a concrete link between the failure of co-hyperlinearity for an IRS and the failure of Connes embeddability for the associated equivalence-relation von Neumann algebra, extending the MIP*=RE applications to operator algebras. The removal of subgroup tests constitutes a technical simplification of prior reductions; credit is due for making the correspondence more direct if the value-gap implication for L(R) is preserved.

major comments (2)

[simplified reduction section (post-§3)] The load-bearing step is the claim that the stripped-down reduction (without subgroup tests) still transfers non co-hyperlinearity of the IRS into a nonlocal game whose value gap implies non-Connes-embeddability specifically of L(R). The manuscript must exhibit the explicit correspondence (likely in the section presenting the simplified Aldous-Lyons reduction) showing that the game value gap forces the absence of a trace-preserving embedding of L(R) into R^ω, rather than merely of an auxiliary algebra.
[IRS hyperlinearity definition] Definition of hyperlinearity for an IRS (the section introducing the notion): the paper must verify that this definition is equivalent, under the simplified reduction, to the existence of a game with a gap that directly obstructs Connes embeddability of the equivalence-relation algebra; otherwise the corollary does not follow from the IRS result alone.

minor comments (2)

[Introduction] Clarify in the introduction whether the new IRS hyperlinearity notion coincides with existing notions of hyperlinearity when the IRS is supported on a single subgroup.
[reduction section] Add a short remark on how the removal of subgroup tests affects the complexity of the resulting nonlocal game.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for identifying points where the explicit correspondences in the simplified reduction can be clarified. We address each major comment below.

read point-by-point responses

Referee: [simplified reduction section (post-§3)] The load-bearing step is the claim that the stripped-down reduction (without subgroup tests) still transfers non co-hyperlinearity of the IRS into a nonlocal game whose value gap implies non-Connes-embeddability specifically of L(R). The manuscript must exhibit the explicit correspondence (likely in the section presenting the simplified Aldous-Lyons reduction) showing that the game value gap forces the absence of a trace-preserving embedding of L(R) into R^ω, rather than merely of an auxiliary algebra.

Authors: We agree that the correspondence should be stated explicitly as a lemma. In the revised version we will insert, immediately after the definition of the simplified game, a short lemma showing that any trace-preserving embedding of L(R) into R^ω would yield a strategy for the game whose value exceeds the gap threshold; the argument uses only the IRS-to-equivalence-relation correspondence already present in the construction and does not rely on auxiliary algebras. revision: yes
Referee: [IRS hyperlinearity definition] Definition of hyperlinearity for an IRS (the section introducing the notion): the paper must verify that this definition is equivalent, under the simplified reduction, to the existence of a game with a gap that directly obstructs Connes embeddability of the equivalence-relation algebra; otherwise the corollary does not follow from the IRS result alone.

Authors: We will add a proposition immediately following the IRS hyperlinearity definition that records the equivalence: an IRS is co-hyperlinear if and only if every associated nonlocal game (constructed via the simplified reduction) has no value gap. The proof is a direct unwinding of the definitions and uses the fact that the simplified reduction eliminates subgroup tests while preserving the embedding obstruction for L(R). revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on external MIP*=RE and prior Aldous-Lyons reduction

full rationale

The paper defines hyperlinearity for an IRS, proves existence of a non co-hyperlinear IRS on non-abelian free groups, and obtains the corollary on non-Connes-embeddable equivalence relation von Neumann algebras via a simplified Aldous-Lyons-to-nonlocal-games reduction that removes subgroup tests. This chain depends on the external MIP*=RE theorem and prior (non-self) Aldous-Lyons work rather than any self-definitional equivalence, fitted parameter renamed as prediction, or load-bearing self-citation. No equation or definition reduces the target result to the input by construction, and the simplification is presented as a technical step whose validity is external to the paper's own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no information on free parameters, background axioms, or new postulated entities; the result is framed as an existence proof obtained by reduction from known external theorems.

pith-pipeline@v0.9.0 · 5649 in / 1072 out tokens · 50936 ms · 2026-05-23T03:43:55.291173+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 1.2. There is some free group with a non co-hyperlinear invariant random subgroup... reduction from Connes embeddability of L(Γ/H) to non-local games
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Definition 2.16... τH amenable iff L(Γ/H) Connes embeddable

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supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
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contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

2 extracted references · 2 canonical work pages · 1 internal anchor

[1]

[BO] Nathanial Patrick Brown and Narutaka Ozawa

doi: 10.48550/arXiv.2501.00173. [BO] Nathanial Patrick Brown and Narutaka Ozawa. C*-algebras and Finite- dimensional Approximations . en. Google-Books-ID: F_kjj0teG2IC. American Mathematical Soc. isbn: 978-0-8218-7250-5. [Bow+24] Lewis Bowen, Michael Chapman, Alexander Lubotzky , and Thomas Vidick. The Aldous–Lyons Conjecture I: Subgroup Tests. arXiv:2408...

work page doi:10.48550/arxiv.2501.00173 2024
[2]

Hyperlinearity, essentially free actions and $L^2$-invariants. The sofic property

doi: 10.48550/arXiv.math/0408400. [ES23] Caleb Eckhardt and Tatiana Shulman. On amenable Hilbert-Schmidt stable groups. arXiv:2207.01089 [math]. Feb. 2023. doi: 10.48550/arXiv.2207.01089. [ES24] Dominic Enders and Tatiana Shulman. On the (Local) Lifting Prop- erty. arXiv:2403.12224 [math] version: 1. Mar. 2024. doi: 10.48550/arXiv.2403.12224. [Gol21] Isaa...

work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.math/0408400 2023

[1] [1]

[BO] Nathanial Patrick Brown and Narutaka Ozawa

doi: 10.48550/arXiv.2501.00173. [BO] Nathanial Patrick Brown and Narutaka Ozawa. C*-algebras and Finite- dimensional Approximations . en. Google-Books-ID: F_kjj0teG2IC. American Mathematical Soc. isbn: 978-0-8218-7250-5. [Bow+24] Lewis Bowen, Michael Chapman, Alexander Lubotzky , and Thomas Vidick. The Aldous–Lyons Conjecture I: Subgroup Tests. arXiv:2408...

work page doi:10.48550/arxiv.2501.00173 2024

[2] [2]

Hyperlinearity, essentially free actions and $L^2$-invariants. The sofic property

doi: 10.48550/arXiv.math/0408400. [ES23] Caleb Eckhardt and Tatiana Shulman. On amenable Hilbert-Schmidt stable groups. arXiv:2207.01089 [math]. Feb. 2023. doi: 10.48550/arXiv.2207.01089. [ES24] Dominic Enders and Tatiana Shulman. On the (Local) Lifting Prop- erty. arXiv:2403.12224 [math] version: 1. Mar. 2024. doi: 10.48550/arXiv.2403.12224. [Gol21] Isaa...

work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.math/0408400 2023