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Marrakchi's relative bicentralizer conjecture

Connes asked whether the bicentralizer of a faithful normal state on a type III1\mathrm{III}_1 factor with separable predual is always trivial; Haagerup settled the amenable case. Ando, Haagerup, Houdayer and Marrakchi built a relative bicentralizer and its flow for expected inclusions. Marrakchi formulated the relative bicentralizer conjecture and proved that, for separable preduals, it is equivalent to: for every inclusion N⊂MN\subset M of von Neumann algebras with separable preduals and a faithful normal conditional expectation, there is an expected amenable subalgebra P⊂NP\subset N with P′∩c(M)=N′∩c(M)P'\cap c(M)=N'\cap c(M), where c(M)c(M) is the continuous core. Is the relative bicentralizer conjecture true?

Result
Proved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Argument
Field
Operator algebras; type III von Neumann algebras, bicentralizers
Posed by
A. Marrakchi, Kadison's problem for type III subfactors and the bicentralizer conjecture, Invent. Math. 239 (2025), Conjecture C (arXiv 2308.15163)
Year posed
2023
Years open
3y
Solved
2026-09-23
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Unreviewed
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
32 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Theorem 1.1: for every unital inclusion N⊂MN\subset M with separable preduals and a faithful normal conditional expectation there are an amenable P⊂NP\subset N and an expectation N→PN\to P with P′∩c(M)=N′∩c(M)P'\cap c(M)=N'\cap c(M); with Marrakchi's Theorem D this is the relative bicentralizer conjecture, including arbitrary centers and types, and gives an expected MASA corollary relevant to Kadison's problem for subfactors. Along the way it reproves Connes' bicentralizer conjecture for separable type III1\mathrm{III}_1 factors; priority for that absolute case belongs to Houdayer and Marrakchi (arXiv 2609.11462, 10 Sep 2026), whose statement has no separability assumption. The companion gives a second absolute proof via bounded recovery. Non-separable preduals are not treated.

What the AI did

The release README says all results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author.

Verification

No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against Marrakchi's Conjecture C via his Theorem D equivalence, as the paper states it. The proof was not refereed. Not Lean-checked: the family's only formal statement, ComparatorChallenges/BoundedRecovery.lean (OAI.BoundedRecovery.bounded_recovery), was read here and covers the companion's bounded-recovery lemma for a state with scalar centralizer, not the relative or the absolute bicentralizer statement, so under the tier rule this entry stays unreviewed. The argument uses as inputs Marrakchi's Theorem D equivalence and his 2026 binormal-state lemmas.

Sources

Changelog1 change

Discussion