Browse › Operator algebras › 290 Relative bicentralizers and modular spectral recovery
Expected amenable subalgebras preserving core commutants
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Abstract
We prove that every inclusion $N\subset M$ of von Neumann algebras with separable preduals and a faithful normal conditional expectation contains an expected amenable subalgebra $P\subset N$ such that $P'\cap c(M)=N'\cap c(M)$, where $c(M)$ denotes the continuous core of M. This resolves the relative bicentralizer conjecture in this setting.
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Notes
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work in progress
Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Follow-up works
- Shuoxing Zhou, Poisson boundaries of type III factors and the relative bicentralizer conjecture (hexagon:2610.00134v2, v1 6 Oct 2026, v2 7 Oct 2026). Using Poisson boundary theory, it proves the weak Dixmier property for an expected inclusion into the continuous core and, via Marrakchi's Theorem D, the relative bicentralizer conjecture for separable preduals; it cites this paper and its provenance statement says the result "overlaps with OpenAI's concurrent work, but the proofs use fundamentally different methods." [AI-generated per source]
Comments and discussion
- Cyril Houdayer, reaction in 100+ reactions to 100+ solutions, Proofs and Prompts (post dated 8 Oct 2026; collection updated through 19 October). Houdayer says that "The proof announced by OpenAI builds on the methods developed in Amine's earlier work" and on a resonance mechanism found with Marrakchi, and that "OpenAI's result extends ours by establishing the relative version." [family-level]
Formalization
- OpenAI math release, Lean scope note for family 290 (
lean/docs/290.md) (release of 6 Oct 2026). The release's Lean development covers only the bounded recovery lemma of the companion paper; neither this paper's main theorem nor the absolute bicentralizer conjecture is formalized. [family-level] - Shuoxing Zhou, Poisson boundaries of type III factors and the relative bicentralizer conjecture, verification disclosure on p. 5 (hexagon:2610.00134v2, 7 October 2026). Zhou reports a Lean check of a short note covering Section 2, the existence argument in Example 3.9, Theorem 5.1 and Section 7's alternative relative bicentralizer proof, conditional on explicitly assumed published results and without formalizing their dependencies.
Related prior literature mentioned by others
- Cyril Houdayer and Amine Marrakchi, The classification of flows on II_1 factors and Connes' bicentralizer problem (arXiv:2609.11462, v1 10 Sep 2026, v2 5 Oct 2026). In the Proofs and Prompts post, Houdayer names this joint work as settling the absolute (nonrelative) problem that this paper extends and as the source of the resonance mechanism; the paper cites v1. [family-level]
- Amine Marrakchi, Kadison's problem for type III subfactors and the bicentralizer conjecture (arXiv:2308.15163; Inventiones Mathematicae 239, 2025). Zhou invokes Theorem D to pass from the weak Dixmier property in the continuous core to the relative bicentralizer conjecture, the characterization also underlying this paper's amenable-subalgebra conclusion.
Possibly relevant literature (found by the agent, not checked)
- Amine Marrakchi, On the weak relative Dixmier property (arXiv:1907.00615, 2019; Proc. London Math. Soc.). Proves the weak relative Dixmier property for every inclusion with a faithful normal conditional expectation; related to the paper's final-section claim that $N$-unitary orbits in $c(M)$ have weak* closed convex hulls meeting $N'\cap c(M)$, the core analogue of that property.
Elsewhere
- Hexagon: Poisson boundaries of type III factors and the relative bicentralizer conjecture (human and AI; human understanding: all)
- VibeMathed: Problem record
- Emergent Mind: family 290
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Expected-amenable-subalgebras-preserving-core-commutants-September-23-2026,
author = {{OpenAI}},
title = {{Expected amenable subalgebras preserving core commutants}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Expected-amenable-subalgebras-preserving-core-commutants-September-23-2026/Expected-amenable-subalgebras-preserving-core-commutants-September-23-2026.pdf}{OAI:Expected-amenable-subalgebras-preserving-core-commutants-September-23-2026}},
year = {2026}
}Lean formalization
OpenAI provides Lean files for this family: lean/docs/290.md. The Lean statements have not been checked by people against the paper's claims.
Family 290: Relative bicentralizers and modular spectral recovery
Proves Connes' bicentralizer conjecture for every type III1 factor with separable predual and every faithful normal state. More generally, for every inclusion $N\subset M$ of von Neumann algebras with separable preduals admitting a faithful normal conditional expectation, constructs an amenable expected subalgebra $P\subset N$ with $P'\cap c(M)=N'\cap c(M)$, resolving the relative bicentralizer conjecture.
Other papers in this family:
- Bounded recovery for modular spectral averages · September 23, 2026