Browse › Operator algebras › 290 Relative bicentralizers and modular spectral recovery
Bounded recovery for modular spectral averages
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Abstract
We prove a bounded spectral recovery theorem for a von Neumann algebra with a faithful normal state whose centralizer consists only of scalars. A positive averaged squared norm for an arbitrary bounded Hilbert-space operator on shrinking modular spectral bands can be recovered on uniformly bounded algebra elements with shrinking spectral support. We apply this theorem to spectral intertwining rigidity and obtain an alternative proof of bicentralizer triviality for type III1 factors with separable predual.
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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.
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). Formalizes this paper's bounded recovery theorem for a faithful normal state with scalar centralizer, recovering uniformly bounded algebra elements from positive modular spectral averages in spectral bands up to four times wider; the note states that the absolute bicentralizer conjecture is not included.
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 having solved the bicentralizer problem for all type III_1 factors and as the source of the resonance mechanism; this paper gives an alternative proof of its separable-predual case and cites v1. [family-level]
Possibly relevant literature (found by the agent, not checked)
- Cyril Houdayer and Sorin Popa, Singular MASAs in type III factors and Connes' Bicentralizer Property (arXiv:1704.07255, 2017). Shows that separable type III_1 factors with Connes' bicentralizer property contain a singular MASA with expectation; related to the paper's remark that its main theorem yields an expected maximal abelian subalgebra in every separable type III_1 factor.
- Cyril Houdayer and Amine Marrakchi, Selfless W-probability spaces and Connes' bicentralizer problem* (arXiv:2511.11409, v1 Nov 2025, v2 May 2026). Shows that a separable type III_1 factor with trivial bicentralizer is selfless with respect to every faithful normal state, which bears on consequences of the paper's main theorem.
- Hiroshi Ando and Isaac Goldbring, Model theory and Connes' bicentralizer problem (arXiv:2605.12776, May 2026). Proves trivial bicentralizer for a class of "pseudoperiodic" type III_1 factors and recasts the problem model-theoretically; earlier partial work on the statement of the paper's main theorem, not in its overview of prior results.
Digests and expositions
- Claude Opus 5.5, Modular averaging and bounded recovery (Open Mathematics Courses, October 2026). This lesson follows the paper's bounded recovery theorem through modular averaging, fourth-moment estimates, polar truncation and Fourier filtering, and identifies itself as self-checked by its writing AI. [AI-generated per source]
- Claude Opus 5.5, Spectral intertwining rigidity (Open Mathematics Courses, October 2026). The companion lesson develops the paper's passage from bounded recovery to interval estimates and joint spectral rigidity, including its application to the bicentralizer flow. [AI-generated per source]
Elsewhere
- Emergent Mind: family 290
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Bounded-recovery-for-modular-spectral-averages-September-23-2026,
author = {{OpenAI}},
title = {{Bounded recovery for modular spectral averages}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Bounded-recovery-for-modular-spectral-averages-September-23-2026/Bounded-recovery-for-modular-spectral-averages-September-23-2026.pdf}{OAI:Bounded-recovery-for-modular-spectral-averages-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:
- Expected amenable subalgebras preserving core commutants · September 23, 2026