# Relative bicentralizers and modular spectral recovery

> Claims every expected inclusion of von Neumann algebras holds an amenable piece with the same commutant in the core.

Canonical: https://closedform.nishachay.tech/e/openai-math-2026-290/
Last updated: 2026-10-14

**Claims every expected inclusion of von Neumann algebras holds an amenable piece with the same commutant in the core.**

**Trust:** Formal proof listed · **Papers:** 2 · **First paper:** September 23, 2026 · **Field:** Operator algebras · **Entry:** 290

*Explained by AI.* Drafted by one AI model and checked sentence by sentence against the paper by a different one. Spot a mistake? Report an error.

A commutant is everything that commutes with a given set of operators. This paper asks whether a tame, amenable part of an algebra N can force the same commutation rules as all of N. It claims yes, inside a larger setting called the continuous core, for every expected inclusion with separable preduals. That settles Marrakchi's relative bicentralizer conjecture, if the proof holds. A companion paper proves, with a Lean check, a bounded recovery theorem used for the absolute bicentralizer problem.

## At a glance

- **First step:** 1982 (Alain Connes, bicentralizer problem in the classification of injective factors)
- **Significance:** Major. It claims to settle the relative bicentralizer conjecture, a recent open problem in the structure of type III von Neumann algebras.
- **Trust:** Formal proof listed
- **Field:** Operator algebras (2 papers)

## The explanation

Start with 2-by-2 matrices. Which matrices commute with D = diag(1, 2)? Write a general matrix with entries a, b in the top row and c, d below. Multiplying by D on the right gives a, 2b, c, 2d. Multiplying on the left gives a, b, 2c, 2d. These agree only when b = 0 and c = 0. So the matrices commuting with D form a 2-dimensional set, the diagonal ones. Now also ask them to commute with the swap matrix, which has 1s off the diagonal. That forces a = d. Only multiples of the identity survive. Two well-chosen matrices impose the same rule as all four dimensions of matrices.

The set of everything commuting with a collection is its commutant. In infinite dimensions, the analogous question is harder. Take a von Neumann algebra N inside a larger one, M. Can a tame part P of N impose exactly the same commutation constraints as N? Tame here means amenable, also called injective. Popa's 1981 work gave a model for finding such pieces in one setting[1]. The inclusion must also be expected: there is a well-behaved averaging map from M onto N.

The harder setting involves the continuous core. This is a larger algebra that adds unitaries driving the modular flow, a built-in time evolution of type III algebras. Connes introduced the bicentralizer problem in classifying injective factors[2]. Haagerup solved the amenable type III₁ case[3]. Marrakchi formulated the relative bicentralizer conjecture. He proved it is equivalent, for separable preduals, to the statement this paper claims[4]. The obstacle is that the core holds those extra unitaries, so matching commutants inside M alone is not enough.

This paper claims that for every such inclusion there is an amenable P inside N with an averaging map onto P[5]. Its commutant inside the core matches that of N. This would resolve the conjecture, including inclusions with arbitrary centers and types[6]. Houdayer and Marrakchi have announced that every type III₁ factor with a faithful normal state has trivial bicentralizer[7]. The paper gives another proof of the separable case. The companion proves a bounded spectral recovery theorem for states whose centralizer is only scalars[8]. It uses this for another proof of bicentralizer triviality for separable type III₁ factors[9].

How sure is this? A computer proof checker called Lean has a formal proof of the bounded recovery theorem. It turns positive spectral averages into uniformly bounded algebra elements. The main core-commutant theorem and the absolute bicentralizer result are not part of that Lean statement. No specialist review of the papers has been published.

**What it doesn't show.** This does not cover algebras without separable preduals, and Lean does not check the main theorem or the absolute bicentralizer result.

## Where it came from

This result comes from an unreleased OpenAI model, published in the openai/math release on 6 October 2026.

## Hype check

Major if the main proof holds. Lean checks only the bounded recovery theorem, not the core-commutant theorem or bicentralizer triviality, and no specialist review is published.

## Words used

- **Von Neumann algebra:** An algebra of operators on a Hilbert space, closed under adjoints and under weak limits.
- **Amenable (injective):** A tameness property of von Neumann algebras that allows averaging arguments.
- **Conditional expectation:** A well-behaved averaging map from a larger algebra onto a smaller one inside it.
- **Modular flow:** A built-in one-parameter time evolution that every faithful state gives a von Neumann algebra.
- **Continuous core:** A larger algebra built from M and the unitaries that drive its modular flow.
- **Type III₁ factor:** A kind of von Neumann algebra with no trace, whose modular flow is as rich as possible.
- **Bicentralizer:** The elements commuting asymptotically with every bounded sequence that asymptotically commutes with a given state.
- **Separable predual:** A size limit: the algebra is determined by countably many states.

## The original papers (2)

- [Expected amenable subalgebras preserving core commutants](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) - September 23, 2026
- [Bounded recovery for modular spectral averages](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) - September 23, 2026 · Formal proof listed

- [Lean notes](https://github.com/openai/math/blob/main/lean/docs/290.md)

## Quoted lines from the paper (9)

[1] From [Expected amenable subalgebras preserving core commutants](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):

> Popa's solution of Kadison's problem for irreducible expected inclusions with separable preduals and type II bottom factor provides a model for finding amenable subalgebras that retain prescribed commutation properties.

[2] From [Expected amenable subalgebras preserving core commutants](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):

> Connes introduced the problem of its triviality in the classification of injective factors.

[3] From [Expected amenable subalgebras preserving core commutants](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):

> Haagerup solved the amenable type $\mathrm{III}_1$ case, completing the uniqueness theorem for the injective type $\mathrm{III}_1$ factor.

[4] From [Expected amenable subalgebras preserving core commutants](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):

> Marrakchi formulated the relative bicentralizer conjecture and proved that, for separable preduals, it is equivalent to the conclusion of Theorem ….

[5] From [Expected amenable subalgebras preserving core commutants](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):

> There exist a unital amenable von Neumann subalgebra $P\subset N$ and a faithful normal conditional expectation $F:N\to P$ such that \[ P'\cap c(M)=N'\cap c(M). \]

[6] From [Expected amenable subalgebras preserving core commutants](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):

> Our theorem therefore resolves that conjecture positively, including inclusions with arbitrary centers and types.

[7] From [Expected amenable subalgebras preserving core commutants](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):

> Houdayer and Marrakchi have announced that every type $\mathrm{III}_1$ factor admitting a faithful normal state has trivial bicentralizer; their theorem statement has no separability restriction.

[8] From [Bounded recovery for modular spectral averages](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):

> We prove a bounded spectral recovery theorem for a von Neumann algebra with a faithful normal state whose centralizer consists only of scalars.

[9] From [Bounded recovery for modular spectral averages](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):

> We apply this theorem to spectral intertwining rigidity and obtain an alternative proof of bicentralizer triviality for type $\mathrm{III}_1$ factors with separable predual.

## Other sources

- [openai/math, Lean scope note for family 290](https://github.com/openai/math/blob/main/lean/docs/290.md): “For a faithful normal state with scalar centralizer in the stated standard-space representation, the formalized result converts positive modular spectral averages into uniformly bounded algebra elements.”
- [openai/math release notes (2026): model provenance and review status](https://github.com/openai/math/blob/main/README.md): “manuscripts and supporting proof artifacts produced by an internal OpenAI model”

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