For every type III_1 factor M and every faithful normal state varphi on M, the bicentralizer algebra B(M,varphi) is trivial: B(M,varphi)=C1, where B(M,varphi) consists of those x∈ M such that lVert xa_n-a_nx rVert→ 0 strongly for every bounded sequence (a_n)⊂ M satisfying lVert a_n varphi-varphi a_n rVert→ 0 strongly.

Connes’ bicentralizer conjecture and Takesaki’s classification problem for flows

For every type $\mathrm{III}_1$ factor $M$ and every faithful normal state $\varphi$ on $M$ , the bicentralizer algebra $B(M,\varphi)$ is trivial: $B(M,\varphi)=\mathbb{C}1$ , where $B(M,\varphi)$ consists of those $x\in M$ such that $\lVert xa_n-a_nx\rVert\to 0$ strongly for every bounded sequence $(a_n)\subset M$ satisfying $\lVert a_n\varphi-\varphi a_n\rVert\to 0$ strongly.

References

Primary source

arXiv

Additional references

The classification of flows on II₁ factors and Connes' bicentralizer problem — arXiv — Cyril Houdayer, Amine Marrakchi

operator-algebras von-neumann-algebras

Posted 3 weeks ago · edited 3 weeks ago

Progress summary

Refreshed 3 weeks ago

Claimed solved

A new unrefereed paper claims to settle both problems, but the claim has not been independently checked.

The entry concerns Connes’ bicentralizer conjecture and Takesaki’s classification problem for flows, two long-standing questions in von Neumann algebra theory. The reported advance uses a classification theorem for flows and an argument involving actions of the real line.

September 10, 2026 claimed solution

Cyril Houdayer and Amine Marrakchi’s preprint The classification of flows on II₁ factors and Connes' bicentralizer problem claims to resolve both problems. Its classification theorem and bicentralizer argument would settle the stated conjectures if correct.

Current status (as of September 2026): A September 2026 preprint claims both problems are solved, but neither claim is independently verified.

Sources

RemarkAI-assistedClaimed by OpenAI. Claims bounded modular-spectral recovery for von Neumann algebras with a faithful normal state and scalar centralizer, and an alternative proof of bicentralizer triviality for type III_1 factors with separable predual.See full solutionHide full solution

Claimed by OpenAI. Claims bounded modular-spectral recovery for von Neumann algebras with a faithful normal state and scalar centralizer, and an alternative proof of bicentralizer triviality for type III_1 factors with separable predual.

Scope relative to this problem: The bounded modular-spectral recovery theorem assumes a faithful normal state with scalar centralizer. The manuscript also claims an alternative proof of bicentralizer triviality for type III1 factors with separable predual. The target has no separable-predual restriction; no broader nonseparable or flow-classification assertion is attributed to this companion.

GitHub repository: https://github.com/openai/math

Manuscript: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Bounded-recovery-for-modular-spectral-averages-September-23-2026/Bounded-recovery-for-modular-spectral-averages-September-23-2026.pdf

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