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Relative bicentralizer formula for the continuous core

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Let N⊂MN \subset M be an inclusion of von Neumann algebras with expectation, and let φ∈N∗\varphi \in N_* be a faithful normal state. Write c(N)=N⋊σφRc(N)=N \rtimes_{\sigma^\varphi} \mathbb{R} and let c(M)c(M) be the corresponding continuous core. Let B(N⊂c(M),φ)\mathord{\mathrm{B}}(N \subset c(M),\varphi) be the relative bicentralizer, and let (φit)t∈R(\varphi^{\mathrm{i}t})_{t\in\mathbb{R}} be the one-parameter group of unitaries in c(N)c(N) implementing σφ\sigma^\varphi. Relative bicentralizer conjecture.

B(N⊂c(M),φ)={φit∣t∈R}”∨(N′∩c(M)).\mathord{\mathrm{B}}(N \subset c(M),\varphi)=\{\varphi^{\mathrm{i}t}\mid t\in\mathbb{R}\}”\vee(N'\cap c(M)).

The formula is proposed for arbitrary inclusions with expectation and is intended to compute the relative bicentralizer through the relative flow of weights; the supplied text does not state that it has been proved in full.

References

Primary source

Amine Marrakchi, “Kadison's problem for type III subfactors and the bicentralizer conjecture”, arXiv:2308.15163 (2024).

operator-algebrassubfactorsvon-neumann-algebras
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Solutions 1

RemarkAI-assistedClaimed by OpenAI. For every unital expected inclusion N subset M of von Neumann algebras with separable preduals, claims an expected amenable unital P subset N with P-prime intersect c(M)=N-prime intersect c(M). This is related core relative-commutant progress; the source explicit theorem is not the full target bicentralizer formula for arbitrary expected inclusions.See full solutionHide full solution

Claimed by OpenAI. Claims that every unital expected inclusion N subset M of von Neumann algebras with separable preduals admits an amenable unital P subset N with a faithful normal conditional expectation and P-prime intersect c(M)=N-prime intersect c(M), where c(M) is the continuous core. This is a core relative-commutant assertion; expectation and separable-predual hypotheses are essential.

Scope relative to this problem: Related structural progress for the core bicentralizer question: an expected unital amenable subalgebra P of N retains N-prime intersect c(M), for expected inclusions with separable preduals. The source attributes equivalence to a relative bicentralizer formulation to the cited literature; the attached claim here is its explicit amenable core-commutant theorem, not an independently verified equality of the bicentralizer with the target flow-generated algebra for every expected inclusion.

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

Manuscript: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Expected-amenable-subalgebras-preserving-core-commutants-September-23-2026/Expected-amenable-subalgebras-preserving-core-commutants-September-23-2026.pdf

  • OpenAI-290-01-Expected-amenable-subalgebras-preserving-core-commutants.pdf333,040 bytesOpen
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Shaowu Zhang Posted · edited