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Poisson boundaries of type III factors and the relative bicentralizer conjecture

A mix of human-written and AI-generated textHuman understanding: all partsmath.OA — Operator Algebras

Contributed by Shuoxing Zhou

Version 2 / Oct 06, 2026 / CC BY 4.0

Abstract

We develop the Poisson boundary theory for general von Neumann algebras with separable preduals. As applications, we generalize and give new proofs of the results on the weak Dixmier property for inclusions with expectations or operator-valued weights \cite{Pop99,Mar19,Iso25}. We also prove that every continuous factor admits irreducible embeddings into the hyperfinite factors of type IIIλ\mathrm{III}_\lambda, 0<λ≤10<\lambda\leq1, and into some hyperfinite factor of type III0\mathrm{III}_0. Finally, using the Poisson boundary techniques developed above, we prove that for every inclusion with faithful normal conditional expectation M⊂NM\subset N, M⊂c(N)M\subset c(N) has the weak Dixmier property. By Marrakchi's characterization \cite[Theorem~D]{Ma25}, this gives the relative bicentralizer conjecture for von Neumann algebras with separable preduals.

Provenance statement

This version is posted to provide a public record of independently developed, AI-assisted work. Our result on the relative bicentralizer conjecture overlaps with OpenAI’s concurrent work, but the proofs use fundamentally different methods. This part (Section 7) has been fully polished, while Section 8 on irreducible incusions has not yet been fully polished.

Tools used

OpenAI
ChatGPTVersion 6 Astra
Google
GeminiVersion 3.1 Pro

References

  1. Ando, Hiroshi; Haagerup, Uffe; Houdayer, Cyril; Marrakchi, Amine. Structure of bicentralizer algebras and inclusions of type III factors. Math. Ann., vol. 376, pp. 1145–1194. 2020arXivDOI
  2. Connes, Alain. Classification of injective factors. Cases II1II_1, II∞II_\infty, IIIλIII_\lambda, λ≠1\lambda\ne1. Ann. of Math. (2), vol. 104, no. 1, pp. 73–115. 1976DOI
  3. Connes, Alain; Takesaki, Masamichi. The flow of weights on factors of type III. Tohoku Math. J. (2), vol. 29, no. 4, pp. 473–575. 1977DOI
  4. Das, Sayan; Peterson, Jesse. Poisson boundaries of II1\rm II_1 factors. Compos. Math., vol. 158, no. 8, pp. 1746–1776. 2022DOI
  5. Giordano, Thierry; Skandalis, Georges. Krieger factors isomorphic to their tensor square and pure point spectrum flows. J. Funct. Anal., vol. 64, no. 2, pp. 209–226. 1985link
  6. Haagerup, Uffe. On the dual weights for crossed products of von Neumann algebras. I. Math. Scand., vol. 43, pp. 99–118. 1978DOI
  7. Haagerup, Uffe. LpL^p-spaces associated with an arbitrary von Neumann algebra. Algèbres d'opérateurs et leurs applications en physique mathématique, vol. 274, pp. 175–184. 1979link
  8. Haagerup, Uffe. Operator valued weights in von Neumann algebras. I. J. Funct. Anal., vol. 32, no. 2, pp. 175–206. 1979DOI
  9. Haagerup, Uffe. Connes' bicentralizer problem and uniqueness of the injective factor of type III1III_1. Acta Math., vol. 158, no. 1-2, pp. 95–148. 1987DOI
  10. Houdayer, Cyril; Marrakchi, Amine. The classification of flows on II1II_1 factors and Connes' bicentralizer problem. 2026arXivDOI
  11. Yusuke Isono. Weak relative Dixmier property and Popa's intertwining technique for type III subfactors. 2025arXiv
  12. Izumi, Masaki. Non-commutative Poisson boundaries and compact quantum group actions. Adv. Math., vol. 169, no. 1, pp. 1–57. 2002DOI
  13. Marrakchi, Amine. Full factors, bicentralizer flow and approximately inner automorphisms. Invent. Math., vol. 222, pp. 375–398. 2020arXivDOI
  14. Marrakchi, Amine. On the weak relative Dixmier property. Proc. Lond. Math. Soc. (3), vol. 122, no. 1, pp. 118–123. 2021DOI
  15. Marrakchi, Amine. Kadison's problem for type III subfactors and the bicentralizer conjecture. Invent. Math., vol. 239, pp. 79–163. 2025arXivDOI
  16. OpenAI. Expected amenable subalgebras preserving core commutants. 2026. September 23, 2026link
  17. Popa, Sorin. On a problem of R. V. Kadison on maximal abelian ∗*-subalgebras in factors. Invent. Math., vol. 65, pp. 269–281. 1981DOI
  18. Popa, Sorin. Hyperfinite subalgebras normalized by a given automorphism and related problems. Operator Algebras and their Connections with Topology and Ergodic Theory, vol. 1132, pp. 421–433. 1985DOI
  19. Popa, Sorin. On the relative Dixmier property for inclusions of C∗C^*-algebras. J. Funct. Anal., vol. 171, no. 1, pp. 139–154. 2000DOI
  20. Popa, Sorin. On ergodic embeddings of factors. Comm. Math. Phys., vol. 384, pp. 971–996. 2021arXivDOI
  21. Sekine, Yoshihiro. Flows of weights of crossed products of type III factors by discrete groups. Publ. Res. Inst. Math. Sci., vol. 26, no. 4, pp. 655–666. 1990link
  22. Sutherland, Colin E.; Takesaki, Masamichi. Right inverse of the module of approximately finite dimensional factors of type III and approximately finite ergodic principal measured groupoids. Operator Algebras and Their Applications, II, vol. 20, pp. 149–159. 1998link
  23. Takesaki, M. Theory of operator algebras. I. Springer-Verlag, Berlin, vol. 124, pp. xx+415. 2002. Reprint of the first (1979) edition, Operator Algebras and Non-commutative Geometry, 5
  24. Takesaki, M. Theory of operator algebras. II. Springer-Verlag, Berlin, vol. 125, pp. xxii+518. 2003. Operator Algebras and Non-commutative Geometry, 6DOI
  25. Van Daele, Alfons. Locally compact quantum groups. A von Neumann algebra approach. SIGMA, vol. 10, pp. 082. 2014. 41 pp.DOI
  26. Zhou, Shuoxing. Noncommutative Poisson Boundaries, Ultraproducts, and Entropy. Int. Math. Res. Not. IMRN, vol. 2024, no. 10, pp. 8794-8818. 2024DOI

Version history

  1. v1Submitted by Shuoxing ZhouInitial depositSupersededOct 06, 2026
  2. v2Submitted by Shuoxing ZhouSubmitted as a new versionCurrentOct 07, 2026