HEXAGONbeta

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 1 / 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.

Tools used

OpenAI
ChatGPTVersion 6 Astra
Google
GeminiVersion 3.1 Pro

References

  1. Hiroshi Ando, Uffe Haagerup, Cyril Houdayer, and Amine Marrakchi. Structure of bicentralizer algebras and inclusions of type III factors. Math. Ann. , 376:1145--1194, 2020. https://arxiv.org/abs/1804.05706 arXiv:1804.05706 , https://doi.org/10.1007/s00208-019-01939-9 doi:10.1007/s00208-019-01939-9 .arXivDOI
  2. Alain Connes. Classification of injective factors. cases II1 II _1 , $ II _ $ , $ III _ $ , 1 1 . Ann. of Math. (2) , 104(1):73--115, 1976. https://doi.org/10.2307/1971057 doi:10.2307/1971057 .DOI
  3. Alain Connes and Masamichi Takesaki. The flow of weights on factors of type III . Tohoku Math. J. (2) , 29(4):473--575, 1977. https://doi.org/10.2748/tmj/1178240493 doi:10.2748/tmj/1178240493 .DOI
  4. Sayan Das and Jesse Peterson. Poisson boundaries of II1 II _1 factors. Compos. Math. , 158(8):1746--1776, 2022. https://doi.org/10.1112/S0010437X22007539 doi:10.1112/S0010437X22007539 .DOI
  5. Thierry Giordano and Georges Skandalis. Krieger factors isomorphic to their tensor square and pure point spectrum flows. J. Funct. Anal. , 64(2):209--226, 1985. URL: r̆l https://webusers.imj-prg.fr/ georges.skandalis/Publications/Publi10.pdf .link
  6. Uffe Haagerup. On the dual weights for crossed products of von Neumann algebras. I . Math. Scand. , 43:99--118, 1978. https://doi.org/10.7146/math.scand.a-11768 doi:10.7146/math.scand.a-11768 .DOI
  7. Uffe Haagerup. LpL^p -spaces associated with an arbitrary von Neumann algebra. In Alg è bres d'op é rateurs et leurs applications en physique math é matique , volume 274 of Colloques Internationaux du CNRS , pages 175--184. CNRS, Paris, 1979. URL: r̆l https://dmitripavlov.org/scans/haagerup.pdf .link
  8. Uffe Haagerup. Operator valued weights in von N eumann algebras. I . J. Funct. Anal. , 32(2):175--206, 1979. https://doi.org/10.1016/0022-1236(79)90053-3 doi:10.1016/0022-1236(79)90053-3 .DOI
  9. Uffe Haagerup. Connes' bicentralizer problem and uniqueness of the injective factor of type III1 III _1 . Acta Math. , 158(1-2):95--148, 1987. https://doi.org/10.1007/BF02392257 doi:10.1007/BF02392257 .DOI
  10. Yusuke Isono. Weak relative dixmier property and popa's intertwining technique for type iii subfactors, 2025. URL: r̆l https://arxiv.org/abs/2508.17592 , https://arxiv.org/abs/2508.17592 arXiv:2508.17592 .arXiv
  11. Masaki Izumi. Non-commutative P oisson boundaries and compact quantum group actions. Adv. Math. , 169(1):1--57, 2002. https://doi.org/10.1006/aima.2001.2053 doi:10.1006/aima.2001.2053 .DOI
  12. Amine Marrakchi. Full factors, bicentralizer flow and approximately inner automorphisms. Invent. Math. , 222:375--398, 2020. https://arxiv.org/abs/1811.10253 arXiv:1811.10253 , https://doi.org/10.1007/s00222-020-00971-w doi:10.1007/s00222-020-00971-w .arXivDOI
  13. Amine Marrakchi. On the weak relative D ixmier property. Proc. Lond. Math. Soc. (3) , 122(1):118--123, 2021. URL: r̆l https://doi-org.proxy.rubens.ens.fr/10.1112/plms.12347 , https://doi.org/10.1112/plms.12347 doi:10.1112/plms.12347 .DOI
  14. Amine Marrakchi. Kadison's problem for type III subfactors and the bicentralizer conjecture. Invent. Math. , 239:79--163, 2025. https://arxiv.org/abs/2308.15163 arXiv:2308.15163 , https://doi.org/10.1007/s00222-024-01299-5 doi:10.1007/s00222-024-01299-5 .arXivDOI
  15. OpenAI. Expected amenable subalgebras preserving core commutants. OpenAI Math Release preprint, 2026. 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 Available in the OpenAI Math repository .link
  16. OpenAI. Bounded recovery for modular spectral averages. OpenAI Math Release preprint, 2026. 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 Available in the OpenAI Math repository .link
  17. Sorin Popa. On a problem of R. V. Kadison on maximal abelian ∗* -subalgebras in factors. Invent. Math. , 65:269--281, 1981. https://doi.org/10.1007/BF01389015 doi:10.1007/BF01389015 .DOI
  18. Sorin Popa. Hyperfinite subalgebras normalized by a given automorphism and related problems. In Operator Algebras and their Connections with Topology and Ergodic Theory , volume 1132 of Lecture Notes in Mathematics , pages 421--433. Springer, 1985. https://doi.org/10.1007/BFb0074899 doi:10.1007/BFb0074899 .DOI
  19. Sorin Popa. On the relative D ixmier property for inclusions of C∗C^* -algebras. J. Funct. Anal. , 171(1):139--154, 2000. URL: r̆l https://doi-org.proxy.rubens.ens.fr/10.1006/jfan.1999.3536 , https://doi.org/10.1006/jfan.1999.3536 doi:10.1006/jfan.1999.3536 .DOI
  20. Sorin Popa. On ergodic embeddings of factors. Comm. Math. Phys. , 384:971--996, 2021. https://arxiv.org/abs/1910.06923 arXiv:1910.06923 , https://doi.org/10.1007/s00220-020-03865-3 doi:10.1007/s00220-020-03865-3 .arXivDOI
  21. Yoshihiro Sekine. Flows of weights of crossed products of type III factors by discrete groups. Publ. Res. Inst. Math. Sci. , 26(4):655--666, 1990. URL: r̆l https://ems.press/content/serial-article-files/42525 .link
  22. Colin E. Sutherland and Masamichi Takesaki. Right inverse of the module of approximately finite dimensional factors of type III and approximately finite ergodic principal measured groupoids. In Operator Algebras and Their Applications, II , volume 20 of Fields Institute Communications , pages 149--159. American Mathematical Society, 1998. URL: r̆l https://dmitripavlov.org/scans/sutherland-takesaki.pdf .link
  23. M. Takesaki. Theory of operator algebras. I , volume 124 of Encyclopaedia of Mathematical Sciences . Springer-Verlag, Berlin, 2002. Reprint of the first (1979) edition, Operator Algebras and Non-commutative Geometry, 5.
  24. M. Takesaki. Theory of operator algebras. II , volume 125 of Encyclopaedia of Mathematical Sciences . Springer-Verlag, Berlin, 2003. Operator Algebras and Non-commutative Geometry, 6. https://doi.org/10.1007/978-3-662-10451-4 doi:10.1007/978-3-662-10451-4 .DOI
  25. Alfons Van Daele. Locally compact quantum groups. A von Neumann algebra approach. SIGMA , 10:082, 2014. 41 pp. https://doi.org/10.3842/SIGMA.2014.082 doi:10.3842/SIGMA.2014.082 .DOI
  26. Shuoxing Zhou. Noncommutative Poisson Boundaries, Ultraproducts, and Entropy . Int. Math. Res. Not. IMRN , 2024(10):8794--8818, 2024. https://doi.org/10.1093/imrn/rnae022 doi:10.1093/imrn/rnae022 .DOI

Version history

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