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Approximate Unitary Equivalence of Finite Index Endomorphisms of AFD Factors

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Abstract

We consider two finite index endomorphisms \({\rho}\), \({\sigma}\) of any AFD factor M. We characterize the condition for there being a sequence \({\{ u_n\}}\) of unitaries of the factor M with \({\mathrm{Ad}u_n \circ \rho \to \sigma}\). The characterization is given by using the canonical extension of endomorphisms, which is introduced by Izumi. Our result is a generalization of the characterization of approximate innerness of endomorphisms of AFD factors, obtained by Kawahiashi–Sutherland–Takesaki and Masuda–Tomatsu. Our proof, which does not depend on the types of factors, is based on recent development on the Rohlin property of flows on von Neumann algebras.

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References

  1. Araki H.: Some properties of modular conjugation operator of von Neumann algebras and noncommutative Radon–Nikodym theorem with a chain rule. Pac. J. Math. 50(2), 309–354 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  2. Ando H., Haagerup U.: Ultraproducts of von Neumann algebras. J. Funct. Anal. 266, 6842–6913 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  3. Connes A.: Classification of injective factors. Cases \({\mathrm{II}_1}\), \({\mathrm{II}_\infty}\), \({\mathrm{III}_\lambda}\), \({\lambda \not= 1}\). Ann. Math. (2) 104(1), 73–115 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  4. Connes A.: Une classification des facteurs de type III. Ann. Sci. Ecole Norm. Sup. (4) 6, 133–252 (1973)

    MathSciNet  MATH  Google Scholar 

  5. Connes A., Takesaki M.: The flow of weights on factors of type III. Tohoku Math. J. (2) 29(4), 473–575 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  6. Guido D., Longo R.: The conformal spin and statistics theorem. Commun. Math. Phys. 181(1), 11–35 (1996)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  7. Haagerup U., Størmer E.: Equivalence of normal states on von Neumann algebras and the flow of weights. Adv. Math. 83(2), 180–262 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  8. Izumi M.: Canonical extension of endomorphisms of type III factors. Am. J. Math. 125(1), 11–56 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  9. Jones, V.F.R.: Actions of finite groups on the hyperfinite type \({\mathrm{II}_1}\) factor. Mem. Am. Math. Soc. 237 (1980)

  10. Jones V.F.R., Takesaki M.: Actions of compact abelian groups on semifinite injective factors. Acta. Math. 153(3–4), 213–258 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  11. Kosaki H., Longo R.: A remark on the minimal index of subfactors. J. Funct. Anal. 107(2), 458–470 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  12. Katayama Y., Sutherland C.E., Takesaki M.: The characteristic square of a factor and the cocycle conjugacy of discrete group actions on factors. Invent. Math. 132(2), 331–380 (1998)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  13. Kawahigashi Y., Sutherland C.E., Takesaki M.: The structure of the automorphism group of an injective factor and the cocycle conjugacy of discrete abelian group actions. Acta. Math. 169, 105–130 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  14. Kawahigashi Y., Takesaki M.: Compact abelian group actions on injective factors. J. Funct. Anal. 105(1), 112–128 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  15. Longo R.: Index of subfactors and statistics of quantum fields. I. Commun. Math. Phys. 126(2), 217–247 (1989)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  16. Masuda T.: Unified approach to classification of actions of discrete amenable groups on injective factors. J. Reine Anguew. Math. 683, 1–47 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  17. Masuda T., Tomatsu R.: Approximate innerness and central triviality of endomorphisms. Adv. Math. 220(4), 1075–1134 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  18. Masuda, T., Tomatsu, R.: Classification of actions of discrete Kac algebras on injective factors, Preprint (2013). arXiv:1306.5046, to appear in Memo. Amer. Math. Soc.

  19. Masuda T., Tomatsu R.: Classification of minimal actions of a compact Kac algebra with amenable dual. Commun. Math. Phys. 274(2), 487–551 (2007)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  20. Masuda T., Tomatsu R.: Classification of minimal actions of a compact Kac algebra with amenable dual on injective factors of type III. J. Funct. Anal. 258(6), 1965–2025 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  21. Masuda, T., Tomatsu, R.: Rohlin flows on von Neumann algebras. arXiv:1206.0955, to appear in Memo. Amer. Math. Soc.

  22. Ocneanu, A.: Actions of discrete amenable groups on von Neumann algebras, Lecture Notes in Mathematics, 1138. Springer-Verlag, Berlin (1985). iv+115 pp.

  23. Popa S., Wassermann A.: Actions of compact Lie groups on von Neumann algebras. C. R. Acad. Sci. Paris Ser. I Math. 315, 421–426 (1992)

    MathSciNet  MATH  Google Scholar 

  24. Sutherland C.E., Takesaki M.: Actions of discrete amenable groups on injective factors of type \({\mathrm{III}_\lambda}\), \({\lambda \not=1}\). Pac. J. Math. 137, 405–444 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  25. Sutherland, C.E., Takesaki, M.: 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 (Waterloo, ON, 1994/1995), 149–159, Fields Inst. Commun., 20, Amer. Math. Soc. Providence, RI (1998)

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  1. Department of Mathematical Sciences, University of Tokyo, Komaba, Tokyo, 153-8914, Japan

    Koichi Shimada

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  1. Koichi Shimada
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Correspondence to Koichi Shimada.

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Communicated by Y. Kawahigashi

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Shimada, K. Approximate Unitary Equivalence of Finite Index Endomorphisms of AFD Factors. Commun. Math. Phys. 344, 507–529 (2016). https://doi.org/10.1007/s00220-015-2505-7

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