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Isometries between non-commutative symmetric spaces associated with semi-finite von Neumann algebras

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We show that positive surjective isometries between symmetric spaces associated with semi-finite von Neumann algebras are projection disjointness preserving if they are finiteness preserving. This is subsequently used to obtain a structural description of such isometries. Furthermore, it is shown that if the initial symmetric space is a strongly symmetric space with absolutely continuous norm, then a similar structural description can be obtained without requiring positivity of the isometry.

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  1. Banach, S.: Théorie des Opérations Linéaires. Chelsea, Warsaw (1932)

    MATH  Google Scholar 

  2. Bikchentaev, A.: Block projection operator on normed solid spaces of measurable operators (Russian). Izv. Vyssh. Uchebn. Zaved. Mat. 2, 86–91 (2012) [English translation in Russian Math. (Iz. VUZ) 56(2), 75–79 (2012)]

  3. Chilin, V.I., Medzhitov, A.M., Sukochev, F.A.: Isometries of non-commutative Lorentz spaces. Math. Z. 200, 527–545 (1989)

    Article  MathSciNet  Google Scholar 

  4. Conway, J.B.: A Course in Functional Analysis, 2nd edn. Springer, New York (2007)

    Book  Google Scholar 

  5. de Jager, P.: Isometries on symmetric spaces associated with semi-finite von Neumann algebras. Ph.D. Thesis, University of Cape Town (2017).

  6. de Jager, P., Conradie, J.J.: Extension of projection mappings. Quaest. Math. (2019).

    Article  Google Scholar 

  7. de Jager, P., Conradie, J.J.: Isometries between non-commutative (quantum) Lorentz spaces associated with semi-finite von Neumann algebras. Preprint (in preparation). arXiv:1907.07619

  8. de Pagter, B.: Non-commutative Banach Function Spaces, Positivity: Trends Mathematics, pp. 197–227. Birkhäuser, Basel (2007)

    MATH  Google Scholar 

  9. Dodds, P.G., Dodds, T.K.-Y., de Pagter, B.: Fully symmetric operator spaces. Integr. Equ. Oper. Theory 15, 942–972 (1992)

    Article  MathSciNet  Google Scholar 

  10. Dodds, P.G., de Pagter, B.: The non-commutative Yosida–Hewitt decomposition revisited. Trans. Am. Math. Soc. 364, 6425–6457 (2012)

    Article  MathSciNet  Google Scholar 

  11. Dodds, P.G., de Pagter, B.: Normed Köthe spaces: a non-commutative viewpoint. Indag. Math. 25, 206–249 (2014)

    Article  MathSciNet  Google Scholar 

  12. Fleming, R.J., Jamison, J.E.: Isometries on Banach spaces: Function Spaces, vol. 1. Chapman and Hall, Boca Raton (2003)

    MATH  Google Scholar 

  13. Goldstein, S., Lindsay, J.M.: Markov semigroups KMS-symmetric for a weight. Math. Ann. 313, 39–67 (1999)

    Article  MathSciNet  Google Scholar 

  14. Hong, G., Kumar, S.K., Wang, S.: Maximal ergodic inequalities for some positive operators on noncommutative \(L^p\)-spaces. Preprint arXiv:1907.12967

  15. Huang, J., Sukochev, F., Zanin, D.: Logarithmic submajorisation and order-preserving linear isometries. Preprint arXiv:1808.10557v2

  16. Huang, J., Sukochev, F.: Interpolation between \(L_0(\cal{M}\tau )\) and \(L_\infty (\cal{M},\tau )\). Preprint arXiv:1902.05907v1

  17. Kaddison, R.V.: Isometries of operator algebras. Ann. Math. 54(2), 325–338 (1951)

    Article  MathSciNet  Google Scholar 

  18. Kaddison, R.V., Ringrose, J.R.: Fundamentals of the Theory of Operator Algebras, vol. 1. Birkhäuser, Providence (1997)

    Google Scholar 

  19. Kaddison, R.V., Ringrose, J.R.: Fundamentals of the Theory of Operator Algebras, Volume 2, Advanced Theory. Birkhäuser, Providence (1997)

    Google Scholar 

  20. Kalton, N.J., Sukochev, F.A.: Symmetric norms and spaces of operators. J. Reine Angew. Math. 621, 81–121 (2008)

    MathSciNet  MATH  Google Scholar 

  21. Kan, C.-H.: Ergodic properties of Lamperti operators. Can. J. Math. 30(6), 1206–1214 (1978)

    Article  MathSciNet  Google Scholar 

  22. Lamperti, J.: On the isometries of certain function spaces. Pac. J. Math. 8, 459–466 (1958)

    Article  MathSciNet  Google Scholar 

  23. Le Merdy, C., Zadeh, S.: \(\ell ^1\)-contractive maps on noncommutative \(L^p\)-spaces. Preprint arXiv:1907.03995

  24. Sukochev, F.A.: Isometries of symmetric operator spaces associated with AFD factors of type \(II\) and symmetric vector-valued spaces. Integr. Equ. Oper. Theory 26, 102–124 (1996)

    Article  MathSciNet  Google Scholar 

  25. Sukochev, F., Veksler, A.: Positive linear isometries in symmetric operator spaces. Integr. Equ. Oper. Theory 90(5), 58 (2018)

    Article  MathSciNet  Google Scholar 

  26. Terp, M.: \(L^p\)-spaces associated with von Neumann algebras. Rapport No. 3a, University of Copenhagen (1981)

  27. Yeadon, F.J.: Isometries of non-commutative \(L^p\)-spaces. Math. Proc. Camb. Phil. Soc. 90, 41–50 (1981)

    Article  MathSciNet  Google Scholar 

  28. Zaidenberg, M.G.: A representation of isometries on function spaces. Mat. Fiz. Anal. Geom. 4(3), 339–347 (1997)

    MathSciNet  MATH  Google Scholar 

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The greater part of this research was conducted during the first author’s doctoral studies at the University of Cape Town. The first author would like to thank his Ph.D. supervisor, Dr. Robert Martin, for his input and guidance and the NRF for funding towards this project in the form of scarce skills and grantholder-linked bursaries. Furthermore, the support of the DST-NRF Centre of Excellence in Mathematical and Statistical Sciences (CoE-MaSS) towards this research is hereby acknowledged. Opinions expressed and conclusions arrived at, are those of the authors and are not necessarily attributed to the CoE. The authors would also like to thank the reviewer for pointing out the preprints [14, 15, 23] and for numerous useful comments and suggestions.

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Correspondence to Pierre de Jager.

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de Jager, P., Conradie, J. Isometries between non-commutative symmetric spaces associated with semi-finite von Neumann algebras. Positivity 24, 815–835 (2020).

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