Interpolation between \(L_0({\mathcal M},\tau )\) and \(L_\infty ({\mathcal M},\tau )\)

  • J. HuangEmail author
  • F. Sukochev


Let \({\mathcal M}\) be a semifinite von Neumann algebra with a faithful semifinite normal trace \(\tau \). We show that the symmetrically \(\Delta \)-normed operator space \(E({\mathcal M},\tau )\) corresponding to an arbitrary symmetrically \(\Delta \)-normed function space \(E(0,\infty )\) is an interpolation space between \(L_0({\mathcal M},\tau )\) and \({\mathcal M}\), which is in contrast with the classical result that there exist symmetric operator spaces \(E({\mathcal M},\tau )\) which are not interpolation spaces between \(L_1({\mathcal M},\tau )\) and \({\mathcal M}\). Besides, we show that the \({\mathcal K}\)-functional of every \(X\in L_0({\mathcal M},\tau )+{\mathcal M}\) coincides with the \({\mathcal K}\)-functional of its generalized singular value function \(\mu (X)\). Several applications are given, e.g., it is shown that the pair \((L_0({\mathcal M},\tau ),{\mathcal M})\) is \({\mathcal K}\)-monotone when \({\mathcal M}\) is a non-atomic finite factor.


Interpolation Orbits \({\mathcal K}\)-orbits Symmetrically \(\Delta \)-normed spaces Von Neumann algebras 

Mathematics Subject Classification

46L10 46E30 47A57 



The authors would like to thank Sergei Astashkin and Dima Zanin for helpful discussions.

J. Huang acknowledges the support of University International Postgraduate Award (UIPA). F. Sukochev was supported by the Australian Research Council.


  1. 1.
    Acosta, M., Kamińska, A.: Weak neighborhoods and the Daugavet property of the interpolation spaces \(L^1+L^\infty \) and \(L^1 \cap L^\infty \). Indiana Univ. Math. J. 57(1), 77–96 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Astashkin, S.V.: Interpolation of operators in quasinormed groups of measurable functions. Sib. Math. J. 35(6), 1075–1082 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Astashkin, S.V., Maligranda, L.: Interpolation between \(L_1\) and \(L_p\), \(1<p<\infty \). Proc. Am. Math. Soc. 132(10), 2929–2938 (2004)CrossRefzbMATHGoogle Scholar
  4. 4.
    Bergh, J., Löfström, J.: Interpolation Spaces, An Introduction. Springer, Berlin (1976)CrossRefzbMATHGoogle Scholar
  5. 5.
    Calderón, A.: Spaces between \(L^1\) and \(L^\infty \) and the theorem of Marcinkiewicz. Stud. Math. 26, 273–299 (1966)CrossRefzbMATHGoogle Scholar
  6. 6.
    Chilin, V., Krygin, A., Sukochev, F.: Local uniform and uniform convexity of non-commutative symmetric spaces of measurable operators. Math. Proc. Camb. Philos. Soc. 111, 355–368 (1992)CrossRefzbMATHGoogle Scholar
  7. 7.
    Dixmier, J.: Les algebres d’operateurs dans l’Espace Hilbertien, 2nd edn. Gauthier-Vallars, Paris (1969)zbMATHGoogle Scholar
  8. 8.
    Dodds, P., Dodds, T., de Pagter, B.: Fully symmetric operator spaces. Integr. Equ. Oper. Theory 15, 942–972 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Dodds, P., Dodds, T., de Pagter, B.: Noncommutative Köthe duality. Trans. Am. Math. Soc. 339(2), 717–750 (1993)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Dodds, P., Dodds, T., Sukochev, F.: On \(p\)-convexity and \(q\)-concavity in non-commutative symmetric spaces. Integr. Equ. Oper. Theory 78, 91–114 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Dodds, P., de Pagter, B.: Normed Köthe spaces: a non-commutative viewpoint. Indag. Math. 25, 206–249 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Dodds, P., de Pagter, B., Sukochev, F.: Theory of noncommutative integration. unpublished manuscriptGoogle Scholar
  13. 13.
    Dykema, K., Sukochev, F., Zanin, D.: An upper triangular decomposition theorem for some unbounded operators affiliated to \(II_1\)-factors. Israel J. Math. 222(2), 645–709 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Fack, T., Kosaki, H.: Generalized \(s\)-numbers of \(\tau \)-measurable operators. Pacific J. Math. 123(2), 269–300 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Gohberg, I., Krein, M.: Introduction to the Theory of Linear Nonselfadjoint Operators. Translations of mathematical monographs, vol. 18. American Mathematical Society, Providence, R.I. (1969)zbMATHGoogle Scholar
  16. 16.
    Gohberg, I., Krein, M.: Theory and Applications of Volterra Operators in Hilbert Space. Translations of mathematical monographs, vol. 24. American Mathematical Society, Providence, R.I. (1970)zbMATHGoogle Scholar
  17. 17.
    Huang, J., Levitina, G., Sukochev, F.: Completeness of symmetric \(\Delta \)-normed spaces of \(\tau \)-measurable operators. Stud. Math. 237(3), 201–219 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Hudzik, H., Maligranda, L.: An interpolation theorem in symmetric function \(F\)-spaces. Proc. Am. Math. Soc. 110(1), 89–96 (1990)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Kadison, R., Ringrose, J.: Fundamentals of the Theory of Operator Algebras I. Academic Press, Orlando (1983)zbMATHGoogle Scholar
  20. 20.
    Kadison, R., Ringrose, J.: Fundamentals of the Theory of Operator Algebras II. Academic Press, Orlando (1986)zbMATHGoogle Scholar
  21. 21.
    Kalton, N., Peck, N., Rogers, J.: An F-space Sampler. London Math. Soc. Lecture Note Ser., vol.89, Cambridge University Press, Cambridge (1985)Google Scholar
  22. 22.
    Kalton, N., Sukochev, F.: Symmetric norms and spaces of operators. J. Reine Angew. Math. 621, 81–121 (2008)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Krein, S., Petunin, Yu., Semenov, E.: Interpolation of Linear Operators. Translated from Russian by J. Szũcs. Translations of Mathematical Monographs, Vol. 54. American Mathematical Society, Providence, R.I., (1982)Google Scholar
  24. 24.
    Lord, S., Sukochev, F., Zanin, D.: Singular traces: theory and applications, De gruyter studies in Mathematical. Physics 46, (2012)Google Scholar
  25. 25.
    Maligranda, L.: The \({\cal{K}}\)-functional for symmetric spaces. Lect. Notes Math. 1070, 169–182 (1984)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Maligranda, L., Ovchinnikov, V.: On interpolation between \(L_1+L_\infty \) and \(L_1\cap L_\infty \). J. Funct. Anal. 107, 342–351 (1992)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Maligranda, L., Persson, L.: The \(E\)-functional for some pairs of groups. Results Math. 20, 538–553 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Mitjagin, B.: An interpolation theorem for modular spaces. Mat. Sb. 66(108), 473–482 (1965)MathSciNetGoogle Scholar
  29. 29.
    Muratov, M., Chilin, V.: Algebras of measurable and locally measurable operators, Kyiv. Pratsi In-ty matematiki NAN Ukraini 69, (2007) (in Russian)Google Scholar
  30. 30.
    Nelson, E.: Notes on non-commutative integration. J. Funct. Anal. 15, 103–116 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Rotfel’d, S.: Analogues of the Interpolation Theorems of Mitjagin and Semenov for Operators in Non-normed Symmetric Spaces, Problems of Mathematical Analysis Izdat. Leningrad University, Leningrad (1973)Google Scholar
  32. 32.
    Segal, I.: A non-commutative extension of abstract integration. Ann. Math. 57, 401–457 (1953)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Simon, B.: Trace ideals and their applications, Second edition. Mathematical Surveys and Monographs, 120. American Mathematical Society, Providence, RI, (2005)Google Scholar
  34. 34.
    Sukochev, F.: Completeness of quasi-normed symmetric operator spaces. Indag. Math. 25, 376–388 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Sukochev, F., Tulenov, K., Zanin, D.: Nehari-type theorem for non-commutative Hardy spaces. J. Geom. Anal 27, 1789–1802 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Takesaki, M.: Theory of Operator Algebras I. Springer, New York (1979)CrossRefzbMATHGoogle Scholar
  37. 37.
    Takesaki, M.: Theory of Operator Algebras II. Springer, Berlin (2003)zbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.School of Mathematics and StatisticsUniversity of New South WalesKensingtonAustralia

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