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Symmetric Functionals and Singular Traces

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Abstract

We study the construction and properties of positive linear functionals on symmetric spaces of measurable functions which are monotone with respect to submajorization. The construction of such functionals may be lifted to yield the existence of singular traces on certain non-commutative Marcinkiewicz spaces which generalize the notion of Dixmier trace.

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References

  1. S. Albeverio, D. Guido, A. Ponosov and S. Scarlatti, Singular traces and compact operators, J. Funct. Anal. 137(1996), 281-302.

    Google Scholar 

  2. M. Braverman and A. Mekler, The Hardy-Littlewood property for symmetric spaces, Sib. Mat. Zhur. 18(1977) 522-540 (Russian).

    Google Scholar 

  3. C. Bennett and R. Sharpley, Interpolation of Operators,Academic Press, 1988.

  4. A. Connes, Noncommutative Geometry, Academic Press, 1994.

  5. V. I. Chilin and F. A. Sukochev, Symmetric spaces over semifinite von Neumann algebras, Soviet Math. Dokl. 42(1991), 97-101.

    Google Scholar 

  6. V. I. Chilin and F. A. Sukochev, Weak convergence in non-commutative symmetric spaces, J. Operator Theory 31(1994), 35-65.

    Google Scholar 

  7. J. Dixmier, Existence de traces non normales, C. R. Acad. Sci. Paris 262(1966) A1107-A1108.

    Google Scholar 

  8. P. G. Dodds, T. K. Dodds and B. de Pagter, Non-commutative Banach function spaces, Math. Z. 201(1989), 583-597.

    Google Scholar 

  9. P. G. Dodds, T. K. Dodds and B. de Pagter, Fully symmetric operator spaces, Integr. Equat. Oper. Th. 15(1992), 942-972.

    Google Scholar 

  10. T. Fack and H. Kosaki, Generalized s-numbers of τ-measurable operators, Pacific J. Math. 123(1986), 269-300.

    Google Scholar 

  11. I. C. Gohberg and M. G. Krein, Introduction to the theory of non-selfadjoint operatorsTranslations of Mathematical Monographs, vol.18, AMS (1969).

  12. D. Guido and T. Isola, Singular traces on semifinite von Neumann algebras, J. Funct. Anal. 134(1995), 451-485.

    Google Scholar 

  13. A. Kaminska, Some remarks on Orlicz-Lorentz spaces, Math. Nachr. 147(1990) 29-38.

    Google Scholar 

  14. S. G. Krein, Ju. I. Petunin and E. M. Semenov, Interpolation of linear operators, Translations of Mathematical Monographs, Amer. Math. Soc. 54(1982).

  15. G. Ya. Lozanovskii, On localizable functionals in vector lattices,in: Theory of Functions, Functional Analysis and their Applications, 19(1974) 66-80 (Russian).

    Google Scholar 

  16. G. Ya. Lozanovskii, On the representation of linear functionals in Marcinkiewicz spaces,Izvestiya VUZov. Matematika N1 (188) (1978) 43–53. (Russian)

    Google Scholar 

  17. G. Ya. Lozanovskii, A supplement to the paper “On localizable functionals in vector lattices”, J. Soviet Math. 14(1980), 1170-1173.

    Google Scholar 

  18. E. Nelson, Notes on non-commutative integration, J. Functional Anal. 15(1974), 103-116.

    Google Scholar 

  19. R. Prinzis, Traces Residuelles et Asymptotique du Spectre d'Operateurs Pseudo-Differentiels, Ph. D. Thesis, Lyon, 1995.

  20. G. Russu, Symmetric spaces not having the majorant property, Mat. Issled. 4(1969), 82-93 (Russian).

    Google Scholar 

  21. A. A. Sedaev, A symmetric space which is intermediate for the couple {L 1 ;L } but which is not a subspace of any interpolation space, in Studies in the theory of functions of many real variables, Collection of papers, Yaroslavl (1990), (134-139).

  22. K. Sundaresan, Additive functionals on Orlicz spaces, Studia Math. 32(1969), 269-276.

    Google Scholar 

  23. M. Terp, L p -spaces associated with von Neumann algebras,Notes, Copenhagen Univ. (1981).

  24. J. V. Varga, Traces on irregular ideals, Proc. Amer. Math. Soc. 107(1989) 715-723.

    Google Scholar 

  25. A. C. Zaanen, Riesz Spaces II, (North-Holland), Amsterdam-New York-Oxford (1983).

    Google Scholar 

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Dodds, P., de Pagter, B., Semenov, E. et al. Symmetric Functionals and Singular Traces. Positivity 2, 47–75 (1998). https://doi.org/10.1023/A:1009720826217

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