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Topological Algebras of Measurable and Locally Measurable Operators

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Abstract

In this paper, we review the results on topological ∗-algebras S(M), S(M, τ), and LS(M) of measurable, τ -measurable, and locally measurable operators affiliated with the von Neumann algebra M. Also, we consider relations between those algebras for different classes of von Neumann algebras and establish the continuity of operator-valued functions with respect to the local convergence in measure. We describe maximal commutative ∗-subalgebras of the algebra LS(M) as well.

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References

  1. J.P. Antoine, A. Inoue, and C. Trapani, Partial ∗-Algebras and Their Operator Realizations, Kluwer Academic Publishers, Dordrecht (2002).

  2. A. F. Ber, V. I. Chilin, and F. A. Sukochev, “Continuous derivations on algebras of locally measurable operators are inner,” Proc. London Math. Soc., 109, No. 3, 65–89 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  3. A. M. Bikchentaev, “Local convergence in measure on semifinite von Neumann algebras,” Tr. Mat. Inst. Steklova, 255, 41–54 (2006).

  4. A. M. Bikchentaev, “Local convergence in measure on semifinite von Neumann algebras. II,” Mat. Zametki, 82, No. 5, 703–707 (2007).

  5. U. Bratelli and D. Robinson, Operator Algebras and Quantum Statistical Mechanics [Russian translation], Mir, Moscow (1982).

  6. V. I. Chilin and M.A. Muratov, “Comparison of topologies on ∗-algebras of locally measurable operators,” Positivity, 17, No. 1, 111–132 (2013).

  7. V. I. Chilin and M.A. Muratov, “Continuity of operator-valued functions in the ∗-algebra of locally measurable operators,” Methods Funct. Anal. Topology, 20, No. 2, 124–134 (2014).

  8. E. B. Davies, “A generalization of Kaplansky’s theory,” J. London Math. Soc., 4, 435-436 (1972).

    Article  MATH  Google Scholar 

  9. J. Dieudonne, Foundations of Modern Analysis, Acad. Press, New York–London (1960).

  10. J. Dixmier, Les Algebres d’Operateurs dans l’Espace Hilbertien (Algebres de von Neumann), Gauthier-Villars, Paris (1969).

  11. P.G. Dixon, “Unbounded operator algebras,” Proc. London Math. Soc., 23, No. 3, 53–59 (1973).

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

    Article  MathSciNet  MATH  Google Scholar 

  13. I. M. Gel’fand and M. A. Naymark, “On the inclusion of the normed ring into the ring of operators in a Hilbert space,” Mat. Sb., 12, 197–213 (1943).

    Google Scholar 

  14. R. V. Kadison, “Strong continuity of operator functions,” Pacific J. Math., 26, 121–129 (1968).

    Article  MathSciNet  MATH  Google Scholar 

  15. I. Kaplansky, “Projections in Banach algebras,” Ann. of Math., 53, 235–249 (1951).

    Article  MathSciNet  MATH  Google Scholar 

  16. I. Kaplansky, “A theorem on rings operators,” Pacific J. Math., 1, 227–232 (1951).

    Article  MathSciNet  MATH  Google Scholar 

  17. R. A. Kunce, “L p Fourier transforms on locally compact unimodular groups,” Trans. Amer. Math. Soc., 89, 519–540 (1958).

  18. M. A. Muratov and V. I. Chilin, “∗-Algebras of measurable and locally measurable operators affiliated with the von Neumann algebra,” Dopov. Nats. Akad. Nauk Ukr. Mat. Prirodozn. Tekh. Nauki, 9, 28–30 (2005).

    MATH  Google Scholar 

  19. M. A. Muratov and B. I. Chilin, “∗-Algebras of unbounded operators affiliated with the von Neumann algebra,” Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 326, 183–197 (2005).

  20. M. A. Muratov and V. I. Chilin, Algebras of Measurable and Locally Measurable Operators [in Russian], Pratsi In-t. Matem. NAN Ukraïni, Kiïv (2007).

  21. M. A. Muratov and V. I. Chilin, “∗-Algebras of unbounded operators affiliated with a von Neumann algebra,” J. Math. Sci. (N.Y.), 140, No. 3, 445–451 (2007).

  22. M. A. Muratov and V. I. Chilin, “Central extensions of ∗-algebra of bounded operators,” Dopov. Nats. Akad. Nauk Ukr., Mat. Prirodozn. Tekh. Nauky, 7, 24–28 (2009).

  23. G. J. Murphy, C ⋆ -Algebras and Operator Theory, Academic Press, Inc., New York–London (1990).

  24. F. J. Murrey and J. von Neumann, “On ring of operators,” Ann. of Math., 37, 116–229 (1936).

  25. F. J. Murrey and J. von Neumann, “On ring of operators. II,” Trans. Amer. Math. Soc., 41, 208–248 (1937).

  26. F. J. Murrey and J. von Neumann, “On ring of operators. IV,” Ann. of Math., 44, 716–808 (1943).

    Article  MathSciNet  Google Scholar 

  27. M. A. Naymark, Normed Rings [in Russian], Nauka, Moscow (1968).

  28. E. Nelson, “Notes on noncommutative integration,” J. Funct. Anal., 15, 103–116 (1974).

    Article  MATH  Google Scholar 

  29. J. von Neumann, “On ring of operators. III,” Ann. of Math., 41, 94–161 (1940).

    Article  MathSciNet  MATH  Google Scholar 

  30. A. R. Padmanabhan, “Convergence in measure and related results in finite rings of operators,” Trans. Amer. Math. Soc., 128, 359–388 (1967).

    Article  MathSciNet  MATH  Google Scholar 

  31. M. Reed and B. Simon, Methods of Modern Mathematical Physics. I. Functional Analysis, Academic Press, New York (1980).

  32. F. Riesz and B. Sz.-Nagy, Lectures in Functional Analysis [Russian translation], Mir, Moscow (1979).

  33. W. Rudin, Functional Analysis [Russian translation], Mir, Moscow (1975).

  34. S. Sakai, C ⋆ -Algebras and W ⋆ -Algebras, Springer, New York (1971).

  35. S. Sankaran, “The ∗-algebra of unbounded operators,” J. London Math. Soc., 343, 337–344 (1959).

  36. S. Sankaran, “Stochastic convergence for operators,” Q. J. Math., 2, No. 15, 97–102 (1964).

  37. T. A. Sarymsakov, Sh. A. Ayupov, D. Khadzhiev, and V. I. Chilin, Ordered Algebras [in Russian], FAN, Tashkent (1983).

  38. K. Schmudgen, Unbounded Operator Algebras an Representation Theory, Birkhäuser, Basel (1990).

  39. I.E. Segal, “A non-commutative extension of abstract integration,” Ann. of Math., 57, 401–457 (1953).

    Article  MathSciNet  MATH  Google Scholar 

  40. W. E. Stinespring, “Integration theorems for gages and duality for unimodular groups,” Trans. Amer. Math. Soc., 90, 15–56 (1959).

    Article  MathSciNet  MATH  Google Scholar 

  41. S. Strătilă and L. Zsidó, Lectures on von Neumann Algebras, Abacus Press, Bucharest (1979).

  42. F. A. Sukochev and V. I. Chilin, “The triangle inequality for measurable operators with respect to the Hardy–Littlewood ordering,” Izv. Akad. Nauk UzSSR, Ser. Fiz.-Mat. Nauk, 4, 44–50 (1988).

    MATH  Google Scholar 

  43. M. Takesaki, Theory of Operator Algebras. I, Springer, New York (1979).

  44. O.E. Tikhonov, “Continuity of operator functions in topologies connected to the trace on the Neumann algebra,” Izv. Vyssh. Uchebn. Zaved. Mat., 1, 77–79 (1987).

  45. F. J. Yeadon, “Convergence of measurable operators,” Math. Proc. Cambridge Philos. Soc., 74, 257–268 (1973).

    Article  MathSciNet  MATH  Google Scholar 

  46. F. J. Yeadon, “Non-commutative L p-spaces,” Math. Proc. Cambridge Philos. Soc., 77, 91–102 (1975).

    Article  MathSciNet  MATH  Google Scholar 

  47. B. S. Zakirov and V. I. Chilin, “Abstract characterization of EW ∗-algebras,” Funktsional. Anal. i Prilozhen., 25, No. 1, 76–78 (1991).

  48. B. S. Zakirov and V. I. Chilin, “Description of GB ∗-algebras that have a W ∗ algebra as a bounded part,” Uzbek. Mat. Zh., No. 2, 24–29 (1991).

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Authors and Affiliations

  1. V. I. Vernadsky Crimean Federal University, Simferopol, Russia

    M. A. Muratov

  2. M. Ulugbek National University of Uzbekistan, Tashkent, Uzbekistan

    V. I. Chilin

Authors
  1. M. A. Muratov
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  2. V. I. Chilin
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Correspondence to M. A. Muratov.

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Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 61, Proceedings of the Crimean Autumn Mathematical School-Symposium, 2016.

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Muratov, M.A., Chilin, V.I. Topological Algebras of Measurable and Locally Measurable Operators. J Math Sci 239, 654–705 (2019). https://doi.org/10.1007/s10958-019-04320-y

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