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On the bundle convergence of double orthogonal series in noncommutative $L_2$-spaces

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The notion of bundle convergence in von Neumann algebras and their $L_2$-spaces for single (ordinary) sequences was introduced by Hensz, Jajte, and Paszkiewicz in 1996. Bundle convergence is stronger than almost sure convergence in von Neumann algebras. Our main result is the extension of the two-parameter Rademacher-Men'shov theorem from the classical commutative case to the noncommutative case. To our best knowledge, this is the first attempt to adopt the notion of bundle convergence to multiple series. Our method of proof is different from the classical one, because of the lack of the triangle inequality in a noncommutative von Neumann algebra. In this context, bundle convergence resembles the regular convergence introduced by Hardy in the classical case. The noncommutative counterpart of convergence in Pringsheim's sense remains to be found.
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Bibliografia
  • [1] R. P. Agnew, On double orthogonal series, Proc. London Math. Soc. (2) 33 (1932), 420-434.
  • [2] G. Alexits, Convergence Problems of Orthogonal Series, Pergamon Press, Oxford, 1961.
  • [3] J. Dixmier, Les algèbres d'opérateurs dans l'espace Hilbertien. (Algèbres de von Neumann), deuxième edition, Gauthier-Villars, Paris, 1969.
  • [4] G. H. Hardy, On the convergence of certain multiple series, Proc. Cambridge Philos. Soc. 19 (1916-1919), 86-95.
  • [5] E. Hensz and R. Jajte, Pointwise convergence theorems in $L^2$ over a von Neumann algebra, Math. Z. 193 (1986), 413-429.
  • [6] E. Hensz, R. Jajte, and A. Paszkiewicz, The bundle convergence in von Neumann algebras and their $L_2$-spaces, Studia Math. 120 (1996), 23-46.
  • [7] R. Jajte, Strong Limit Theorems in Non-Commutative Probability, Lecture Notes in Math. 1110, Springer, Berlin, 1985.
  • [8] R. Jajte, Strong Limit Theorems in Noncommutative $L_2$-Spaces, Lecture Notes in Math. 1477, Springer, Berlin, 1991.
  • [9] B. Le Gac and F. Móricz, Two-parameter SLLN in noncommutative $L_2$-spaces in terms of bundle convergence, J. Funct. Anal., submitted.
  • [10] F. Móricz, On the convergence in a restricted sense of multiple series, Anal. Math. 5 (1979), 135-147.
  • [11] F. Móricz, Some remarks on the notion of regular convergence of multiple series, Acta Math. Acad. Sci. Hungar. 41 (1983), 161-168.
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bwmeta1.element.bwnjournal-article-smv140i2p177bwm
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