On the bundle convergence of double orthogonal series in noncommutative L 2 -spaces

Ferenc Móricz; Barthélemy Le Gac

Studia Mathematica (2000)

  • Volume: 140, Issue: 2, page 177-190
  • ISSN: 0039-3223

Abstract

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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.

How to cite

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Móricz, Ferenc, and Le Gac, Barthélemy. "On the bundle convergence of double orthogonal series in noncommutative $L_2$-spaces." Studia Mathematica 140.2 (2000): 177-190. <http://eudml.org/doc/216761>.

@article{Móricz2000,
abstract = {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.},
author = {Móricz, Ferenc, Le Gac, Barthélemy},
journal = {Studia Mathematica},
keywords = {von Neumann algebra; faithful and normal state; completion; Gelfand-Naimark-Segal representation theorem; bundle convergence; almost sure convergence; regular convergence; orthogonal sequence of vectors in $L_2$; Rademacher-Men'shov theorem; convergence in Pringsheim's sense; orthogonal sequence of vectors in ; bundle convergence in von Neumann algebras; two-parameter Rademacher-Men'shov theorem},
language = {eng},
number = {2},
pages = {177-190},
title = {On the bundle convergence of double orthogonal series in noncommutative $L_2$-spaces},
url = {http://eudml.org/doc/216761},
volume = {140},
year = {2000},
}

TY - JOUR
AU - Móricz, Ferenc
AU - Le Gac, Barthélemy
TI - On the bundle convergence of double orthogonal series in noncommutative $L_2$-spaces
JO - Studia Mathematica
PY - 2000
VL - 140
IS - 2
SP - 177
EP - 190
AB - 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.
LA - eng
KW - von Neumann algebra; faithful and normal state; completion; Gelfand-Naimark-Segal representation theorem; bundle convergence; almost sure convergence; regular convergence; orthogonal sequence of vectors in $L_2$; Rademacher-Men'shov theorem; convergence in Pringsheim's sense; orthogonal sequence of vectors in ; bundle convergence in von Neumann algebras; two-parameter Rademacher-Men'shov theorem
UR - http://eudml.org/doc/216761
ER -

References

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  1. [1] R. P. Agnew, On double orthogonal series, Proc. London Math. Soc. (2) 33 (1932), 420-434. Zbl0004.10702
  2. [2] G. Alexits, Convergence Problems of Orthogonal Series, Pergamon Press, Oxford, 1961. 
  3. [3] J. Dixmier, Les algèbres d'opérateurs dans l'espace Hilbertien. (Algèbres de von Neumann), deuxième edition, Gauthier-Villars, Paris, 1969. Zbl0175.43801
  4. [4] G. H. Hardy, On the convergence of certain multiple series, Proc. Cambridge Philos. Soc. 19 (1916-1919), 86-95. 
  5. [5] E. Hensz and R. Jajte, Pointwise convergence theorems in L 2 over a von Neumann algebra, Math. Z. 193 (1986), 413-429. Zbl0613.46056
  6. [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. Zbl0856.46033
  7. [7] R. Jajte, Strong Limit Theorems in Non-Commutative Probability, Lecture Notes in Math. 1110, Springer, Berlin, 1985. Zbl0554.46033
  8. [8] R. Jajte, Strong Limit Theorems in Noncommutative L 2 -Spaces, Lecture Notes in Math. 1477, Springer, Berlin, 1991. 
  9. [9] B. Le Gac and F. Móricz, Two-parameter SLLN in noncommutative L 2 -spaces in terms of bundle convergence, J. Funct. Anal., submitted. Zbl0980.46043
  10. [10] F. Móricz, On the convergence in a restricted sense of multiple series, Anal. Math. 5 (1979), 135-147. Zbl0428.40001
  11. [11] F. Móricz, Some remarks on the notion of regular convergence of multiple series, Acta Math. Acad. Sci. Hungar. 41 (1983), 161-168. Zbl0525.40002

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