Banach principle in the space of τ-measurable operators

Michael Goldstein; Semyon Litvinov

Studia Mathematica (2000)

  • Volume: 143, Issue: 1, page 33-41
  • ISSN: 0039-3223

Abstract

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We establish a non-commutative analog of the classical Banach Principle on the almost everywhere convergence of sequences of measurable functions. The result is stated in terms of quasi-uniform (or almost uniform) convergence of sequences of measurable (with respect to a trace) operators affiliated with a semifinite von Neumann algebra. Then we discuss possible applications of this result.

How to cite

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Goldstein, Michael, and Litvinov, Semyon. "Banach principle in the space of τ-measurable operators." Studia Mathematica 143.1 (2000): 33-41. <http://eudml.org/doc/216808>.

@article{Goldstein2000,
abstract = {We establish a non-commutative analog of the classical Banach Principle on the almost everywhere convergence of sequences of measurable functions. The result is stated in terms of quasi-uniform (or almost uniform) convergence of sequences of measurable (with respect to a trace) operators affiliated with a semifinite von Neumann algebra. Then we discuss possible applications of this result.},
author = {Goldstein, Michael, Litvinov, Semyon},
journal = {Studia Mathematica},
keywords = {von Neumann algebra; faithful semifinite normal trace; complete lattice of all projections; closed operator; topological -algebra of all -measurable operators; bilateral almost uniform convergence; noncommutative Banach principle; noncommutative ergodic theorems},
language = {eng},
number = {1},
pages = {33-41},
title = {Banach principle in the space of τ-measurable operators},
url = {http://eudml.org/doc/216808},
volume = {143},
year = {2000},
}

TY - JOUR
AU - Goldstein, Michael
AU - Litvinov, Semyon
TI - Banach principle in the space of τ-measurable operators
JO - Studia Mathematica
PY - 2000
VL - 143
IS - 1
SP - 33
EP - 41
AB - We establish a non-commutative analog of the classical Banach Principle on the almost everywhere convergence of sequences of measurable functions. The result is stated in terms of quasi-uniform (or almost uniform) convergence of sequences of measurable (with respect to a trace) operators affiliated with a semifinite von Neumann algebra. Then we discuss possible applications of this result.
LA - eng
KW - von Neumann algebra; faithful semifinite normal trace; complete lattice of all projections; closed operator; topological -algebra of all -measurable operators; bilateral almost uniform convergence; noncommutative Banach principle; noncommutative ergodic theorems
UR - http://eudml.org/doc/216808
ER -

References

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  1. [BJ] A. Bellow and R. L. Jones, A Banach Principle for L ∞ , Adv. Math. 120 (1996), 115-172. 
  2. [BR] O. Bratteli and D. N. Robinson, Operator Algebras and Quantum Statistical Mechanics, Springer, Berlin, 1979. 
  3. [DS] N. Dunford and J. T. Schwartz, Linear Operators I, Wiley, New York, 1958. 
  4. [FK] T. Fack and H. Kosaki, Generalized s-numbers of τ-mesurable operators, Pacific J. Math. 123 (1986), 269-300. Zbl0617.46063
  5. [Ga] A. Garsia, Topics in Almost Everywhere Convergence, Lectures in Adv. Math. 4, Markham, Chicago, 1970. 
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  7. [La] E. C. Lance, Non-commutative ergodic theory, in: Proc. Meeting on C*-Algebras and Their Applications to Theoretical Physics, Roma, 1975, 70-79. 
  8. [Li] S. Litvinov, On individual ergodic theorems with operator-valued Besicovitch's weights, to be submitted. 
  9. [LM] S. Litvinov and F. Mukhamedov, On individual subsequential ergodic theorem in von Neumann algebras, Studia Math, to appear. Zbl0973.46052
  10. [Ne] E. Nelson, Notes on non-commutative integration, J. Funct. Anal. 15 (1974), 103-116. Zbl0292.46030
  11. [Pa] A. Paszkiewicz, Convergences in W*-algebras, ibid. 69 (1986), 143-154. 
  12. [Se] I. Segal, A non-commutative extension of abstract integration, Ann. of Math. 57 (1953), 401-457. Zbl0051.34201
  13. [Ta] M. Takesaki, Theory of Operator Algebras I, Springer, Berlin, 1979. Zbl0436.46043
  14. [Ye] F. J. Yeadon, Ergodic theorems for semifinite von Neumann algebras I, J. London Math. Soc. (2) 16 (1977), 326-332. Zbl0369.46061

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