The unconditional pointwise convergence of orthogonal seriesin L 2 over a von Neumann algebra

Ewa Hensz; Ryszard Jajte; Adam Paszkiewicz

Colloquium Mathematicae (1996)

  • Volume: 69, Issue: 2, page 167-178
  • ISSN: 0010-1354

Abstract

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The paper is devoted to some problems concerning a convergence of pointwise type in the L 2 -space over a von Neumann algebra M with a faithful normal state Φ [3]. Here L 2 = L 2 ( M , Φ ) is the completion of M under the norm x → | x | 2 = Φ ( x * x ) 1 / 2 .

How to cite

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Hensz, Ewa, Jajte, Ryszard, and Paszkiewicz, Adam. "The unconditional pointwise convergence of orthogonal seriesin $L_2$ over a von Neumann algebra." Colloquium Mathematicae 69.2 (1996): 167-178. <http://eudml.org/doc/210333>.

@article{Hensz1996,
abstract = {The paper is devoted to some problems concerning a convergence of pointwise type in the $L_2$-space over a von Neumann algebra M with a faithful normal state Φ [3]. Here $L_2 = L_2(M,Φ)$ is the completion of M under the norm $x → |x|^2 = Φ(x*x)^\{1/2\}$.},
author = {Hensz, Ewa, Jajte, Ryszard, Paszkiewicz, Adam},
journal = {Colloquium Mathematicae},
keywords = {-finite von Neumann algebra; faithful normal state; almost surely convergent; Tandori theorem},
language = {eng},
number = {2},
pages = {167-178},
title = {The unconditional pointwise convergence of orthogonal seriesin $L_2$ over a von Neumann algebra},
url = {http://eudml.org/doc/210333},
volume = {69},
year = {1996},
}

TY - JOUR
AU - Hensz, Ewa
AU - Jajte, Ryszard
AU - Paszkiewicz, Adam
TI - The unconditional pointwise convergence of orthogonal seriesin $L_2$ over a von Neumann algebra
JO - Colloquium Mathematicae
PY - 1996
VL - 69
IS - 2
SP - 167
EP - 178
AB - The paper is devoted to some problems concerning a convergence of pointwise type in the $L_2$-space over a von Neumann algebra M with a faithful normal state Φ [3]. Here $L_2 = L_2(M,Φ)$ is the completion of M under the norm $x → |x|^2 = Φ(x*x)^{1/2}$.
LA - eng
KW - -finite von Neumann algebra; faithful normal state; almost surely convergent; Tandori theorem
UR - http://eudml.org/doc/210333
ER -

References

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  1. [1] G. Alexits, Convergence Problems of Orthogonal Series, Pergamon Press, New York, 1961. 
  2. [2] C. J. K. Batty, The strong law of large numbers for states and traces of a W*-algebra, Z. Wahrsch. Verw. Gebiete 48 (1979), 177-191. Zbl0395.60033
  3. [3] O. Bratteli and D. W. Robinson, Operator Algebras and Quantum Statistical Mechanics I, Springer, New York, 1979. Zbl0421.46048
  4. [4] M. S. Goldstein, Theorems in almost everywhere convergence, J. Operator Theory 6 (1981), 233-311 (in Russian). Zbl0488.46053
  5. [5] E. Hensz, Strong laws of large numbers for orthogonal sequences in von Neumann algebras, in: Proc. Probability Theory on Vector Spaces IV, Łańcut 1987, Lecture Notes in Math. 1391, Springer, 1989, 112-124. 
  6. [6] E. Hensz and R. Jajte, Pointwise convergence theorems in L 2 over a von Neumann algebra, Math. Z. 193 (1986), 413-429. Zbl0613.46056
  7. [7] E. Hensz, R. Jajte and A. Paszkiewicz, Topics in pointwise convergence in L 2 over a von Neumann algebra, Quantum Probab. Related Topics 9 (1994), 239-271. 
  8. [8] R. Jajte, Strong limit theorems for orthogonal sequences in von Neumann algebras, Proc. Amer. Math. Soc. 94 (1985), 229-236. Zbl0601.46058
  9. [9] R. Jajte, Strong Limit Theorems in Noncommutative Probability, Lecture Notes in Math. 1110, Springer, Berlin, 1985. 
  10. [10] R. Jajte, Almost sure convergence of iterates of contractions in noncommutative L 2 -spaces, Math. Z. 205 (1990), 165-176. Zbl0739.46048
  11. [11] R. Jajte, Strong Limit Theorems in Noncommutative L 2 -Spaces, Lecture Notes in Math. 1477, Springer, Berlin, 1991. Zbl0743.46069
  12. [12] E. C. Lance, Ergodic theorem for convex sets and operator algebras, Invent. Math. 37 (1976), 201-214. Zbl0338.46054
  13. [13] W. Orlicz, Zur Theorie der Orthogonalreihen, Bull. Internat. Acad. Polon. Sci. Sér. A (1927), 81-115. Zbl53.0265.05
  14. [14] A. Paszkiewicz, Convergence in W*-algebras, J. Funct. Anal. 69 (1986), 143-154. Zbl0612.46060
  15. [15] I. E. Segal, A non-commutative extension of abstract integration, Ann. of Math. 57 (1953), 401-457. Zbl0051.34201
  16. [16] Ya. G. Sinai and V. V. Anshelevich, Some problems of non-commutative ergodic theory, Russian Math. Surveys 31 (1976), 157-174. Zbl0365.46053
  17. [17] K. Tandori, Über die orthogonalen Funktionen X (unbedingte Konvergenz), Acta Sci. Math. (Szeged) 23 (1962), 185-221. Zbl0144.32103

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