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On limit theorems in JW-algebras

Wybrane pełne teksty z tego czasopisma
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Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
In the present paper, we study bundle convergence in JW- algebra and prove certain ergodic theorems with respect to such convergence. Moreover, conditional expectations of reversible JW-algebras are considered. Using such expectations, the convergence of supermartingales is established.
Rocznik
Strony
153--165
Opis fizyczny
Bibliogr. 29 poz.
Twórcy
autor
  • Institute of Textile and Light Industry, Tashkent, 100100, Uzbekistan
  • Department of Computational & Theoretical Sciences, Faculty of Science, International Islamic University Malaysia, P.O. Box, 141, 25710, Kuantan Pahang, Malaysia
Bibliografia
  • [1] Sh. A. Ayupov, Martingale convergence and strong laws of large numbers in Jordan algebras, An. Univ. Craiova (Romania) Ser. Mat. Fiz.-Chim. 9 (1981), pp. 29-36.
  • [2] Sh. A. Ayupov, Statistical ergodic theorems in Jordan algebras, Uspekhi Mat. Nauk 36 (6) (1981), pp. 201-202.
  • [3] Sh. A. Ayupov, Ergodic theorems for Markov operators in Jordan algebras. I; II, Izv. Akad. Nauk UzSSR Ser. Fiz.-Mat. Nauk 3 (1982), pp. 12-15; 5 (1982), pp. 7-12.
  • [4] Sh. A. Ayupov, Extension of traces and type criterions for Jordan algebras of self-adjoint operators, Math. Z. 181 (1982), pp. 253-268.
  • [5] Sh. A. Ayupov, Supermartingales on Jordan algebras, in: Random Processes and Mathematical Statistics, Fan, Tashkent 1983, pp. 20-31.
  • [6] Sh. A. Ayupov, Classification and Representation of Ordered Jordan Algebras, Fan, Tashkent 1986.
  • [7] Sh. Ayupov, A. Rakhimov and Sh. Usmanov, Jordan; Real; and Lie Structures in Operator Algebras, Kluwer Academic Publishers, Dordrecht-Boston 1997.
  • [8] C. Barnett, Supermartingales on semifinite von Neumann algebras, J. London Math. Soc. 24 (1981), pp. 175-181.
  • [9] C. J. K. Batty, The strong laws of large numbers for state and traces of a W*-algebra, Z. Wahrsch. Verw. Gebiete 48 (1979), pp. 177-191.
  • [10] M. A. Berdikulov, Conditional expectations and martingales on Jordan algebras, Dokl. AN UzSSR 6 (1983), pp. 3-4.
  • [11] N. Etemadi and A. Lenzhen, Convergence of sequences of pairwise independent random variables, Proc. Amer. Math. Soc. 132 (2004), pp. 1201-1202.
  • [12] B. Le Gac and F. Moricz, Bundle convergence in von Neumann algebra and in a von Neumann subalgebra, Bull. Polish Acad. Sci. Math. 52 (2004), pp. 283-295.
  • [13] B. Le Gac and F. Moricz, Bundle convergence of sequences of pairwise uncorrelated operators in von Neumann algebras and vectors in their L2-spaces, Indag. Math. 17 (2006), pp. 221-230.
  • [14] M. Goldstein, Theorems of almost everywhere convergence in von Neumann algebras, J. Operator Theory 6 (1981), pp. 223-311.
  • [15] U. Haagerup and E. Stormer, Positive projections of von Neumann algebras onto JW-algebras, Rep. Math. Phys. 36 (1995), pp. 317-330.
  • [16] H. Hanche-Olsen and E. Stormer, Jordan Operator Algebras, Monographs and Studies Math. 21, Pitman, 1984.
  • [17] E. Hensz, R. Jajte and A. Paszkiewicz, The bundle convergence in von Neumann algebras and their L2-spaces, Studia Math. 120 (1996), pp. 23-46.
  • [18] R. Jajte, Strong Limit Theorems in Non-commutative Probability, Lecture Notes in Math. No 1110, Springer, Berlin 1985.
  • [19] A. K. Karimov, Convergence in JW-algebras and its enveloping von Neumann algebras, Siberian Adv. Math. 18 (2008), pp. 176-184.
  • [20] A. K. Karimov and F. M. Mukhamedov, An individual ergodic theorem with respect to a uniform sequence and the Banach principle in Jordan algebras, Sb. Math. 194 (2003), pp. 73-86.
  • [21] E. C. Lance, Ergodic theorem for convex sets and operator algebras, Invent. Math. 37 (1976), pp. 151-169.
  • [22] F. Mukhamedov, S. Temir and H. Akin, On asymptotical stability for positive L1-contractions of finite Jordan algebras, Siberian Adv. Math. 15 (2005), pp. 28-43.
  • [23] F. Mukhamedov, S. Temir and H. Akin, A note on dominant contractions of Jordan algebras, Turkish J. Math. (in press) (www.arxiv.org/arXiv:0806.2926).
  • [24] A. Paszkiewicz, Convergence in W*-algebras, J. Funct. Anal. 69 (1986), pp. 143-154.
  • [25] D. Petz, Quasi-uniform ergodic theorem in von Neumann algebras, Bull. London Math. Soc. 16 (1984), pp. 151-156.
  • [26] A. Skalski, On the unconditional bundle convergence in L2-space over a von Neumann algebra, Probab. Math. Statist. 22 (2002), pp. 19-27.
  • [27] O. Topping, Jordan algebras of self-adjoint operators, Mem. Amer. Math. Soc. 53 (1965), pp. 1-48.
  • [28] H. Umegaki, Conditional expectation in an operator algebra II, Tohoku Math. J. 8 (1956), pp. 86-100.
  • [29] F. J. Yeadon, Ergodic theorem for semifinite von Neumann algebras I, J. London Math. Soc. 16 (1977), pp. 326-332.
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Bibliografia
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