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The strong limit of von Neumann subalgebras with conditional expectations


Author: Makoto Tsukada
Journal: Proc. Amer. Math. Soc. 94 (1985), 259-264
MSC: Primary 46L50; Secondary 46L10, 46L30
DOI: https://doi.org/10.1090/S0002-9939-1985-0784175-2
MathSciNet review: 784175
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Abstract: The strong lower limit and the weak upper limit of a net of von Neumann subalgebras on which the conditional expectations exist with respect to a fixed faithful normal state are defined. The limits coincide if and only if the corresponding conditional expectations converge strongly.


References [Enhancements On Off] (What's this?)

  • [1] H. Araki, Some properties of modular conjugation operator of von Neumann algebras and a non-commutative Radon-Nikodym theorem with a chain rule, Pacific J. Math. 50 (1974), 309-354. MR 0350437 (50:2929)
  • [2] F. Hiai and M. Tsukada, Strong martingale convergence of generalized conditional expectations on von Neumann algebras, Trans. Amer. Math. Soc. 282 (1984), 791-798. MR 732120 (85k:46070)
  • [3] M. Takesaki, Conditional expectations in von Neumann algebras, J. Funct. Anal. 9 (1972), 306-321. MR 0303307 (46:2445)
  • [4] M. Tsukada, Convergence of closed convex sets and $ \sigma $-fields, Z. Wahrsch. Verw. Gebiete 62 (1983), 137-146. MR 684213 (84e:60061)
  • [5] -, Strong convergence of martingales in von Neumann algebras, Proc. Amer. Math. Soc. 88 (1983), 537-540. MR 699429 (85b:46076)
  • [6] -, Convergence of best approximations in a smooth Banach space, J. Approx. Theory 40 (1984), 301-309. MR 740641 (86a:41034)
  • [7] H. Umegaki, Conditional expectation in an operator algebra, Tôhoku Math. J. (2) 6 (1954), 177-181. MR 0068751 (16:936b)
  • [8] -, Conditional expectation in an operator algebra. II, Tôhoku Math. J. (2) 8 (1956), 86-100. MR 0090789 (19:872b)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1985-0784175-2
Keywords: Faithful normal state, conditional expectation, martingale
Article copyright: © Copyright 1985 American Mathematical Society

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