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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
MathSciNet review:
784175
Full-text PDF Free Access
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Additional Information
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.
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(1983), no. 1, 137–146. MR 684213
(84e:60061), http://dx.doi.org/10.1007/BF00532167
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Makoto
Tsukada, Strong convergence of martingales in
von Neumann algebras, Proc. Amer. Math.
Soc. 88 (1983), no. 3, 537–540. MR 699429
(85b:46076), http://dx.doi.org/10.1090/S0002-9939-1983-0699429-6
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Makoto
Tsukada, Convergence of best approximations in a smooth Banach
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Hisaharu
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Tôhoku Math. J. (2) 8 (1956), 86–100. MR 0090789
(19,872b)
- [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
 -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:
http://dx.doi.org/10.1090/S0002-9939-1985-0784175-2
PII:
S 0002-9939(1985)0784175-2
Keywords:
Faithful normal state,
conditional expectation,
martingale
Article copyright:
© Copyright 1985 American Mathematical Society
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