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Martingale convergence in von Neumann algebras

Published online by Cambridge University Press:  24 October 2008

E. Christopher Lance
E. Christopher Lance
Affiliation:
University of Manchester
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Extract

Let N be a von Neumann subalgebra of a von Neumann algebra M. A linear mapping π: M → N is called a retraction if it is idempotent and has norm one. By a result of Tomiyama(15) a retraction is a positive mapping and is a module homo-morphism over N. A retraction is normal if it is ultraweakly continuous, and faithful if it does not annihilate any nonzero positive element of M. Suppose that (Nn)n≥1 is an increasing sequence of von Neumann subalgebras of M whose union is weakly dense in M and that, for each n, πn: M → Nn is a faithful normal retraction. The sequence (πn) is called a martingale if, whenever m ≥ n,

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 84 , Issue 1 , July 1978 , pp. 47 - 56
Copyright
Copyright © Cambridge Philosophical Society 1978

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References

REFERENCES

(1)Baez-Duarte, L.Another look at the martingale theorem. J. Math. Anal. Appl. 23 (1968), 551–558.CrossRefGoogle Scholar
(2)Connes, A.Une classification des facteurs de type III. Ann. Sci. École Norm. Sup. 6 (1973), 133–252.CrossRefGoogle Scholar
(3)Connes, A.On hyperfinite factors of type III and Krieger's factors. J. Functional Analysis 18 (1975), 318–327.CrossRefGoogle Scholar
(4)Cuculescu, I.Martingales on von Neumann algebras. J. Multivariate Anal. 1 (1971), 17–27.CrossRefGoogle Scholar
(5)Dixmier, J.Lea algèbres d'opérateurs dans l'espace Hilbertien, 2nd ed. (Paris, Gauthier-Villars, 1969).Google Scholar
(6)Effros, E. and Lance, C.Tensor products of operator algebras. Advances in Math. 25, (1977), 1–34.CrossRefGoogle Scholar
(7)Isaac, R.A proof of the martingale convergence theorem. Proc. Amer. Math. Soc. 16 (1965), 842–844.Google Scholar
(8)Lance, C.Ergodic theorems for convex sets and operator algebras. Invent. Math. 37 (1976), 201–214.CrossRefGoogle Scholar
(9)Radin, C. Pointwise ergodic theory and operator algebras (to appear).Google Scholar
(10)Sakai, S. C*-algebras and W*-algebras (Berlin, Springer-Verlag, 1971).Google Scholar
(11)Segal, I.A non-commutative extension of abstract integration. Ann. of Math. 57 (1953), 401–457.CrossRefGoogle Scholar
(12)Stinespring, F.Positive functions on C*-algebras. Proc. Amer. Math. Soc. 6 (1955), 211–216.Google Scholar
(13)Takesaki, M.Conditional expectations in von Neumann algebras. J. Functional Analysis 9 (1972), 306–321.CrossRefGoogle Scholar
(14)Takesaki, M.Duality for crossed products and the structure of von Neumann algebras of type III. Acta Math. 131 (1973), 243–310.CrossRefGoogle Scholar
(15)Tomiyama, J.On the projection of norm one in W*-algebras. Proc. Japan Acad. 33 (1957), 608–612.Google Scholar
(16)Tomiyama, J.On the projection of norm one in W*-algebras III. Tohoku Math. J. 11 (1959), 125–129.Google Scholar
(17)Wassermann, S.Injective W*-algebras. Math. Proc. Cambridge Philos. Soc. 82 (1977), 39–47.CrossRefGoogle Scholar