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Ergodic theorems for semifinite von Neumann algebras: II

Published online by Cambridge University Press:  24 October 2008

F. J. Yeadon
Affiliation:
University of Hull

Extract

In (7) we proved maximal and pointwise ergodic theorems for transformations a of a von Neumann algebra which are linear positive and norm-reducing for both the operator norm ‖ ‖ and the integral norm ‖ ‖1 associated with a normal trace ρ on . Here we introduce a class of Banach spaces of unbounded operators, including the Lp spaces defined in (6), in which the transformations α reduce the norm, and in which the mean ergodic theorem holds; that is the averages

converge in norm.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1980

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References

REFERENCES

(1)Fremlin, D. H.Stable subspaces of L 1 + L . Proc. Cambridge Philos. Soc. 64 (1968), 625643.Google Scholar
(2)Luxemburg, W. A. J. and Zaanen, A. C.Notes on Banach function spaces. I-IV; XIII. Nederl. Akad. Wetensch. 66 (A) (1963); 67 (A) (1964).Google Scholar
(3)Segal, I. E.A non-commutative extension of abstract integration. Ann of Math. 57 (1952), 401457.Google Scholar
(4)Yeadon, F. J. Modules of measurable operators. Dissertation, Cambridge, 1968.Google Scholar
(5)Yeadon, F. J.Convergence of measurable operators, Proc. Cambridge Philos. Soc. 74 (1973), 257268.CrossRefGoogle Scholar
(6)Yeadon, F. J.Non-commutative -L-spaces, Proc. Cambridge Philos. Soc. 77 (1975), 91102.Google Scholar
(7)Yeadon, F. J.Ergodic theorems for semifinite von Neumann algebras: I. J. London Math. Soc. (2) 16 (1977), 326332.CrossRefGoogle Scholar