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Ergodic theorems for semifinite von Neumann algebras: II

  • F. J. Yeadon (a1)

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.

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(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).
(3)Segal I. E. A non-commutative extension of abstract integration. Ann of Math. 57 (1952), 401457.
(4)Yeadon F. J. Modules of measurable operators. Dissertation, Cambridge, 1968.
(5)Yeadon F. J. Convergence of measurable operators, Proc. Cambridge Philos. Soc. 74 (1973), 257268.
(6)Yeadon F. J. Non-commutative -L-spaces, Proc. Cambridge Philos. Soc. 77 (1975), 91102.
(7)Yeadon F. J. Ergodic theorems for semifinite von Neumann algebras: I. J. London Math. Soc. (2) 16 (1977), 326332.
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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
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