Ergodic theorems for semifinite von Neumann algebras: II

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 L p 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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