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Pointwise convergence theorems in \(L_ 2\) over a von Neumann algebra. (English) Zbl 0613.46056
Let M be a von Neumann algebra with a faithful normal state \(\phi\). We consider the Hilbert space \(H=L_ 2(M,\phi)\) (the completion of M under the norm \(x\mapsto \phi (x^*x)^{1/2})\). In this space we introduce a notion of the almost sure convergence which coincides with the usual almost everywhere convergence in the case \(M=L_{\infty}\) (over a probability space).
Using this type of convergence we prove an individual ergodic theorem for unitary operators in H induced by *-automorphisms of M. We also give some (almost sure) asymptotic formulae for the Cesàro means and the ergodic Hilbert transform of a unitary operator in H. A non-commutative extension of the Rademacher-Menshov theorem is proved as well.

MSC:
46L51 Noncommutative measure and integration
46L53 Noncommutative probability and statistics
46L54 Free probability and free operator algebras
46L55 Noncommutative dynamical systems
28D05 Measure-preserving transformations
81P20 Stochastic mechanics (including stochastic electrodynamics)
60F15 Strong limit theorems
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