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.
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 |
Keywords:
von Neumann algebra with a faithful normal state; almost sure convergence; almost everywhere convergence; individual ergodic theorem for unitary operators; Cesàro means; ergodic Hilbert transform; non- commutative extension of the Rademacher-Menshov theoremReferences:
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