Pointwise convergence theorems in \(L_ 2\) over a von Neumann algebra
type: Item
instance of: scholarly article
title: Pointwise convergence theorems in \(L_ 2\) over a von Neumann algebra
zbMATH Open document ID: 0613.46056
DOI: 10.1007/BF01229809
author: Ryszard Jajte
author: Ewa Hensz-Chądzyńska
published in: Mathematische Zeitschrift
publication date: 1986-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/173797
review text: 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.
Mathematics Subject Classification ID: 46L51
Mathematics Subject Classification ID: 46L53
Mathematics Subject Classification ID: 46L54
Mathematics Subject Classification ID: 46L55
Mathematics Subject Classification ID: 28D05
Mathematics Subject Classification ID: 81P20
Mathematics Subject Classification ID: 60F15
zbMATH DE Number: 3992200
zbMATH Keywords: von Neumann algebra with a faithful normal state
zbMATH Keywords: almost sure convergence
zbMATH Keywords: almost everywhere convergence
zbMATH Keywords: individual ergodic theorem for unitary operators
zbMATH Keywords: Cesàro means
zbMATH Keywords: ergodic Hilbert transform
zbMATH Keywords: non- commutative extension of the Rademacher-Menshov theorem
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3992200