Pointwise convergence theorems in \(L_ 2\) over a von Neumann algebra
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.
von Neumann algebra with a faithful normal state
individual ergodic theorem for unitary operators
non- commutative extension of the Rademacher-Menshov theorem
scientific article; zbMATH DE number 3992200
scientific article