Norm convergence of moving averages for τ-integrable operators

It is shown that if α is a positive linear map on L¹(M, τ) of a von Neumann algebra M with a faithful normal (semi-)finite trace τ which is norm-reducing for both the operator norm and the integral norm associated with τ, then the moving averages converge in L^p-norm, 1 ≤ p ≤ ∞. Using this result it has been shown that similar norm convergence results hold for some super-additive processes in L^p(M, τ) relative to τ-preserving α.