Almost Uniform Convergence in Noncommutative Dunford-Schwartz Ergodic Theorem for $p>1$

We prove that the ergodic Ces' aro averages generated by a positive Dunford-Schwartz operator in a noncommutative space $L^p(\mathcal{M},\tau )$, $1<p<\mathrm{\infty }$, converge almost uniformly (in Egorov’s sense). This problem goes back to the original paper of Yeadon \cite{ye}, where bilaterally almost uniform convergence of these averages was established for $p=1$.