  We prove a pointwise ergodic theorem for actions of amenable groups on
noncommutative measure spaces. To do so, we establish the maximal ergodic
inequality for averages of operator-valued functions on amenable groups. The
main arguments are a geometric construction of martingales based on the
Ornstein-Weiss quasi-tilings and harmonic analytic estimates coming from
noncommutative Calder\'on-Zygmund theory.
