  In this paper, we provide a counterexample to show that in sharp contrast to
the classical case, the almost uniform convergence may not happen for truly
noncommutative $L_p$-martingales when $1\leq p<2$. The same happens to ergodic
averages. The proof consists of some sharp estimates of the distributional
function of a sequence of matrices and some non standard transference
techniques, which might admit further applications.
