Failure of almost uniformly convergence for noncommutative martingales

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 $1\le p<2$. 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.