  In this paper we study Johnson-Schechtman inequalities for noncommutative
martingales. More precisely, disjointification inequalities of noncommutative
martingale difference sequences are proved in an arbitrary symmetric operator
space $E(\mathcal{M})$ of a finite von Neumann algebra $\mathcal{M}$ without
making any assumption on the Boyd indices of $E$. We show that we can obtain
Johnson-Schechtman inequalities for arbitrary martingale difference sequences
and that, in contrast with the classical case of independent random variables
or the noncommutative case of freely independent random variables, the
inequalities are one-sided except when $E=L_2(0,1)$. As an application, we
partly resolve a problem stated by Randrianantoanina and Wu in \cite{NW}. We
also show that we can obtain sharp $\Phi$-moment analogues for Orlicz functions
satisfying $p$-convexity and $q$-concavity for $1 \leq p \leq 2$, $q=2$ and
$p=2$, $2< q < \infty$. This is new even for the classical case. We also extend
and strengthen the noncommutative Burkholder-Gundy inequalities in symmetric
spaces and in the $\Phi$-moment case.
