Martingale inequalities in noncommutative symmetric spaces

We investigate the Burkholder–Gundy inequalities in a noncommutative symmetric space E(M){E(\mathcal{M})} associated with a von Neumann algebra M{\mathcal{M}} equipped with a faithful normal state. The results extend the Pisier–Xu noncommutative martingale inequalities, and generalize the classical inequalities in the commutative case.