Martingale inequalities in noncommutative symmetric spaces

Abstract

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