Non-Commutative Martingale Inequalities

Abstract:

We prove the analogue of the classical Burkholder-Gundy inequalites for non-commutative martingales. As applications we give a characterization for an Ito-Clifford integral to be an L ^p-martingale via its integrand, and then extend the Ito-Clifford integral theory in L ², developed by Barnett, Streater and Wilde, to L ^p for all 1<p<∞. We include an appendix on the non-commutative analogue of the classical Fefferman duality between $H ¹ and BMO.