abstract
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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