We investigate martingale inequalities
in noncommutative $L^p$-spaces associated with a von Neumann
algebra equipped with a faithful normal state. We prove the
noncommutative analogue of the classical Burkholder inequality
on the conditioned (or little) square function and extend the
noncommutative Burkholder--Gundy inequalities from Comm. Math. Phys.
189 (1997) 667--698 to this
nontracial setting. We include several related results.
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URBANA, ILLINOIS 61801 E-MAIL: junge@math.uiuc.edu WEB: http://www.math.uiuc.edu/ mjunge/ LABORATOIRE DE MATHÉMATIQUES UNIVERSITÉ DE FRANCHE-COMTÉ 25030 BESANÇON CEDEX FRANCE E-MAIL: qx@math.univ-fcomte.fr