non-commutative conditional expectations and martingales arise inthe setting of von neumann algebras, which are the naturalframework for non-commutative measure theory and integration.analogues of classical martingale inequalities such asburkholder-gundy's square function inequalities and doob'sinequality have recently been established for martingales innon-commutative $l_p$-spaces by junge, pisier and xu. they alsoproved the analogue of the classical duality between $h^1$ and$bmo$ of martingales. we will discuss interpolation properties ofnon-commutative $bmo$ and show that it is a natural substitute for$l_infty,.$ as an application we establish boundedness ofnon-commutative martingale transforms.