ABSTRACT

Non-commutative conditional expectations and martingales arise in the setting of von Neumann algebras, which are the natural framework for non-commutative measure theory and integration. Analogues of classical martingale inequalities such as Burkholder-Gundy's square function inequalities and Doob's inequality have recently been established for martingales in non-commutative Lp-spaces by Junge, Pisier and Xu. They also proved the analogue of the classical duality between H1 and BMO of martingales. We will discuss interpolation properties of non-commutative BMO and show that it is a natural substitute for L-infinity. As an application we establish boundedness of non-commutative martingale transforms.