On the operator space UMD property and non-commutative martingale inequalities.

摘要

We prove that for 1 ≤ p q infinity the analogue of the classical result BMO,Lp pq = Lq holds in the setting of a finite von Neumann algebra M , equipped with an increasing filtration ( M n)n≥1 of von Neumann subalgebras. We also obtain the corresponding results for the real method of interpolation. We discuss the appropriate operator space matrix norms and show that these interpolation results hold in the category of operator spaces. We apply further interpolation techniques to the study of the operator space UMD property, introduced by Pisier in the context of non-commutative vector-valued Lp-spaces, associated to a hyperfinite (and finite) von Neumann algebra. We discuss basic stability properties of UMDp operator spaces. It is unknown whether the property is independent of p in this setting. We show that for 1 p, q infinity, the Schatten q-classes Sq are UMDp. We provide further examples of UMDp (independent of p) operator spaces, including the non-commutative Lorentz spaces associated to a hyperfinite (and finite) von Neumann algebra.