Operator martingale decompositions and the Radon-Nikodym property in Banach spaces

We consider submartingales and uniform amarts of maps acting between a Banach lattice and a Banach lattice or a Banach space. In this measure-free setting of martingale theory, it is known that a Banach space Y has the Radon-Nikodym property if and only if every uniformly norm bounded martingale defined on the Chaney-Schaefer l-tensor product E (circle times) over tilde (t) Y, where E is a suitable Banach lattice, is norm convergent. We present applications of this result. Firstly, an analogues characterization for Banach lattices Y with the Radon-Nikodym property is given in terms of a suitable set of submartingales (supermartingales) on E (circle times) over tilde (t) Y. Secondly, we derive a Riesz decomposition for uniform amarts of maps acting between a Banach lattice and a Banach space. This result is used to characterize Banach spaces with the Radon-Nikodym property in terms of uniformly norm bounded uniform amarts of maps that are norm convergent. In the case 1 < p < infinity, our results yield L(p)(mu, Y)-space analogues of some of the well-known results oil uniform amarts in L(1) (mu, Y)-spaces. (C) 2009 Elsevier Inc. All rights reserved.