Stochastic integration in finite von Neumann algebras

Abstract

We investigate various aspects of stochastic integration in finite von Neumann algebras. For integration with respect to a bounded L2 -martingale the idea of treating the integral as a bounded operator is developed. Several classes of integrable processes are defined, it turns out that some of them form a Banach or C *-algebra. We find representations of these algebras and establish relations between the von Neumann algebras generated by these representations. Finally, we characterize the range of the stochastic integration operator.