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We investigate various aspects of stochastic integration in finite von Neumann algebras. For integration with respect to a bounded L 2 -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.
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