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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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Appl. Logic , 83 ( 1997 ), 225 – 247 . [6] B lanck , J. , Domain representations of topological spaces , Theoretical Comp. Sci. , 247 ( 2000 ), 229 – 255

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. J. 1981 Geometry of radix representations The Geometric Vein: The Coxeter Festschrift Springer New York 129 – 139

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, Electron. J. Linear Algebra , 30 ( 2015 ), 118 - 134 . [3] CHEN, J . L., Zhu , H. H. , Patrício , P. and Zhang , Y. L. , Characterizations and representations of core and dual core inverses , Canad

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, Linear and Multilinear Algebra , 14 ( 1983 ), 195 – 198 . [4] Castro-González , N. , and Dopazo , E. , Representations of the Drazin

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Proceedings of School-Seminar on the Spectral Theory of Operators and Representations of Group Theory Elm, Baku 46 – 71 [in Russian]. [11] Gasymov , M. G. 1968 Inverse

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] Besov , O. V. , Ilin , V. P. , Nikolskii , S. M. 1975 Integral Representations of Functions and Embedding Theorems Nauka Moscow . [6] Burkholder , D. L. 1983 A

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] Gelbart , S. Jacquet , H. 1978 A relation between automorphic representations of GL (2) and GL (3) Ann. Sci. École Norm. Sup. 11 471 – 542

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Networks , Appl. Comput. Harmon. Anal ., 6 ( 2 ) ( 1999 ), 197 – 218 . [6] Candès , E. J . and Donoho , D. L ., New tight frames of curvelets and optimal representations of objects with piesewise- C 2 singularities , Comm. Pure Appl. Math ., 57

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processes: compound and mixed Poisson representations , Journal of Applied Probability , 51 ( 2014 ), 19 – 36 . [5] Beurling , A

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