ABSTRACT In this work the author introduces and studies the construction of the crossed product of a von Neumann algebra M decomposed into the direct integral M = Jx M(x)d/j,(x) by an equivalence relation on X with countable cosets. This construction is the generalization of the construction of the crossed prod- uct of an abelian von Neumann algebra by an equivalence relation introduced by J. Feldman and C. C. Moore in [15]. Many properties of this construction are studied. In particular, the structure theorem generalizing Theorem 1 in [15] is proved. The generalizations of the Spec- tral Theorem on Bimodules (see [25, Theorem 2.5]) and of the theorem on dilations (see [26, Theorem 1]) are proved too. 1991 Mathematics Subject Classification. 47D25, 46L10, 47A20. Key words and phrases. Von Neumann algebra, crossed product, equivalence relation, bimodule, subalgebra, dilation. IX

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