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 .
Many properties of this construction are studied. In particular, the structure
theorem generalizing Theorem 1 in  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,