The crossed product of an operator algebra M by a group G of its automorphisms is a well known concept used for describing physical systems.
Let M be abelian, that is 
or 
where
X is a measure space or a locally compact topological space, respectively.
In both cases, any automorphism g of the algebra M corresponds to an
automorphism 
of X. Let
Then R is an equivalence relation on the set X. Many properties of the
crossed product 
depend on the equivalence relation R rather
than on the group G itself. So, the notion of the crossed product of an
operator algebra by an equivalence relation arises.
In the von Neumann algebra case (i. e. for a measure space) this construction was introduced and studied by J. Feldman and C. C. Moore [6, 7]. In particular, Feldman and Moore proved the structure theorem, i. e. gave necessary and sufficient conditions for a von Neumann algebra to be a crossed product of its Abelian subalgebra by some equivalence relation.
In the C*-algebra case (i. e. for a topological space) the construction was introduced and studied by J. Renault [18]. He has also proved the structure theorem in this case.
In both cases, non-selfadjoint subalgebras were studied by P. Muhly, B. Solel, and others. (See [15, 16].) In the case where the equivalence relation is hyperfinite, the so-called spectral theorem for bimodules was proved. This theorem gives a spectral description of subalgebras.
We introduced and studied a generalization of these constructions
to the case where the algebra M is not commutative, but rather it is
decomposed into a direct integral 
(in the von
Neumann algebra case) or arises from a continuous field
(in the C*-algebra case). For these cases we
defined and studied the crossed product of an algebra by an equivalence
relation. In each case we proved the structure theorem, analogous to the
theorem by Feldman and Moore and to the theorem by Renault, respectively.
Also, in each case we proved an alanogue of the spectral theorem for bimodules,
again under the condition of hyperfiniteness. The results concerning the
von Neumann algebra case are published in [9].