We, thus, obtain a class of algebras equipped with coordinates (with "non com-
mutative fibers"). This allows us to represent the elements of such an algebra
as "generalized operator matrices". In Sections 10, 11 and 13 we use these co-
ordinates to study the automorphisms and isomorphisms of these von Neumann
algebras. We are also able to describe the analytic algebras associated with flows
of automorphisms (see Section 16).
In our case, under the condition that the equivalence relation is hyperfinite, we
also prove the spectral theorem on bimodules (see Theorem 15.18 below). As in
the paper [25], we use this theorem to describe subalgebras of the crossed product
algebras (in particular, non-selfadjoint subalgebras).
In Section 18 we study the case where the equivalence relation is hyperfinite.
We prove that, in this case, every contractive a-weakly continuous representation
of a cr-Dirichlet subalgebra is completely contractive, hence has a dilation to a
representation of the von Neumann algebra. In the Feldman-Moore setting this
was proved by Muhly and Solel in [26]. Such a dilation property was used in the
study of representations of non selfadjoint algebras (see [1, 26, 28, 11]).
Finally, we show the interrelations of our construction with the construction
of the crossed product of a von Neumann algebra by a groupoid, introduced by
Takehiko Yamanouchi in his paper [37]. His construction is different from ours, and
he studied not only actions of groupoids, but also the dual coactions. Nevertheless,
it turns to be that when our case and the case of Yamanouchi "intersect", the
construction of Yamanouchi in essential coincides with our construction.
We wish to note that our algebras are also a special case of algebras obtained
by Fell bundles over measured equivalence relations (see [36]).
The author expresses his sincere graditude to his scientific superviser, Professor
Baruch Solel, for his helpful guidance and constant encouragement. The author
also wish to thank Professor P. Muhly for several helpful discussions.
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