Igor Fulman
September 1997
This report is devoted to the crossed products of von Neumann algebras and C* -algebras by equivalence relations, and to Fell bundles as a generalization of the C* -algebra crossed products.
Let D be a commutative operator algebra. If it is a von Neumann algebra, one
can assume that 
where 
is a measure space.
If D is a C* -algebra, one can assume that 
where X is a
locally compact topological space.
In either case let R be an equivalence relation on the set X with
countable cosets and possessing some additional properties (measurability or
some kind of continuity etc.). Then one can define an algebra
which is a crossed product of D by R. The algebra D
becomes the ``diagonal subalgebra'' of A.
In the case of von Neumann algebras, such a construction was introduced by Feldman and Moore [3,4] in 1977, in the case of C* -algebras --- it was introduced by Renault [13] in 1980.
In both cases, elements of the algebra 
are represented by
functions 
which are ``generalized matrices'' in the
following sense: for 
:
where c is some function with values in T which is a bicocycle.
Many properties of such algebras were studied in the articles mentioned above. In particular, structure theorems were proved, that is, given algebra A and its subalgebra D, there are sufficient and necessary conditions for A to be isomorphic to the crossed product of D by some equivalence relation R.
The conditions
involve: (a) the algebra D to be Abelian; (b) existence of a faithful
conditional expactation 
; (c) the normalizer
to be total in A.
If these conditions are satisfied, the subalgebra D is called a Cartan
subalgebra of 
.
In addition, many properties of such algebras were studied by Muhly, Solel and others [9,10,11,12] in 1988, 1989, ...
In particular, the spectral theorem was proved. This theorem allows to describe a D-bimodule of such an algebra as the set of all functions from A supported on some subset of R.
( Example: An algebra of upper triangular matrices contains all functions (matrices) supported on the upper triangle.)
There was also proved the theorem on dilations, i. e. on possibility to extend a representation of some (non-selfadjoint) subalgebra of A to a representation of the whole algebra A.
We studied an analogue of this theory in a more general situation where
the abelian operator algebra is replaced by the direct integral
(for von Neumann algebras) or by the algebra
generated by a continuous family of algebras 
(for
C* -algebras).
In these cases we succeeded to develop an analogous theory and to get the results analogous to the results in the commutative case.
In particular, the elements of the crossed product in these cases are
represented by functions on R such that for 
:
. Multiplication and involution are defined as follows:
For von Neumann algebras:
where 
is an isomorphism from 
onto 
, 
is a unitary operator.
For C* -algebras there are analogous formulas:
where 
is an automorphism of 
corresponding to the
conjugation by 
, and
is a unitary element in 
.
Thus, here also the elements are like ``generalized matrices''.
In both cases, the algebra D is embedded in the crossed product A as the algebra of functions supported on the diagonal of R.
In these cases also there exist structure theorems which give sufficient
and necessary conditions for an algebra 
to be isomorphic to the crossed
product of its subalgebra M by an equivalence relation.
We have also proved an analogue of the spectral theorem which allows to describe D-bimodules in A. For von Neumann algebras, D-bimodules in the crossed product are described by projections of the following abelian von Neumann algebra:
where 
, 
is the measure on R corresponding
(in some natural way) to the measure 
on X.
In the case of C* -algebras, D-bimodules in the crossed product are described by closed subsets of the set
where 
is a copy of the spectrum of the algebra 
, and
the union is equipped with some natural topology.
We have also proved an analogue of the theorem on dilations.
The results concerning von Neumann algebras are submitted to publication, and the article should appear in May 1997 (see [6]).
Now let us introduce shortly the Fell bundles.
Let R be an equivalence relation as above. Let 
be a Banach
bundle over R with the following properties:
, there exist a multiplication:
, there exists an involution:
is a C* -algebra. Such a bundle is called a Fell bundle.
Here, again, one can define the C* -algebra 
. Its
elements are represented by continuous sections of the bundle 
.
One can also define the ``diagonal subalgebra'' D as the set of functions
supported on the diagonal of R.
This construction was described and studied by A. Kumjian in 1994 in [8].
The spectral theorem holds in this case, too. (Joint with Professor P. Muhly.)
The D-bimodules in A in this case are described by the following
subbundles 
of the bundle 
: each fiber 
is
a subspace of the corresponding fiber 
, and in addition
is a 
-
-subbimodule in
.
This theorem is proved by reducing to the case of C* -algebra crossed products using some technique from [8] and so-called Haagerup tensor product (see [2], [1]).
The structure theorem (or ``coordinatization theorem'') is also proved in this case. This theorem gives necessary and sufficient conditions for a C* -algebra to be isomorphic to the C* -algebra generated by some Fell bundle, such that its given subalgebra is isomorpfic to the diagonal subalgebra.
The conditions in this case are more complicated. The algebra D is to be
generated by a continuous field of C* -algebras. Each fiber of this field
is naturally identified with an ideal of D (namely, the ideal of all sections
vanishing at this fiber). The normalizer here is the set of all elements a
such that 
takes an ideal of the above type to another ideal of
the above type.
The conditions involve existence of a faithful conditional expectation on D and the normalizer defined above to be total in A.
The results conserning the Fell bundle case are submitted to publication (see [7], [5]).
There are other constructions of crossed products of operator algebras by equivalence relations.
In particular, T. Yamanouchi [15] in 1993 described the construction of a crossed product of a von Neumann algebra by a groupoid. It turns to be that in the case where both the constructions work, they essentially coincide.
N. Sieben [14] in 1996 described the construction of an action of an inverse semigroup. Our construction appears to be the particular case of the Sieben's construction.