Abstract
The study of modules over a finite von Neumann algebra
A
can be advanced by the use of torsion theories. In this work, some torsion theories for
A
are presented, compared and studied. In particular, we prove that the torsion theory
(T,P)
(in which a module is torsion if it is zero-dimensional) is equal to both Lambek and Goldie torsion theories for
A
.
Using torsion theories, we describe the injective envelope of a finitely generated projective
A
-module and the inverse of the isomorphism
K
0
(A)→
K
0
(U),
where
U
is the algebra of affiliated operators of
A.
Then, the formula for computing the capacity of a finitely generated module is obtained. Lastly, we study the behavior of the torsion and torsion-free classes when passing from a subalgebra
B
of a finite von Neumann algebra
A
to
A
. With these results, we prove that the capacity is invariant under the induction of a
B
-module.