Torsion theories for group von Neumann algebras

The use of group Von Neumann algebras facilitates the study of homology of certain spaces. The study of modules over a group von Neumann algebra NG can be improved by the use of torsion theories. In this work, some torsion theories for NG are presented, compared and studied. The torsion and torsion-free classes of some of these theories are related to the classes studied by other authors. Using the torsion theories, the class of finitely generated modules over NG is described in more detail. From that description, a useful criterion for checking if a finitely generated NG -module is flat and the formula for computing its capacity are obtained. Also, the result on the isomorphism of K0 of NG and its algebra of affiliated operators UG is improved. Then, the behavior of the torsion and torsion-free classes of the torsion theories of interest under the induction of a module with respect to inclusion of a group von Neumann algebra of a subgroup of G in the algebra NG is studied. Using these results, the formula for the capacity of the induced module is improved.;The torsion theories for the algebra UG are studied as well. It is shown that they have the same properties as their analogues for NG plus some additional properties. These additional properties result from the ring-theoretic features of UG that NG does not necessarily have.;If certain torsion theories, different in general, are equal for a particular NG , then such NG and UG have some additional ring-theoretic properties. In particular, the necessary and sufficient conditions for UG to be semisimple are studied.;In the case of UG not being semisimple, the left and right global dimension of UG are calculated and an upper bound for the left and right global dimension of NG given. These results are proven under the assumption of the Continuum Hypothesis.;A group von Neumann algebra is just one example of a finite von Neumann algebra. Most of the results proven here for a group von Neumann algebra hold for any finite von Neumann algebra without any modification. A few of the results have to be modified slightly before stated and proven in this greater generality.