The study of modules over a finite von Neumann algebra ${ \mathcal A}$ can be advanced by the use of torsion theories. In this work, some torsion theories for ${ \mathcal A}$ are presented, compared and studied. In particular, we prove that the torsion theory $(\mathrm{ { \bf T}},\mathrm{ { \bf P}})$ (in which a module is torsion if it is zero-dimensional) is equal to both Lambek and Goldie torsion theories for ${ \mathcal A}$. Using torsion theories, we describe the injective envelope of a finitely generated projective ${ \mathcal A}$-module and the inverse of the isomorphism $K_0({ \mathcal A})\to K_0({ \mathcal U}),$ where ${ \mathcal U}$ is the algebra of affiliated operators of ${ \mathcal A}.$ 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 ${ \mathcal B}$ of a finite von Neumann algebra ${ \mathcal A}$ to ${ \mathcal A}$. With these results, we prove that the capacity is invariant under the induction of a ${ \mathcal B}$-module.
Categories: math.RA math.OA