Torsion theories for finite von Neumann algebras

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.