TORSION THEORIES FOR FINITE VON NEUMANN ALGEBRAS

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.

Key Words:

• Algebra of affiliated operators
• Finite von Neumann algebra
• Torsion theories

Mathematics Subject Classification:

• 16W99
• 46L99
• 16S90
• 19K99

ACKNOWLEDGMENTS

Part of the results were obtained during the time the author was at the University of Maryland, College Park. The author was supported by NSF grant DMS9971648 at that time.