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Torsion theories for finite von Neumann algebras | Lia Vas
; | Date: |
18 Feb 2007 | Journal: | L. Vas, Torsion Theories for Finite von Neumann Algebras,
Communications in Algebra 33 (2005), no. 3, 663 - 688 | Subject: | Rings and Algebras; Operator Algebras | Abstract: | 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 $(mathrm{{f T}},mathrm{{f P}})$ (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 $K_0({mathcal A}) o K_0({mathcal U}),$ 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. | Source: | arXiv, math/0702527 | Services: | Forum | Review | PDF | Favorites |
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