AlgebraicK-theory of von Neumann algebras |
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Authors: | Wolfgang Lück and Mikael R?rdam |
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Institution: | (1) Fachbereich Mathematik, Johannes Gutenberg-Universität, Saarstraße 21, 55099 Mainz, Bundesrepublik Deutschland;(2) Matematisk Institut, Odense Universitet, Campusvej 55, 5230 Odense, Denmark |
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Abstract: | To every von Neumann algebra, one can associate a (multiplicative) determinant defined on the invertible elements of the algebra with range a subgroup of the Abelian group of the invertible elements of the center of the von Neumann algebra. This determinant is a normalization of the usual determinant for finite von Neumann algebras of type I, for the type II1-case it is the Fuglede-Kadison determinant, and for properly infinite von Neumann algebras the determinant is constant equal to 1. It is proved that every invertible element of determinant 1 is a product of a finite number of commutators. This extends a result of T. Fack and P. de la Harpe for II1-factors. As a corollary, it follows that the determinant induces an injection from the algebraicK
1-group of the von Neumann algebra into the Abelian group of the invertible elements of the center. Its image is described. Another group,K
1
w
(A), which is generated by elements in matrix algebras overA that induce injective right multiplication maps, is also computed. We use the Fuglede-Kadison determinant to detect elements in the Whitehead group Wh(G).Partially supported by NSF Grant DMS-9103327. |
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Keywords: | AlgebraicK-theory von Neumann algebras Whitehead groups |
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