Algebraic \(K\)-theory of von Neumann algebras
type: Item
instance of: scholarly article
title: Algebraic \(K\)-theory of von Neumann algebras
zbMATH Open document ID: 0802.19001
DOI: 10.1007/BF00961216
author: Mikael Rørdam
author: Wolfgang Lück
published in: \(K\)-Theory
publication date: 1994-12-15T00:00:00Z
review text: The paper is devoted to the computation of the algebraic \(K\)-group \(K_ 1({\mathcal A})\) and a closely related group \(K_ 1^ w ({\mathcal A})\) of a von Neumann algebra \({\mathcal A}\). First the authors define the algebraic \(K\)-group \(K_ 1 ({\mathcal A})\) as usual as being generated by bijective endomorphisms of finitely generated projective \({\mathcal A}\)-modules with certain relations, and the group \(K_ 1^ w ({\mathcal A})\) as being generated by injective endomorphisms of such modules and the same relations. Next, it is shown that it suffices to compute these groups in the case where \({\mathcal A}\) is of some fixed type \((\text{I}_ f\), \(\text{I}_ \infty\), \(\text{II}_ 1\), \(\text{II}_ \infty\) or III). In the type \(\text{I}_ f\) case, where \({\mathcal A}\) is a product of matrix algebras over abelian von Neumann algebras, the authors show that a certain normalization of the usual determinant induces a bijection of \(K_ 1 ({\mathcal A})\) with the invertible elements in the center of \({\mathcal A}\) and of \(K_ 1^ w ({\mathcal A})\) with the Grothendieck group of the semigroup of all elements \(a\) in a center of \({\mathcal A}\) such that multiplication with \(a\) is an injective mapping. Next, it is shown that for countably composable von Neumann algebras of type \(\text{II}_ 1\) the so called Fuglede-Kadison determinant defines an isomorphism of \(K_ 1 ({\mathcal A})\) with the positive invertible elements in the center of \({\mathcal A}\), and that in this case the group \(K_ 1^ w({\mathcal A})\) is trivial. Then the authors prove that in all other cases, i.e. for properly infinite von Neumann algebras both groups are trivial. Finally, it is shown how the results obtained so far can be used to detect nontrivial elements in the Whitehead groups of finite groups.
Mathematics Subject Classification ID: 19B99
Mathematics Subject Classification ID: 19K99
Mathematics Subject Classification ID: 46L80
zbMATH DE Number: 549706
zbMATH Keywords: determinants
zbMATH Keywords: von Neumann algebras
zbMATH Keywords: Whitehead groups
reviewed by: Andreas Cap
MaRDI profile type: Publication
cites work: Q3730469
cites work: Commutateurs dans le groupe unitaire d'un facteur
cites work: Sommes de commutateurs dans les algèbres de von Neumann finies continues
cites work: Isomorphism Conjectures in Algebraic K-Theory
cites work: Determinant theory in finite factors
cites work: Produits finis de commutateurs dans les \(C^*\)-algèbres
cites work: Q3736359
cites work: Heat kernels on covering spaces and topological invariants
cites work: \(L^ 2\)-topological invariants of 3-manifolds
cites work: Reidemeister torsion and the \(K\)-theory of von Neumann algebras
cites work: Introduction to Algebraic K-Theory. (AM-72)
cites work: Q4039791
cites work: Q3488732
cites work: Norms of Units in Group Rings
full work available at URL: https://doi.org/10.1007/bf00961216
OpenAlex ID: W2312832354
description: scientific article; zbMATH DE number 549706