#### Vol. 3, No. 1, 2018

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Localization, Whitehead groups and the Atiyah conjecture

### Wolfgang Lück and Peter Linnell

Vol. 3 (2018), No. 1, 33–53
##### Abstract

Let ${K}_{1}^{w}\left(ℤG\right)$ be the ${K}_{1}$-group of square matrices over $ℤG$ which are not necessarily invertible but induce weak isomorphisms after passing to Hilbert space completions. Let $\mathsc{D}\left(G;ℚ\right)$ be the division closure of $ℚG$ in the algebra $\mathsc{U}\left(G\right)$ of operators affiliated to the group von Neumann algebra. Let $\mathsc{C}$ be the smallest class of groups which contains all free groups and is closed under directed unions and extensions with elementary amenable quotients. Let $G$ be a torsionfree group which belongs to $\mathsc{C}$. Then we prove that ${K}_{1}^{w}\left(ℤ\left(G\right)\right)$ is isomorphic to ${K}_{1}\left(\mathsc{D}\left(G;ℚ\right)\right)$. Furthermore we show that $\mathsc{D}\left(G;ℚ\right)$ is a skew field and hence ${K}_{1}\left(\mathsc{D}\left(G;ℚ\right)\right)$ is the abelianization of the multiplicative group of units in $\mathsc{D}\left(G;ℚ\right)$.

##### Keywords
localization, algebraic $K$-theory, Atiyah conjecture
##### Mathematical Subject Classification 2010
Primary: 19B99
Secondary: 16S85, 22D25
##### Milestones
Received: 22 February 2016
Revised: 4 November 2016
Accepted: 27 November 2016
Published: 7 September 2017
##### Authors
 Wolfgang Lück Mathematisches Institut Rheinische Wilhelms-Universität Bonn Endenicher Allee 60 D-53115 Bonn Germany http://www.him.uni-bonn.de/lueck Peter Linnell Department of Mathematics Virginia Tech Blacksburg, VA 24061-0123 United States http://www.math.vt.edu/people/plinnell/