## Annals of K-Theory

### Localization, Whitehead groups and the Atiyah conjecture

#### Abstract

Let $K1w(ℤG)$ be the $K1$-group of square matrices over $ℤG$ which are not necessarily invertible but induce weak isomorphisms after passing to Hilbert space completions. Let $D(G;ℚ)$ be the division closure of $ℚG$ in the algebra $U(G)$ of operators affiliated to the group von Neumann algebra. Let $C$ 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 $C$. Then we prove that $K1w(ℤ(G))$ is isomorphic to $K1(D(G;ℚ))$. Furthermore we show that $D(G;ℚ)$ is a skew field and hence $K1(D(G;ℚ))$ is the abelianization of the multiplicative group of units in $D(G;ℚ)$.

#### Article information

Source
Ann. K-Theory, Volume 3, Number 1 (2018), 33-53.

Dates
Revised: 4 November 2016
Accepted: 27 November 2016
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.akt/1513774601

Digital Object Identifier
doi:10.2140/akt.2018.3.33

Mathematical Reviews number (MathSciNet)
MR3695363

Zentralblatt MATH identifier
06775610

#### Citation

Lück, Wolfgang; Linnell, Peter. Localization, Whitehead groups and the Atiyah conjecture. Ann. K-Theory 3 (2018), no. 1, 33--53. doi:10.2140/akt.2018.3.33. https://projecteuclid.org/euclid.akt/1513774601

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