Localization, Whitehead groups and the Atiyah conjecture

Abstract

Let $K_1^w\left(\mathbb{Z}G\right)$ be the $K_1$-group of square matrices over $\mathbb{Z}G$ which are not necessarily invertible but induce weak isomorphisms after passing to Hilbert space completions. Let $\mathcal{D}\left(G;\mathbb{Q}\right)$ be the division closure of $\mathbb{Q}G$ in the algebra $\mathcal{U}\left(G\right)$ of operators affiliated to the group von Neumann algebra. Let $\mathcal{C}$ 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 $\mathcal{C}$. Then we prove that $K_1^w\left(\mathbb{Z}\left(G\right)\right)$ is isomorphic to $K_1\left(\mathcal{D}\left(G;\mathbb{Q}\right)\right)$. Furthermore we show that $\mathcal{D}\left(G;\mathbb{Q}\right)$ is a skew field and hence $K_1\left(\mathcal{D}\left(G;\mathbb{Q}\right)\right)$ is the abelianization of the multiplicative group of units in $\mathcal{D}\left(G;\mathbb{Q}\right)$.