Annals of K-Theory

Localization, Whitehead groups and the Atiyah conjecture

Wolfgang Lück and Peter Linnell

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Let K1w(G) be the K1-group of square matrices over G which are not necessarily invertible but induce weak isomorphisms after passing to Hilbert space completions. Let D(G;) be the division closure of G 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((G)) is isomorphic to K1(D(G;)). Furthermore we show that D(G;) is a skew field and hence K1(D(G;)) is the abelianization of the multiplicative group of units in D(G;).

Article information

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

Received: 22 February 2016
Revised: 4 November 2016
Accepted: 27 November 2016
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 19B99: None of the above, but in this section
Secondary: 16S85: Rings of fractions and localizations [See also 13B30] 22D25: $C^*$-algebras and $W^*$-algebras in relation to group representations [See also 46Lxx]

localization algebraic $K$-theory Atiyah conjecture


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.

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