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2018 Localization, Whitehead groups and the Atiyah conjecture
Wolfgang Lück, Peter Linnell
Ann. K-Theory 3(1): 33-53 (2018). DOI: 10.2140/akt.2018.3.33


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;).


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Wolfgang Lück. Peter Linnell. "Localization, Whitehead groups and the Atiyah conjecture." Ann. K-Theory 3 (1) 33 - 53, 2018.


Received: 22 February 2016; Revised: 4 November 2016; Accepted: 27 November 2016; Published: 2018
First available in Project Euclid: 20 December 2017

zbMATH: 06775610
MathSciNet: MR3695363
Digital Object Identifier: 10.2140/akt.2018.3.33

Primary: 19B99
Secondary: 16S85 , 22D25

Keywords: algebraic $K$-theory , Atiyah conjecture , Localization

Rights: Copyright © 2018 Mathematical Sciences Publishers


Vol.3 • No. 1 • 2018
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