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 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; ).

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/