Abstract :
∞k =1Gk = {1}. Let A ∈ Matd×d (CG) and Ak ∈ Matd×d (C(G/Gk)) be the images of A under the maps
induced by the epimorphisms G→G/Gk. According to the strong form of the Approximation Conjecture
of Lück [W. Lück, L2-Invariants: Theory and Applications to Geometry and K-theory, Ergeb. Math.
Grenzgeb. (3), vol. 44, Springer-Verlag, Berlin, 2002]
dimG(ker A) = lim
k→∞
dimG/Gk (kerAk),
where dimG denotes the von Neumann dimension. In [J. Dodziuk, P. Linnell, V. Mathai, T. Schick, S. Yates,
Approximating L2-invariants and the Atiyah conjecture, Comm. Pure Appl. Math. 56 (7) (2003) 839–873]
Dodziuk et al. proved the conjecture for torsion free elementary amenable groups. In this paper we extend
their result for all amenable groups, using the quasi-tilings of Ornstein and Weiss [D.S. Ornstein, B. Weiss,
Entropy and isomorphism theorems for actions of amenable groups, J. Anal. Math. 48 (1987) 1–141].
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