  Let G be a torsion free discrete group and let \bar{Q} denote the field of
algebraic numbers in C. We prove that \bar{Q}[G] fulfills the Atiyah conjecture
if G lies in a certain class of groups D, which contains in particular all
groups which are residually torsion free elementary amenable or which are
residually free. This result implies that there are no non-trivial
zero-divisors in C[G]. The statement relies on new approximation results for
L2-Betti numbers over \bar{Q}[G], which are the core of the work done in this
paper.
  Another set of results in the paper is concerned with certain number
theoretic properties of eigenvalues for the combinatorial Laplacian on
L2-cochains on any normal covering space of a finite CW complex.
  We establish the absence of eigenvalues that are transcendental numbers,
whenever the covering transformation group is either amenable or in the Linnell
class \mathcal{C}. We also establish the absence of eigenvalues that are
Liouville transcendental numbers whenever the covering transformation group is
either residually finite or more generally in a certain large bootstrap class
\mathcal{G}. Please take the errata to Schick: "L2-determinant class and
approximation of L2-Betti numbers" into account, which are added at the end of
the file, rectifying some unproved statements about "amenable extension". As a
consequence, throughout, amenable extensions should be extensions with normal
subgroups.
