  The so-called Atiyah conjecture states that the von Neumann dimensions of the
L2-homology modules of free G-CW-complexes belong to a certain set of rational
numbers, depending on the finite subgroups of G. In this article we extend this
conjecture to a statement for the center-valued dimensions. We show that the
conjecture is equivalent to a precise description of the tructure as a
semisimple Artinian ring of the division closure D(QG) of Q[G] in the ring of
affiliated operators. We prove the conjecture for all groups in Linnell's class
C, containing in particular free-by-elementary amenable groups.
  The center-valued Atiyah conjecture states that the center-valued L2-Betti
numbers of finite free G-CW-complexes are contained in a certain discrete
subset of the center of C[G], the one generated as an additive group by the
center-valued traces of all projections in C[H], where H runs through the
finite subgroups of G.
  Finally, we use the approximation theorem of Knebusch for the center-valued
$L^2$-Betti numbers to extend the result to many groups which are residually in
C, in particular for finite extensions of products of free groups and of pure
braid groups.
