On the center-valued Atiyah conjecture for L2-Betti numbers

Abstract

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.