On the Center-Valued Atiyah Conjecture for L^2-Betti Numbers

The so-called Atiyah conjecture states that the $\NG$-dimensions of the $L²$-homology modules of finite 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 structure as a semisimple Artinian ring of the division closure $D(\{Q}[G])$ of $\Q[G]$ in the ring of affiliated operators. We prove the conjecture for all groups in Linnell's class $\CCC$, 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 [Knebusch] for the center-valued $L²$-Betti numbers to extend the result to many groups which are residually in $\CCC$, in particular for finite extensions of products of free groups and of pure braid groups.

2010 Mathematics Subject Classification: Primary: 46L80. Secondary: 20C07, 46L10, 47A58

Keywords and Phrases: Atiyah conjecture, center-valued trace, von Neumann dimension, L^2-Betti numbers

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