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

The so-called Atiyah conjecture states that the $\NG$-dimensions of
the $L^{2}$-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 $L^{2}$-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 $L^{2}$-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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