Geodesic
On the Center-Valued Atiyah Conjecture for $L^2$-Betti Numbers
Documenta mathematica, Tome 22 (2017), pp. 659-677.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

The so-called Atiyah conjecture states that the $\Cal N(G)$-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 $D(\Bbb{Q}[G])$ of $\Bbb{Q}[G]$ in the ring of affiliated operators. We prove the conjecture for all groups in Linnell's class $\frak{C}$, 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 $\Bbb{C}[G]$, the one generated as an additive group by the center-valued traces of all projections in $\Bbb{C}[H]$, where $H$ runs through the finite subgroups of $G$. Finally, we use the approximation theorem of Knebusch [15] for the center-valued $L^2$-Betti numbers to extend the result to many groups which are residually in $\frak{C}$, in particular for finite extensions of products of free groups and of pure braid groups.
Classification : 46L80, 20C07, 46L10, 47A58
Keywords: Atiyah conjecture, center-valued trace, von Neumann dimension, $L^2$-Betti numbers 
@article{DOCMA_2017__22__a34,
     author = {Knebusch, Anselm and Linnell, Peter and Schick, Thomas},
     title = {On the {Center-Valued} {Atiyah} {Conjecture} for $L^2${-Betti} {Numbers}},
     journal = {Documenta mathematica},
     pages = {659--677},
     publisher = {mathdoc},
     volume = {22},
     year = {2017},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DOCMA_2017__22__a34/}
}
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Knebusch, Anselm; Linnell, Peter; Schick, Thomas. On the Center-Valued Atiyah Conjecture for $L^2$-Betti Numbers. Documenta mathematica, Tome 22 (2017), pp. 659-677. http://geodesic.mathdoc.fr/item/DOCMA_2017__22__a34/