Amenable group rings embeddable in skew fields

I've made this question on math.stackexchange.com (also offering a bounty) but I did not receive any answer:

I'm looking for a reference of the following fact:

given a (countable?) amenable group $G$ and a (skew) field $K$, the following are equivalent:

(1) the group ring $K[G]$ is a domain;

(2) $K[G]$ is a (left and right) Ore domain.

I think to remember that this result is due to Beno Eckmann but, unfortunately, I cannot remember in which paper. I tried to look for this result and I'm not able to find it at the moment. Any reference would be strongly appreciated!

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for completeness, here the link to the math.SE question: #391253 – Julian Kuelshammer Jun 21 '13 at 20:40

1 Answer

See here: Lück, Wolfgang, L2-invariants: theory and applications to geometry and K-theory. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], 44. Springer-Verlag, Berlin, 2002. xvi+595 pp., Example 8.16 on page 324.

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