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!



Amenable group rings embeddable in skew fields
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. 


