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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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I'll make the same question on mathoverflow as I did not receive any answer here. – Simone May 24 '13 at 19:09
This question has an answer on mathoverflow – Julian Kuelshammer Jun 21 '13 at 20:36
@JulianKuelshammer: in view of the Crusade of Answers, could you suggest a way to post a formal answer to this question, please? – Alexander Konovalov Jun 30 '13 at 8:40
I think that the best solution is a CW answer, when the answerer of the crosspost answer has not already duplicated their answer. If we spend too much time trying to get crosspost answererers to crosspost solutions, then I think we will spend too much time and energy, and secondly we might encourage some sort of rep-gaming that we don't want to develop. – rschwieb Jul 29 '13 at 12:53
up vote 1 down vote accepted

This is a CW answer designed to get this question out of the unanswered queue.

Mark Sapir's answer to the crosspost of this question on MO was accepted. Here it is:

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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