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!