  We prove that a compact quantum group is coamenable if and only if its
corepresentation ring is amenable. We further propose a Foelner condition for
compact quantum groups and prove it to be equivalent to coamenability. Using
this Foelner condition, we prove that for a coamenable compact quantum group
with tracial Haar state, the enveloping von Neumann algebra is dimension flat
over the Hopf algebra of matrix coefficients. This generalizes a theorem of
Lueck from the group case to the quantum group case, and provides examples of
compact quantum groups with vanishing L^2-Betti numbers.
