2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/208160We 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.Mistake in the proof of Theorem 6.1 is corrected. To appear in Munster Journal of Mathematics. 42 pagesOperator AlgebrasQuantum Algebra16W30,43A07, 46L89, 16E30text