Operator biflatness of the L-1-algebras of compact quantum groups

We prove that the L-1-algebra of any non-Kac type compact quantum group is not operator biflat. Since operator amenability implies operator biflatness, this result shows that any co-amenable, non-Kac type compact quantum group gives a counterexample to the conjecture that L-1 (G) is operator amenable if and only if G is amenable and co-amenable for any locally compact quantum group G. The result also implies that the L-1-algebra of a locally compact quantum group is operator biprojective if and only if G is compact and of Kac type.