Abstract
Abstract 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.
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