There are many different characterisations of amenability for locally compact groups. One of them involves the amenability of the Banach algebra \(L^1(G)\). The amenability of the Fourier algebra \(A(G)\) does not characterise when a group \(G\) is amenable. Z.-J. Ruan [Am. J. Math. 117, No. 6, 1449–1474 (1995; Zbl 0842.43004)] showed instead that a group \(G\) is amenable if and only if the Fourier algebra \(A(G)\) is operator amenable, that is, amenable as an operator space, if and only if \(L^1(G)\) is operator amenable. This article describes completely for which compact quantum groups \(L^1(G)\) is operator amenable. The result disproves a {conjecture of V.Runde}.
The operator amenability of \(L^1(G)\) is related to operator biflatness because a completely contractive Banach algebra is operator amenable if and only if it is operator biflat and has a bounded approximate unit.
The main theorem of the article says that a compact quantum group \(G\) for which \(L^1(G)\) is operator biflat must be of Kac type. Using previous characterisations of co-amenability for Kac-type quantum groups, this implies that \(L^1(G)\) for a compact quantum group \(G\) is operator amenable if and only if \(G\) is co-amenable and of Kac type. Conversely, if \(G\) is compact and of Kac type, then \(L^1(G)\) is operator biprojective and thus operator biflat. Since \(L^1(G)\) can only be operator biprojective if \(G\) is compact, a locally compact quantum group \(G\) is compact and of Kac type if and only if \(L^1(G)\) is operator biprojective.