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We study weak amenability for locally compact quantum groups in the sense ofKustermans and Vaes. In particular, we focus on non-discrete examples. We provethat a coamenable quantum group is weakly amenable if there exists a net ofpositive, scaling invariant elements in the Fourier algebra A(G) whoserepresenting multipliers form an approximate identity in C_0(G) that is boundedin the M_0A(G) norm; the bound being an upper estimate for the associatedCowling-Haagerup constant.  As an application, we find the appropriate approximation properties of theextended quantum SU(1,1) group and its dual. That is, we prove that it isweakly amenable and coamenable. Furthermore, it has the Haagerup property inthe quantum group sense, introduced by Daws, Fima, Skalski and White.

Weak amenability of locally compact quantum groups and approximation properties of extended quantum SU(1,1)