Amenability of locally compact quantum groups and their unitary co-representations

We prove that amenability of a unitary co-representation $U$ of a locally compact quantum group passes to unitary co-representations that weakly contain $U$. This generalizes a result of Bekka, and answers affirmatively a question of B\'edos, Conti and Tuset. As a corollary, we extend to locally compact quantum groups a result of the first-named author, which characterizes amenability of a locally compact group $G$ by nuclearity of the reduced group $C^{*}$-algebra $C_{r}^{*}(G)$ and an additional condition.
Comment: 9 pages; a new co-author; added to Theorem 3.2 a second part about the case of a trivial scaling group, and gave an alternative proof of one of the implications; to appear in the Bulletin of the London Mathematical Society