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 Bédos, 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 Cr∗(G) and an additional condition.