Completely positive multipliers of quantum groups

We show that any completely positive multiplier of the convolution algebra of the dual of an operator algebraic quantum group G (either a locally compact quantum group, or a quantum group coming from a modular or manageable multiplicative unitary) is induced in a canonical fashion by a unitary corepresentation of G. It follows that there is an order bijection between the completely positive multipliers of L(G) and the positive functionals on the universal quantum group C (G}). We provide a direct link between the Junge, Neufang, Ruan representation result and the representing element of a multiplier, and use this to show that their representation map is always weak*-weak*-continuous.