Reiter's properties (P1) and (P2) for locally compact quantum groups

A locally compact group G is amenable if and only if it has Reiter's property (Pp) for p=1 or, equivalently, all p∈[1,∞), i.e., there is a net (mα)α of non-negative norm one functions in Lp(G) such that limαsupx∈K‖Lx−1mα−mα‖p=0 for each compact subset K⊂G (Lx−1mα stands for the left translate of mα by x−1). We extend the definitions of properties (P1) and (P2) from locally compact groups to locally compact quantum groups in the sense of J. Kustermans and S. Vaes. We show that a locally compact quantum group has (P1) if and only if it is amenable and that it has (P2) if and only if its dual quantum group is co-amenable. As a consequence, (P2) implies (P1).