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

A locally compact group G is amenable if and only if it has Reiter's property (P) for p = 1 or, equivalently, all p ∈ [1, ∞), i.e., there is a net (m) of non-negative norm one functions in L (G) such that lim sup {norm of matrix} L m - m {norm of matrix} = 0 for each compact subset K ⊂ G (L m stands for the left translate of m by x). We extend the definitions of properties (P) and (P) 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 (P) if and only if it is amenable and that it has (P) if and only if its dual quantum group is co-amenable. As a consequence, (P) implies (P).