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 $(P_p)$ for $p=1$ or, equivalently, all $p \in [1,\infty)$, i.e., there is a net $(m_\alpha)_\alpha$ of non-negative norm one functions in $L^p(G)$ such that $\lim_\alpha \sup_{x \in K} \| L_{x^{-1}} m_\alpha - m_\alpha \|_p = 0$ for each compact subset $K \subset G$ ($L_{x^{-1}} m_\alpha$ stands for the left translate of $m_\alpha$ by $x^{-1}$). We extend the definitions of properties $(P_1)$ and $(P_2)$ 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 $(P_1)$ if and only if it is amenable and that it has $(P_2)$ if and only if its dual quantum group is co-amenable. As a consequence, $(P_2)$ implies $(P_1)$.