Reiter's properties (P_1) and (P_2) 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 $(m_α)_α$ of non-negative norm one functions in $L^p(G)$ such that $\lim_α\sup_{x \in K} \| L_{x^{-1}} m_α- m_α\|_p = 0$ for each compact subset $K \subset G$ ($L_{x^{-1}} m_α$ stands for the left translate of $m_α$ 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 $(P_1)$.

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