Reiter's properties \((P_1)\) and \((P_2)\) for locally compact quantum groups (Q847705)

The notions of the Reiter properties \((P_1)\) and \((P_2)\) are extended from locally compact groups to the setting of locally compact quantum groups in the sense of \textit{J. Kustermans} and \textit{S. Vaes} [Ann. Sci. Éc. Norm. Supér. (4) 33, No.~6, 837--934 (2000; Zbl 1034.46508)]. It is shown that a locally compact quantum group has the property \((P_1)\) if and only if it is amenable and that it has the property \((P_2)\) if and only if its dual quantum group is co-amenable. It is also shown that \((P_2)\) implies \((P_1)\).