Reiterʹs properties and for locally compact quantum groups

A locally compact group G is amenable if and only if it has Reiterʹs property ( P p ) for p = 1 or, equivalently, all p ∈ [ 1 , ∞ ) , 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 ∈ K ‖ L x − 1 m α − m α ‖ p = 0 for each compact subset K ⊂ 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 ) .