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 is an element of [1, infinity), 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 is an element of K) parallel to L(x-1)m(alpha) - m(alpha)parallel to(p) = 0 for each compact subset K subset of 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). (C) 2009 Elsevier Inc. All rights reserved.