ON ACTION OF LAU ALGEBRAS ON VON NEUMANN ALGEBRAS

Let G be a von Neumann algebraic locally compact quantum group, in the sense of Kustermans and Vaes. In this paper, as a consequence of a notion of amenability for actions of Lau algebras, we show that G, the dual of G, is co-amenable if and only if there is a state m is an element of L-infinity ((G) over cap)* which is invariant under a left module action of L-1(G) on L-infinity ((G) over cap)*. This is the quantum group version of a result by Stokke [17]. We also characterize amenable action of Lau algebras by several properties such as fixed point property. This yields in particular, a fixed point characterization of amenable groups and H-amenable representation of groups.