Bulletin of the
Korean Mathematical Society BKMS

Let $\mathbb{G}$ 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 $\widehat{\mathbb{G}}$, the dual of $\mathbb{G}$, is co-amenable if and only if there is a state $m\in L^\infty(\widehat{\mathbb{G}})^*$ which is invariant under a left module action of $L^1(\mathbb{G})$ on $ L^\infty(\widehat{\mathbb{G}})^*$. This is the quantum group version of a result by Stokke \cite{S.ar}. 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.

Keywords: Hopf von Neumann algebra, locally compact quantum group, Lau algebra, unitary representation, amenability

MSC numbers: 46L65, 46H25, 22D10, 43A07, 22D15