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 On action of Lau algebras on von Neumann algebras Bull. Korean Math. Soc. 2015 Vol. 52, No. 2, 557-570 https://doi.org/10.4134/BKMS.2015.52.2.557Published online March 31, 2015 Mohammad Ramezanpour Damghan University Abstract : 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 Full-Text :