On action of Lau algebras on von Neumann algebras
Bull. Korean Math. Soc. 2015 Vol. 52, No. 2, 557-570
Published 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
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