ON ACTION OF LAU ALGEBRAS ON VON NEUMANN ALGEBRAS

Title & Authors
ON ACTION OF LAU ALGEBRAS ON VON NEUMANN ALGEBRAS

Abstract
Let $\small{\mathbb{G}}$ 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 $\small{\hat{\mathbb{G}}}$, the dual of $\small{\mathbb{G}}$, is co-amenable if and only if there is a state $\small{m{\in}L^{\infty}(\hat{\mathbb{G}})^*}$ which is invariant under a left module action of $\small{L^1(\mathbb{G})}$ on $\small{L^{\infty}(\hat{\mathbb{G}})^*}$. 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.
Keywords
Hopf von Neumann algebra;locally compact quantum group;Lau algebra;unitary representation;amenability;
Language
English
Cited by
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