On Action of Lau Algebras on Von Neumann Algebras

Abstract

Abstract. 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 ∈ L ∞ ( G) * which is invariant under a left module action of L 1 (G) on L ∞ ( 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.

Character amenability and contractibility of some Banach algebras on left coset spaces

“…(see[8]). The next result follows from Theorems 3.5 and 3.8 and this fact that the left amenability of a Lau algebra is equivalent to its right 1-amenability (see[11, Proposition 2.2]).…”