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For a Hopf von Neumann algebra [Formula: see text], we give a fixed point characterization of Voiculescu amenability of [Formula: see text] in terms of modules over [Formula: see text]. As a consequence, we present some descriptions for amenability of locally compact groups in terms of certain associated Hopf von Neumann algebras. We finally apply this result to some modules of continuous functions on a multiplicative subsemigroup of [Formula: see text].

Abstract. For a locally compact semigroup S, we study a general fixed point property in terms of Banach left S-modules. We then use this property to give our main result which is a new characterization for left amenability of a large class of locally compact semigroups; finally, we investigate several examples which lead us to the conjecture that the main result remains true for all locally compact semigroups. Key Words and Phrases: Banach left S-module, foundation semigroup, left amenability, left fixed point, left invariant mean, locally compact semigroup, weak * -operator topology.

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For locally compact quantum groups [Formula: see text], we initiate an investigation of stable states with respect to unitary co-representations [Formula: see text] of [Formula: see text] on Hilbert spaces [Formula: see text]; in particular, we study the subject on the multiplicative unitary operator [Formula: see text] of [Formula: see text] with some examples on locally compact quantum groups arising from discrete groups and compact groups. As the main result, we consider the one co-dimensional Hilbert subspace of [Formula: see text] associated to a suitable vector [Formula: see text], to present an operator theoretic characterization of stable states with respect to a related unitary co-representation [Formula: see text]. This provides a quantum version of an interesting result on unitary representations of locally compact groups given by Lau and Paterson in 1991.