Amenability properties of unitary co-representations of locally compact quantum groups

Abstract

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.