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

For locally compact quantum groups G, we initiate an investigation of stable states with respect to unitary co-representations U of G on Hilbert spaces H-U; in particular, we study the subject on the multiplicative unitary operator Lire of G 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 H-U associated to a suitable vector to present an operator theoretic characterization of stable states with respect to a related unitary co-representation U-eta. This provides a quantum version of an interesting result on unitary representations of locally compact groups given by Lau and Paterson in 1991.