Separation properties for positive-definite functions on locally compact quantum groups and for associated von Neumann algebras

Using Godement mean on the Fourier-Stieltjes algebra of a locally compact quantum group we obtain strong separation results for quantum positive-definite functions associated to a subclass of representations, strengthening for example the known relationship between amenability of a discrete quantum group and existence of a net of finitely supported quantum positive-definite functions converging pointwise to $I$. We apply these results to show that von Neumann algebras of unimodular discrete quantum groups enjoy a strong form of non-$w^{\ast }$-CPAP, which we call the matrix $\epsilon$-separation property.