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Using Godement mean on the Fourier-Stieltjes algebra of a locally compactquantum group we obtain strong separation results for quantum positive-definitefunctions associated to a subclass of representations, strengthening forexample the known relationship between amenability of a discrete quantum groupand existence of a net of finitely supported quantum positive-definitefunctions converging pointwise to $I$. We apply these results to show that vonNeumann algebras of unimodular discrete quantum groups enjoy a strong form ofnon-$w^*$-CPAP, which we call the matrix $\epsilon$-separation property.

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