We characterize positive convolution operators on a finite quantum group G which are L_p-improving. More precisely, we prove that the convolution operator T_φ:xφ x given by a state φ on C(G) satisfies ∃1<p<2, T_φ:L_p(G)→ L_2(G)=1 if and only if the Fourier series φ̂ satisfy φ̂(α)<1 for all nontrivial irreducible unitary representations α, if and only if the state (φ∘ S)φ is non-degenerate (where S is the antipode). We also prove that these L_p-improving properties are stable under taking free

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... oducts, which gives a method to construct L_p-improving multipliers on infinite compact quantum groups. Our methods for non-degenerate states yield a general formula for computing idempotent states associated to Hopf images, which generalizes earlier work of Banica, Franz and Skalski.