$L_p$-improving convolution operators on finite quantum groups

We characterize positive convolution operators on a finite quantum group $\mathbb{G}$ which are $L_p$-improving. More precisely, we prove that the convolution operator $T_{\phi }:x\mapsto \phi \star x$ given by a state $\phi$ on $C(\mathbb{G})$ satisfies $$\mathrm{\exists }1<p<2,\mathrm{\parallel }{T}_{\phi }:L_p(\mathbb{G})\to L_2(\mathbb{G})\mathrm{\parallel }=1$$ if and only if the Fourier series $\widehat{\phi }$ satisfy $\mathrm{\parallel }\widehat{\phi }(\alpha )\mathrm{\parallel }<1$ for all nontrivial irreducible unitary representations $\alpha$, if and only if the state $(\varphi\circ S)\star\varphi$isnon-degenerate(where$S$ is the antipode). We also prove that these $L_p$-improving properties are stable under taking free products, 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.