$L_{p}$-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 $T_{\varphi}:x\mapsto\varphi\star x$ given by a state $\varphi$ on $C(\mathbb{G})$ satisfies $$\exists1<p<2,\quad\|T_{\varphi}:L_{p}(\mathbb{G})\to L_{2}(\mathbb{G})\|=1$$ if and only if the Fourier series $\hat{\varphi}$ satisfy $\|\hat{\varphi}(\alpha)\|<1$ for all nontrivial irreducible unitary representations $\alpha$, if and only if the state $(\varphi\circ S)\star\varphi$ is non-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.