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We characterize positive convolution operators on a finite quantum group$\mathbb{G}$ which are $L_{p}$-improving. More precisely, we prove that theconvolution 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 unitaryrepresentations $\alpha$, if and only if the state $(\varphi\circS)\star\varphi$ is non-degenerate (where $S$ is the antipode). We also provethat these $L_{p}$-improving properties are stable under taking free products,which gives a method to construct $L_{p}$-improving multipliers on infinitecompact quantum groups. Our methods for non-degenerate states yield a generalformula for computing idempotent states associated to Hopf images, whichgeneralizes earlier work of Banica, Franz and Skalski.

$L_{p}$-improving convolution operators on finite quantum groups