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

Abstract

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 $T_φ:x\mapstoφ\star x$ given by a state $φ$ on $C(\mathbb{G})$ satisfies \[ \exists1<p<2,\quad\|T_φ:L_{p}(\mathbb{G})\to L_{2}(\mathbb{G})\|=1 \] if and only if the Fourier series $\hatφ$ satisfy $\|\hatφ(α)\|<1$ for all nontrivial irreducible unitary representations $α$, if and only if the state $(φ\circ S)\starφ$ 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.