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

TL;DR

This paper characterizes positive convolution operators on finite quantum groups that improve $L_{p}$ spaces, providing criteria based on Fourier series and demonstrating stability under free products, with applications to constructing multipliers and analyzing idempotent states.

Contribution

It offers a complete characterization of $L_{p}$-improving convolution operators on finite quantum groups and extends these results to infinite quantum groups via free products.

Findings

Characterization of $L_{p}$-improving convolution operators using Fourier series conditions.

Proof of stability of $L_{p}$-improving properties under free products.

Development of a formula for computing idempotent states related to Hopf images.

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 $T_{\varphi}:x\mapsto\varphi\star x$ 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.