# Fixed points and limits of convolution powers of contractive quantum measures

## TL;DR

This paper investigates the fixed points of convolution operators linked to contractive quantum measures on locally compact quantum groups, providing characterizations and structural insights into their fixed points across various non-commutative spaces.

## Contribution

It offers a novel characterization of fixed points for convolution operators in quantum group settings, extending classical results to non-commutative $L_p$-spaces.

## Key findings

- Characterization of fixed points on $L^
Infty(G)$ and $C_0(G)$
- Structural description of fixed points on non-commutative $L_p$-spaces
- Implications for classical convolution operators and Herz-Schur multipliers

## Abstract

We study fixed points of contractive convolution operators associated to contractive quantum measures on locally compact quantum groups. We characterise the existence of non-zero fixed points respectively on $L^\infty(\mathbb{G})$ and on $C_0(\mathbb{G})$, and exploit these results to obtain for example the structure of the fixed points on the non-commutative $L_p$-spaces. Some consequences for the fixed points of classical convolution operators and Herz-Schur multipliers are also indicated.