A Limit Theorem for Discrete Quantum Groups

TL;DR

This paper proves that for discrete quantum groups, the convolution powers of an irreducible state diminish to zero in strong operator topology, extending classical limit theorems to the quantum setting.

Contribution

It establishes a limit theorem for convolution powers on discrete quantum groups, generalizing classical results to the quantum group framework.

Findings

Convolution powers of irreducible states tend to zero in strong operator topology.

The result applies to discrete quantum groups and their associated completely positive maps.

Extends classical limit theorems to the quantum group context.

Abstract

We consider the {\it concentration functions problem} for discrete quantum groups; we prove that if $\mathbb{G}$ is a discrete quantum group, and $\mu$ is an irreducible state in $l^1(\mathbb{G})$, then the convolution powers $\mu^n$, considered as completely positive maps on $c_0(\mathbb{G})$, converge to zero in strong operator topology.