Analysis for idempotent states on quantum permutation groups

TL;DR

This paper investigates the structure of idempotent states on quantum permutation groups, extending the understanding of Haar states and their associated convex subsets within the framework of quantum group theory.

Contribution

It adapts Van Daele's proof to establish the existence of idempotent states in convex subsets of the state space of quantum permutation groups.

Findings

Existence of idempotent states in weak*-compact convolution-closed convex subsets

Extension of Haar state existence to broader quantum group contexts

Analysis of convex subsets related to quantum permutation groups

Abstract

Woronowicz proved the existence of the Haar state for compact quantum groups under a separability assumption later removed by Van Daele in a new existence proof. A minor adaptation of Van Daele's proof yields an idempotent state in any non-empty weak*-compact convolution-closed convex subset of the state space. Such subsets, and their associated idempotent states, are studied in the case of quantum permutation groups.