Idempotent states on locally compact groups and quantum groups

TL;DR

This paper surveys the concept of idempotent states across classical groups and quantum groups, exploring their connections to subgroups and invariant C*-subalgebras, with a focus on recent developments in quantum group theory.

Contribution

It provides a comprehensive overview of recent results on idempotent states in locally compact quantum groups, linking classical and quantum perspectives.

Findings

Idempotent states relate to subgroups and invariant C*-subalgebras.

Recent results clarify the structure of idempotent states in quantum groups.

The survey includes the dual case in Fourier--Stieltjes algebra.

Abstract

This is a short survey on idempotent states on locally compact groups and locally compact quantum groups. The central topic is the relationship between idempotent states, subgroups and invariant C*-subalgebras. We concentrate on recent results on locally compact quantum groups, but begin with the classical notion of idempotent probability measure. We also consider the `intermediate' case of idempotent states in the Fourier--Stieltjes algebra: this is the dual case of idempotent probability measures and so an instance of idempotent states on a locally compact quantum group.