# Kawada-It\^o-Kelley Theorem for Quantum Semigroups

## TL;DR

This paper extends the Kawada-Itô-Kelley theorem to quantum semigroups, characterizing idempotent states as Haar states on sub-objects and identifying conditions for these to form compact quantum subgroups.

## Contribution

It introduces a quantum analogue of the classical theorem, providing a new framework for understanding idempotent states in quantum semigroup theory.

## Key findings

- Idempotent states correspond to Haar states on certain operator systems.
- Characterization of when these sub-objects form compact quantum subgroups.
- Reproduction of classical results in the quantum setting.

## Abstract

Idempotent states on locally compact quantum semigroups with weak cancellation properties are shown to be Haar states on a certain sub-object described by an operator system with comultiplication. We also give a characterization of the situation when this sub-object is actually a compact quantum subgroup. In particular we reproduce classical results on idempotent probability measures on locally compact semigroups with cancellation.