The Lattice of Idempotent States on a Locally Compact Quantum Group

TL;DR

This paper explores the structure of idempotent states on locally compact quantum groups, focusing on lattice operations, duality, and conditions for finite dimensionality, providing new insights into their algebraic and dual properties.

Contribution

It introduces a duality between normal idempotent states on quantum groups and their duals, and characterizes normal idempotent states on compact quantum groups.

Findings

Duality between normal idempotent states on quantum groups and their duals.

Characterization of when a left coideal is finite dimensional.

Description of normal idempotent states on compact quantum groups.

Abstract

We study lattice operations on the set of idempotent states on a locally compact quantum group corresponding to the operations of intersection of compact subgroups and forming the subgroup generated by two compact subgroups. Normal ($\sigma$-weakly continuous) idempotent states are investigated and a duality between normal idempotent states on a locally compact quantum group $\mathbb{G}$ and on its dual $\widehat{\mathbb{G}}$ is established. Additionally we analyze the question when a left coideal corresponding canonically to an idempotent state is finite dimensional and give a characterization of normal idempotent states on compact quantum groups.