Lattice of Idempotent States on a Locally Compact Quantum Group

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 $\hat{\mathbb{G}}$ is established. Additionally we analyze the question of 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.