Lattice of Idempotent States on a Locally Compact Quantum Group

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.