Coideals, Quantum Subgroups and Idempotent States

We establish a one-to-one correspondence between idempotent states on a locally compact quantum group $\mathbb{G}$ and integrable coideals in the von Neumann algebra $L^\infty(\mathbb G)$ that are preserved by the scaling group. In particular, we show that there is a one-to-one correspondence between idempotent states on $\mathbb G$ and $\psi_\mathbb G$-expected left-invariant von Neumann subalgebras of $L^\infty(\mathbb G)$. We characterize idempotent states of Haar type as those corresponding to integrable normal coideals preserved by the scaling group. We also establish a one-to-one correspondence between open subgroups of $\mathbb G$ and central idempotent states on the dual $\widehat {\mathbb G}$. Finally, we characterize coideals corresponding to open quantum subgroups of $\mathbb G$ as those that are normal and admit an atom. As a byproduct of this study, we get a number of universal lifting results for Podleś condition, normality and regularity, and we generalize a number of results known before to hold under the coamenability assumption.