Group-like projections for locally compact quantum groups

Abstract: Let $\mathbb{G}$ be a locally compact quantum group. We give a 1-1 correspondence between group-like projections in $L^\infty(\mathbb{G})$ preserved by the scaling group and idempotent states on the dual quantum group $\widehat{\mathbb{G}}$. As a byproduct we give a simple proof that normal integrable coideals in $L^\infty(\mathbb{G})$ which are preserved by the scaling group are in 1-1 correspondence with compact quantum subgroups of $\mathbb{G}$.