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We establish a one to one correspondence between idempotent states on alocally compact quantum group G and integrable coideals in the von Neumannalgebra of bounded measurable functions on G that are preserved by the scalinggroup. In particular we show that there is a 1-1 correspondence betweenidempotent states on G and psi-expected left invariant von Neumann subalgebrasof bounded measurable functions on G. We characterize idempotent states of Haartype as those corresponding to integrable normal coideals preserved by thescaling group. We also establish a one to one correspondence between opensubgroups of G and convolutionally central idempotent states on the dual of G.Finally we characterize coideals corresponding to open quantum subgroups of Gas those that are normal and admit an atom. As a byproduct of this study we geta number of universal lifting results for Podles condition, normality andregularity and we generalize number of results known before to hold under thecoamenability assumption.

Coideals, quantum subgroups and idempotent states