Coideals, quantum subgroups and idempotent states

Abstract

We establish a one-to-one correspondence between idempotent states on a locally compact quantum group G and integrable coideals in the von Neumann algebra L∞ (G) that are preserved by the scaling group. In particular, we show that there is a one-to-one correspondence between idempotent states on G and ΨG-expected left-invariant von Neumann subalgebras of L∞ (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 G and central idempotent states on the dual G. Finally, we characterize coideals corresponding to open quantum subgroups of 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 Podlé condition, normality and regularity, and we generalize a number of results known before to hold under the coamenability assumption. © 2017. Published by Oxford University Press.