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Description

We define what it means for an action of a locally compact quantum group on a discrete quantum space to be proper. Properness of such actions allows for a version of the Peter-Weyl theorem to hold in their equivariant representation categories. In particular, these can be used to show that only algebraic quantum groups can admit proper actions on discrete spaces. Conjecturally, the existence of such actions yields a dynamical characterization of the class of algebraic quantum groups.