Partial compact quantum groups

Compact quantum groups of face type, as introduced by Hayashi, form a class of compact quantum groupoids with a classical, finite set of objects. Using the notions of a weak multiplier bialgebra and weak multiplier Hopf algebra (resp. due to B{"o}hm–G'{o}mez-Torrecillas–L'{o}pez-Centella and Van Daele-Wang), we generalize Hayashi’s definition to allow for an infinite set of objects, and call the resulting objects partial compact quantum groups. We prove a Tannaka-Kre$\breve{\text{ı}}$n-Woronowicz reconstruction result for such partial compact quantum groups using the notion of a partial fusion C${}^{\ast }$-category. As examples, we consider the dynamical quantum $SU(2)$-groups from the point of view of partial compact quantum groups.