C⁎-algebraic partial compact quantum groups

In this paper, we introduce C⁎C⁎-algebraic partial compact quantum groups, which are quantizations of topological groupoids with discrete object set and compact morphism spaces. These C⁎C⁎-algebraic partial compact quantum groups are generalizations of Hayashi's compact face algebras to the case where the object set can be infinite. They form the C⁎C⁎-algebraic counterpart of an algebraic theory of partial compact quantum groups developed in an earlier paper by the author and T. Timmermann, the correspondence between which will be dealt with in a separate paper. As an interesting example to illustrate the theory, we show how the dynamical quantum SU(2)SU(2) group, as studied by Etingof–Varchenko and Koelink–Rosengren, fits into this framework.