  In this paper, we introduce C*-algebraic partial compact quantum groups,
which are quantizations of topological groupoids with discrete object set and
compact morphism spaces. These C*-algebraic partial compact quantum groups are
generalisations of Hayashi's compact face algebras to the case where the object
set can be infinite. They form the 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) group, as studied by Etingof-Varchenko and
Koelink-Rosengren, fits into this framework.
