  First, we review the $C^\ast$-algebraic foundations of loop quantization, in
particular, the construction of quantum configuration spaces and the
implementation of symmetries. Then, we apply these results to loop quantum
gravity, focusing on the space of generalized connections and on measures
thereon. Finally, we study the realm of homogeneous isotropic loop quantum
cosmology: once viewed as the loop quantization of classical cosmology, once
seen as the symmetric sector of loop quantum gravity. It will turn out that
both theories differ, i.e., quantization and symmetry reduction do not commute.
Moreover, we will present a uniqueness result for kinematical measures. These
last two key results have originally been due to Hanusch; here, we give
drastically simplified and direct proofs.
