  Given a group $G$ and an abelian $C^*$-algebra $\mathfrak{A}$, the
antihomomorphisms $\Theta\colon G\rightarrow \mathrm{Aut}(\mathfrak{A})$ are in
one-to-one with those left actions $\Phi\colon G\times
\mathrm{Spec}(\mathfrak{A})\rightarrow \mathrm{Spec}(\mathfrak{A})$ whose
translation maps $\Phi_g$ are continuous; whereby continuities of $\Theta$ and
$\Phi$ turn out to be equivalent if $\mathfrak{A}$ is unital. In particular, a
left action $\phi\colon G \times X\rightarrow X$ can be uniquely extended to
the spectrum of a $C^*$-subalgebra $\mathfrak{A}$ of the bounded functions on
$X$ if $\phi_g^*(\mathfrak{A})\subseteq \mathfrak{A}$ holds for each $g\in G$.
In the present paper, we apply this to the framework of loop quantum gravity.
We show that, on the level of the configuration spaces, quantization and
reduction in general do not commute, i.e., that the symmetry-reduced quantum
configuration space is (strictly) larger than the quantized configuration space
of the reduced classical theory. Here, the quantum-reduced space has the
advantage to be completely characterized by a simple algebraic relation,
whereby the quantized reduced classical space is usually hard to compute.
