Invariant Connections in Loop Quantum Gravity

Given a group $G$ and an abelian $C^{\ast }$-algebra $\mathfrak{A}$, the antihomomorphisms $\mathrm{\Theta }\text{\hspace{0.17em}⁣}:G\to \mathrm{A}\mathrm{u}\mathrm{t}(\mathfrak{A})$ are in one-to-one with those left actions $\Phi\colon G\times $\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}$(\mathfrak{A})\rightarrow $\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}$(\mathfrak{A})$ whose translation maps ${\mathrm{\Phi }}_g$ are continuous; whereby continuities of $\mathrm{\Theta }$ and $\mathrm{\Phi }$ turn out to be equivalent if $\mathfrak{A}$ is unital. In particular, a left action $\varphi \text{\hspace{0.17em}⁣}:G\times X\to X$ can be uniquely extended to the spectrum of a $C^{\ast }$-subalgebra $\mathfrak{A}$ of the bounded functions on $X$ if ${\varphi }_g^{\ast }(\mathfrak{A})\subseteq \mathfrak{A}$ 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.