Proposition: A physical gauge net

The map $q_{\mathrm{phys}}$ is a unital $\ast$-homomorphism on the global constraint-preserving algebra. Each $\mathfrak{A}(S)$ is a unital $C^{\ast }$-subalgebra of $\mathfrak{M}$. The net is isotone and locally commuting on disjoint link sets. Its global regional algebra satisfies $\mathfrak{A}(E)=\mathfrak{M}$. Every state on $\mathfrak{M}$ is an admitted physical state in this model.

Proof. A constraint-preserving observable commutes with $P$. Hence its restriction preserves products and adjoints:

The image is a finite-dimensional unital $\ast$-algebra, therefore norm closed. Isotony follows from kinematic support inclusion. Disjoint kinematic supports commute, and their physical images continue to commute because restriction is multiplicative.

For $c\in \mathcal{B}({\mathcal{H}}_{\mathrm{phys}})$, extend it by zero on ${{\mathcal{H}}_{\mathrm{phys}}}^{\perp }$. Every $G_v$ is the identity on ${\mathcal{H}}_{\mathrm{phys}}$ and preserves its orthogonal complement, so this extension commutes with every $G_v$. Its restriction is $c$, proving $\mathfrak{A}(E)=\mathfrak{M}$.