ScalingStacks

Proof. For any $z\in\mathfrak{M}(\mathcal{A}_{Z})$, the multiplicative algebra seminorm (or the corresponding character) $\chi_{z}$ on $\mathcal{A}_{Z}$ corresponds to a unique multiplicative algebra seminorm on $A_{Z}$ by restriction. Since $A_{Z}$ is dense in $\mathcal{A}_{Z}$, the family of open sets $\{U(f;p,q),\text{ }f\in A_{Z},\text{ }p,q\in\mathbb{R}\}$ form a basis for topology on $\mathfrak{M}(\mathcal{A}_{Z})$, hence the inherited topology coincides with the originial topology. So the embedding is continuous, and the image of $\mathfrak{M}(\mathcal{A}_{Z})$ is compact in $Z^{an}$. Since the topology on $Z^{an}$ is Hausdorff, the image of $\mathfrak{M}(\mathcal{A}_{Z})$ is closed. ∎