ScalingStacks

Theorem 2.81. Let $\phi:\mathcal{A}\to\mathcal{B}$ be a homomorphism of Banach algebras from an affinoid algebra to a Banach algebra. Let $V\subseteq\mathfrak{M}(\mathcal{A})$ be any special domain containing $\Sigma_{\phi}$. Then there exists a Banach algebra homomorphism

$$\theta_{\phi}:\Gamma(V,\mathscr{O}_{\mathfrak{M}(\mathcal{A})})\to\mathcal{B}$$

satisfying $\phi=\theta_{\phi}\circ\iota_{V}$, where $\iota_{V}:\mathcal{A}\to\mathcal{A}_{V}=\Gamma(V,\mathscr{O}_{\mathfrak{M}(\mathcal{A})})$ is the Banach algebra homomorphism corresponding to the inclusion $V\subseteq\mathfrak{M}(\mathcal{A})$. ([Ber, Theorem 7.3.4])