ScalingStacks

Proposition 2.79. Let $\mathcal{A}$ be a $k$-affinoid algebra, $\mathcal{B}$ be a Banach $k$-algebra. Let $\phi:\mathcal{A}\to\mathcal{B}$ be a homomorphism of Banach $k$-algebras. Then for any open neighbourhood $U$ in $\mathfrak{M}(\mathcal{A})$ of the spectrum $\Sigma_{\phi}$, there exists a homomorphism of Banach algebras extending $\phi$

$$\widetilde{\phi}:\widetilde{\mathcal{A}}:=\mathcal{A}\{r_{1}^{-1}T_{1},\dots,r_{n}^{-1}T_{n}\}\to\mathcal{B}$$

such that $\mathrm{pr}((\Sigma_{\phi})^{\mathrm{h}})\subseteq U$, where $\mathrm{pr}:\mathfrak{M}(\widetilde{\mathcal{A}})\to\mathfrak{M}(\mathcal{A})$ is the canonical map of projection. ([Ber, Proposition 7.3.3])