ScalingStacks

Proposition 2.77. 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. Let $\mathcal{B}^{\prime}$ be the closed sub-algebra generated by the image of $\phi$ of $\mathcal{A}$ in $\mathcal{B}$ and let $\phi^{\prime}:\mathcal{A}\to\mathcal{B}^{\prime}$ be the restricted homomorphism. Then $(\Sigma_{\phi})^{\mathrm{h}}=\Sigma_{\phi^{\prime}}$. ([Ber, Proposition 7.3.1])