ScalingStacks

Lemma 2.80. Let $\phi:\mathcal{A}\to\mathcal{B}$ be a Banach algebra homomorphism from an affinoid algebra $\mathcal{A}$ to a Banach algebra $\mathcal{B}$. Then for any Laurent domain neighbourhood $V$ of $\Sigma_{\phi}$, $\phi$ extends to a unique Banach algebra homomorphism $\phi_{V}:\mathcal{A}_{V}\to\mathcal{B}$. ([Ber, Corollary 2.5.16])