ScalingStacks

Definition 2.22. Let $\mathcal{A}$ be a Banach $k$-algebra. A character $\chi$ of $\mathcal{A}$ is a homomorphism of Banach $k$-algebra from $\mathcal{A}$ to some complete valued field extension $(K,\lvert\mathord{\cdot}\rvert_{K})$ of $(k,\lvert\mathord{\cdot}\rvert_{k})$. Two characters $\chi_{1}:\mathcal{A}\to(K_{1},\lvert\mathord{\cdot}\rvert_{K_{1}})$ and $\chi_{2}:\mathcal{A}\to(K_{2},\lvert\mathord{\cdot}\rvert_{K_{2}})$ are said to be equivalent if there exist a character $\chi:\mathcal{A}\to(K,\lvert\mathord{\cdot}\rvert_{K})$ and valued field extensions $\iota_{1}:K\to K_{1}$ and $\iota_{2}:K\to K_{2}$ which preserve norms such that $\chi=i_{1}\circ\chi_{1}=i_{2}\circ\chi_{2}$. Let $[\chi]$ be the equivalence class of $\chi$.