ScalingStacks

Proposition 2.26. Let $\phi:\mathcal{A}_{1}\to\mathcal{A}_{2}$ be a homomorphism of Banach $k$-algebras. It induces a continuous map $\phi^{\star}:\mathfrak{M}(\mathcal{A}_{2})\to\mathfrak{M}(\mathcal{A}_{1})$ by sending an equivalent class of characters $[\chi]$ of $\mathcal{A}_{2}$ to the class of characters $[\chi\circ\psi]$ of $\mathcal{A}_{1}$. ([Ber, Remark 1.2.2 (iii)])