# If $\mathbb{A}$ and $\mathbb{B}$ are unital banach algebras, and $f: \mathbb{A} \to \mathbb{B}$

If $\mathbb{A}$ and $\mathbb{B}$ are unital banach $\mathbb{C}$ algebras, and $f: \mathbb{A} \to \mathbb{B}$ a unital algebra homomorphism$\mathbb{C}$ algebra morphism, and $\text{Sp}(\mathbb{A})$ the set of characters from $\mathbb{A}$ to $\mathbb{B}$. I think that we have a continuous mapping from $\text{Sp}(B)$ to $\text{Sp}(\mathbb{A})$ (equipped with the weak star topology). I am really unsure of my proof. Could you please pinpoint any mistake? If $\phi$ is a charcter on $\mathbb{B}$, then $\phi \circ f$ is also a character on $\mathbb{A}$, so we have a unction from the $\text{Sp}(B)$ to $\text{Sp}(\mathbb{A})$ , say $k$, such that $k(\phi)=\phi \circ f$. Now I need to prove that k is continuous. In the weak star topology, it is enough to check that, for every $a \in \mathbb{A}$, the map $\text{Sp}(\mathbb{B}) \to \mathbb{C}$, defined by $g_{a} \circ k$ is continuous, where $g_{a} : \text{Sp}(\mathbb{A}) \to \mathbb{C}, g_{a}(\rho)=\rho(a)$. Let $\phi \in \text{Sp}(\mathbb{B})$, let $\epsilon>0$. Consider $W=W(\phi,f(a),\frac{\epsilon}{2}) \cap \text{Sp}(\mathbb{B})=\lbrace \psi \in \text{Sp}(\mathbb{B})$, $\vert \phi(f(a))-\psi(f(a)) \vert<\frac{\epsilon}{2} \rbrace$. If $\psi \in U$, then $\vert (g_{a} \circ k(\phi)) -(g_{a} \circ k )(\psi) \vert <\epsilon$, establishing continuity.