Proposition 7

Let A and B be unital Banach algebras and $\varphi$ a continuous unital morphism from A into B. Let $a\in$ A. Then Sp$_B(\varphi (a))\subset$ Sp$_A(a)$ and one has $\varphi (f(a))$ = $f(\varphi (a))$ for every $f\in \mathscr{O}$(Sp$_A(a)$). In particular, for every $\chi \in \mathsf{X}(A)$, one has $\chi (f(a)) =f(\chi (a))$.

This results from prop. 3 of I, p. 66.