Exercise 13, § 6

Let $A$ be a unital Banach algebra over $\mathbf{C},x\in A$, and let S = Sp$_A(x)$. Suppose that there exists a morphism $\varphi$ of $\mathscr{C}(S)$ into A which transforms 1 into 1 and $z$ into $x$, denoting by $z$ the identity mapping of S. Then $\|\varphi (f)\|\geqslant \|f\|$ for every $f\in \mathscr{C}(S)$. In particular, if the morphism $\varphi$ is continuous, then $\varphi$ is a homeomorphism onto its image. (We may suppose A commutative. Let $\lambda \in S$. There exists $\chi \in \mathsf{X}(A)$ such that $\lambda =\chi (x) =\chi (\varphi (z)) = (\mathsf{X}(\varphi )(\chi ))(z)$. Hence $\mathsf{X}(\varphi )(\chi )$ is the character $f\mapsto f(\lambda )$ of $\mathscr{C}(S)$. Let $f\in \mathscr{C}(S)$. From what precedes, there exists $\chi \in \mathsf{X}(A)$ such that $|f|$ attains its maximum at $\mathsf{X}(\varphi )(\chi )$. Then $|\chi (\varphi (f))|=\|f\|$, hence $\|\varphi (f)\|\geqslant \|f\|.$)