Exercise 11, § 4

Let $A$ be a commutative unital Banach algebra and let $x\in A$. Let S be a subset of $\mathsf{X}(A)$ closed for the topology of Jacobson and let $f$ be a complex holomorphic function in a neighbourhood of $(\mathscr{G}x)(S)$. There exists $y\in A$ such that $\mathscr{G}y=f\circ \mathscr{G}x$ on S. (Let I be the intersection of the kernels of the $\chi \in S$. Use the functional calculus in $A/I$.)