Exercise 19, § 3

Let $A$ be a commutative unital Banach algebra, $B$ a closed unital subalgebra of $A$, $T$ the mapping $\chi \mapsto \chi |B$ from $\mathsf{X}(A)$ into $\mathsf{X}(B)$ and $R$ the equivalence relation in $\mathsf{X}(A)$ defined by T. Then T defines by passing to the quotient a homeomorphism of $\mathsf{X}(A)/R$ onto $T(\mathsf{X}(A))$, and for every $x\in B$, the function $f=\mathscr{G}_B(x)$ is such that $|f|$ attains its upper bound on $T(\mathsf{X}(A))$.