Exercise 16, § 3

Let $A$ be a commutative Banach algebra. The following conditions are equivalent:

(i) $A$ has no radical and $\mathscr{G}(A)$ is closed in $\mathscr{C}_0(\mathsf{X}(A))$ ;

(ii) $\mathscr{G}$ is a homeomorphism of A onto $\mathscr{G}(A)$;

(iii) there exists a constant $c >0$ such that $\|x\|^2\leqslant c\|x^2\|$ for all $x\in A$.