Corollary

On a commutative complex algebra without radical, two norms defining Banach algebra structures are equivalent.

It is enough to apply prop. 9 to the identity map of the algebra.

Let A and B be commutative Banach algebras. By No.$^o7$ of I, p. 9, if $h: A\rightarrow B$ is a surjective morphism, $\mathsf{X}'(h)$ is a homeomorphism of $\mathsf{X}'(B)$ onto a closed subspace of $\mathsf{X}'(A)$ which transforms 0 into 0. (In the case where $h$ is the injection of a subalgebra into $\mathscr{C}(X)$, the map $\mathsf{X}'(h)$ is injective under much weaker hypotheses, cf. I, p. 142, prop. 1, d).)

Now let $h: A\rightarrow B$ be an injective morphism. In general, $\mathsf{X}'(h)$ is not surjective, but the following proposition provides a necessary condition for this to be the case.