Exercise 6, § 6

Let $A$ be a commutative unital involutive Banach algebra all of whose characters are hermitian. Suppose that, for every hermitian non-invertible element $x\in A$, one has $\|(x-\lambda )^{-1}\|=o(\mathscr{I}(\lambda )^{-2})$ when $\lambda$ tends to 0 with $\mathscr{I}(\lambda )\not= 0$. Then every closed ideal I of A is intersection of maximal ideals. (Let $\varphi : A\longrightarrow A/I$ be the canonical morphism. Let $x\in A$ be such that $\varphi (x)$ is in the radical of $A/I$. It is necessary to prove that $\varphi (x) = 0$. Reduce to the case where $x$ is hermitian. Use then Exercise 29 of I, p. 161).