Corollaire 2

Let A be a commutative regular Banach algebra, I an ideal of A and $f:\mathsf{X}'(A)\rightarrow \mathbf{C}$ a continuous function. Suppose that, for every $\chi \in \mathsf{X}'(A)$, there exists an element $y_{\chi}\in I$ such that $f=\mathscr{G}'(y_{\chi})$ in a neighbourhood of $\chi$. Then there exists an element $y\in I$ such that $f=\mathscr{G}'(y)$.

Let $\widetilde{A}$ be the Banach algebra obtained from A by adjunction of a unit element. Then $\widetilde{A}$ is regular (remark 1), and $\mathsf{X}'(A) =\mathsf{X}(\widetilde{A})$; it is therefore sufficient to apply cor. 1 to $\widetilde{A}$ and to the ideal I.

If I is an ideal of a commutative Banach algebra, recall (cf. I, p. 30) that we denote by V(I) the set of $\chi \in \mathsf{X}(A)$ whose kernel contains I, in other words the set of $\chi \in \mathsf{X}(A)$ where all the functions $\mathscr{G}(x)$ vanish for $x\in I$. It is a closed subset of $\mathsf{X}(A)$ for the Jacobson topology.