The following comes from Katznelson's An Introduction to Harmonica Analysis Chapter VIII, pages 236 to 239.
Definition : A function algebra $B$ on a compact Hausdorff space $X$ is regular if, given a point $p \in X$ and a compact set $K\subset X$ such that $p \not\in K$, there exists a function $f \in B$ such that $f(p) = 1$ and $f$ vanishes on $K$.
We consider a semisimple Banach algebra $B$ with a unit, and denote its maximal ideal space by $\mathcal{M}$.
Definition : The hull, $h(I)$, of an ideal $I$ in $B$, is the set of all $M \in \mathcal{M}$ such that $I \subset M$. Equivalently : $h(I)$ is the set of all common zeros of $\widehat{x}(M)$ for $x \in I$. Since the set of common zeros of any family of continuous functions is closed, $h(I)$ is always closed in $\mathcal{M}$.
Definition : A semisimple Banach algebra $B$ is regular if $\widehat{B}$ is regular on $\mathcal{M}$.
Let $E$ be a closed subset of $\mathcal{M}$. The set $I_0(E)$ of all $x \in B$ such that $\widehat{x}(M)$ vanishes on a neighborhood of $E$ is clearly an ideal and if $B$ is regular, $h\big( I_0(E)\big)$ = E$.
I managed to convince myself that $h\big( I_0(E)\big) \supset E$, but proving the reverse inclusion is more daunting. I don't see how regularity can be of any help here...
Suppose $M \in h\big( I_0(E)\big) \backslash E$. Regularity tells us that there exists a function $\widehat{x} \in \widehat{B}$ such that $\widehat{x}(M) = 1$ and $\widehat{x}$ vanishes on $E$. So what ?
If our hypothesis was that $\widehat{x}$ vanishes on a neighborhood of $E$ then we would get a contradiction and could draw the conclusion that such an $M \in h\big( I_0(E)\big) \backslash E$ cannot exist.
Did I commit a silly mistake ? Does any one know where to go from here ?