Corollary 1

Let A be a commutative regular Banach algebra without radical. Let J be the set of $x\in A$ such that $\mathscr{G}(x)$ has compact support. Suppose that J = A. Then every closed ideal of A distinct from A is contained in a regular maximal ideal.

If I is a closed ideal of A which is contained in no regular maximal ideal, then $V(I) =\emptyset$, hence $I\supset J$ (Prop. 4 applied to F = $\emptyset$ ), whence $I\supset \overline{J}= A$.