Finite codimensional ideals in Banach algebras

Abstract:

Let $A$ be a commutative, selfadjoint, point spectral Banach algebra and let $M$ be a finite codimensional closed subspace of $A$ such that for each $f$ in $M$ there are $n$ distinct maximal ideals $I_1^f, \ldots ,I_n^f$ of $A$ with $f \in I_j^f$. We prove that then there are distinct maximal ideals ${I_1}, \ldots ,{I_n}$ of $A$ such that $M \subset {I_1} \cap \cdots \cap {I_n}$; in particular if codim$(M) = n$, then $M$ is an ideal.