Proposition 7.4.3.label Let $\frak{m}\tleq A$ be a proper regular ideal of an abelian Banach algebra then the topological closure $\cl{\frak{m}}$ is also a proper regular ideal of $A$. In particular, any maximal regular ideal of $A$ is closed.
Proof. Let $e$ be an identity modulo $\frak{m}$ then by Proposition 7.4.2, $e\nin \cl{\frak{m}}$. $\cl{\frak{m}}$ is regular since $\frak{m}\suf\cl{\frak{m}}$. Lastly $\cl{\frak{m}}$ is an ideal since since multiplication and addtion are continuous in a Banach algebra.$\square$
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