Suppose that $A$ is a natural Banach function algebra on $K$, a compact Hausdorff space. So $A$ is realised as an algebra of continuous functions on $K$, is a Banach algebra for some norm (necessarily dominating the supremum norm) and each character on $A$ is given by evaluation at a point of $K$.
If $F\subseteq K$ is closed, then $$ I(F)=\{f\in A : f(k)=0 \ (k\in F) \}$$ is a closed ideal in $A$. If e.g. $A=C(K)$ then every closed ideal is of this form.
What's a simple example of an $A$ where not every closed ideal is of this form?
If I look in Bonsall+Duncan, I find that the Disc Algebra is an example. But quite a bit of theory is needed to show this. I'd like an easy example which I can explain to students. For bonus marks:
Can we find an $A$ which is conjugate closed?
$C^1([0,1])$ - take the ideal of functions which vanish at $p$ and whose derivatives vanish at $p$.