I'm currently working on a problem that's asking me to give an example of a non self adjoint ideal of $C(D) = \{f: D \longrightarrow \mathbb{C} \: | \: \text{$f$ is continuous}\}$ with $||\cdot||_{\infty}$, and $D = \{z \in \mathbb{C} \: |\: |z| \leq 1\}$.
So far I've been trying to mess with holomorphicity, integration, compact supports, and noninvertible functions.
My issue is that I keep yielding either a vector subspace which isn't self adjoint but not an ideal (like the collection of all holomorphic functions), or I get an ideal thats self adjoint (functions with compact support, functions whose integral over $D$ is $0$, maximal ideals of the form $M_y=\{f \in C(D) \: | \: f(y)=0\}$, etc). I know that whatever ideal I choose, it can't be closed ofcourse.
I would be very appreciative of a $\textbf{small hint that points me in the right direction, not a full answer}$!
Hint: Consider the identity function $z\in C(D)$ and the ideal $I:= C(D)z.$ Does it contain $\overline{z}$?