Let $X$ be a compact Hausdorff topological set, and $Y$ be its closed subset. Is the ideal of functions vanishing on $Y$
$$
I=\{f\in C(X):\ \forall y\in Y\ f(y)=0\}
$$
complementable (as a closed subspace) in $C(X)$ (as a Banach space)?


This is true in the case when $X\subseteq {\mathbb R}^n$ (this follows from: Stein. Singular integrals... VI 2.2), but what about general case?



Is the ideal of functions vanishing at a set complementable in $C(X)$?
Not in general.


It's well-known in Banach space theory that the ideal $c\_0$ in $\ell^\infty$ is not complemented (see e.g. Albiac & Kalton).


By the Gelfand representation, $\ell^\infty \simeq C(\beta \mathbb{N})$ as a $C^\ast$-algebra. This maps $c\_0$ to the ideal of functions on $\beta \mathbb{N}$ that vanish on $\beta \mathbb{N} \setminus \mathbb{N}$.


