A general property of ideals in uniform algebras

Let $I$ and $J$ be to closed ideals in a uniform algebra $A\subset C(S)$. It will be shown that if the complex conjugate $\overline{I\cap J}$ of their intersection is not included in some of them, then the sum $I+\bar{J}$ is not closed in $C(S)$.
The question arose during the joint work of the author and I. Zlotnikov on interpolation properties of coinvariant subspaces of the shift operator. The answer indicated above may be viewed as a far-reaching generalization of the fact that $C_A+\overline{C_A}\neq C(\mathbb{T})$, where $C_A$ is the disk-algebra,
$$ C_A= \{f\in C(\mathbb{T})\colon \hat{f}(n)=0\quad \text{for}\quad n<0\}. $$

The proof is based on the presence of certain very slight traces of analytic structure on an arbitrary proper uniform algebra. A similar technique was used by the author around 1987 for the proof of the Glicksberg conjecture.