  We develop a notion of a non-commutative hull for a left ideal of the
$L^1$-algebra of a compact quantum group $\mathbb{G}$. A notion of
non-commutative spectral synthesis for compact quantum groups is proposed as
well. It is shown that a certain Ditkin's property at infinity (which includes
those $\mathbb{G}$ where the dual quantum group $\widehat{\mathbb{G}}$ has the
approximation property) is equivalent to every hull having synthesis. We use
this work to extend recent work of White that characterizes the weak$^*$ closed
ideals of a measure algebra of a compact group to those of the measure algebra
of a coamenable compact quantum group. In the sequel, we use this work to study
bounded right approximate identities of certain left ideals of
$L^1(\mathbb{G})$ in relation to coamenability of $\mathbb{G}$.
