On Ideals of $L^1$-algebras of Compact Quantum Groups

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 $\mathbb{G}$.