On ideals of L1-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 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 G where the dual quantum group (G) over cap 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(G) in relation to coamenability of G.