On ideals of L1-algebras of compact quantum groups

Abstract

We develop a notion of a non-commutative hull for a left ideal of the [Formula: see text]-algebra of a compact quantum group [Formula: see text]. 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 [Formula: see text] where the dual quantum group [Formula: see text] 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[Formula: see text] 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 [Formula: see text] in relation to coamenability of [Formula: see text].