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

TL;DR

This paper introduces non-commutative hulls and spectral synthesis concepts for ideals in $L^1$-algebras of compact quantum groups, linking these notions to properties like Ditkin's property and coamenability.

Contribution

It develops a non-commutative spectral synthesis framework and extends classical results to the setting of compact quantum groups, including characterizations of ideals and approximate identities.

Findings

A non-commutative hull concept for ideals is established.

Ditkin's property at infinity is shown to be equivalent to synthesis of hulls.

Extension of White's characterization to measure algebras of coamenable quantum groups.

Abstract

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 $\mathbb{G}$.