Beurling--Fourier Algebras and Hull--Synthesis on Compact Quantum Groups

Abstract

We introduce a family of weighted Beurling–Fourier algebras \(A_\omega(\mathbb G)\) on a coamenable compact quantum group \(\mathbb G\), defined via admissible weight functions on its set of irreducible corepresentations. Extending the non‑commutative hull–synthesis framework of Sackaney (2022), we define weighted hulls and synthesis ideals in \(A_\omega(\mathbb G)\), and establish necessary and sufficient conditions on the weight \(\omega\) under which every hull admits synthesis. We further characterize a weighted version of Ditkin’s property at infinity in terms of bounded \(\omega\)‑approximate identities, and examine the impact of coamenability on the existence of such identities. Several illustrative examples—including quantum \(\mathrm{SU}(2)\) with polynomial and exponential weights—demonstrate new phenomena arising in the weighted setting.