An Ideal Correspondence Result for Crossed Products by Quantum Groups
Matthew Gillespie
TL;DR
This work extends classical ideal-correspondence results for crossed products to the quantum-group setting by leveraging a weak Kac system with duality arising from a regular LCQG. It develops a ladder-style framework and uses Morita equivalence (Rieffel correspondence) to transfer invariant ideals across iterative crossed-products, establishing that coaction-invariant ideals of A correspond to dual-invariant ideals in both full and reduced crossed products when coactions satisfy maximality or normality. The approach unifies full S- and 1-coactions and their duals, and provides canonical, algebraic lattice isomorphisms that generalize the Gootman–Lazar/Nilsen results to LCQGs. The results are accompanied by concrete conventions, examples, and links to classical crossed-product theory, highlighting how quantum-group actions broaden ideal-theoretic perspectives in C*-algebraic dynamical systems.
Abstract
Given a weak Kac system with duality $(\mathcal{H},V,U)$ arising from regular $\mathrm{C}^{*}$-algebraic locally compact quantum group $(\mathcal{G},Δ)$, a $\mathrm{C}^{*}$-algebra $A$, and a sufficiently well-behaved coaction $α$, we construct natural lattice isomorphisms from the coaction invariant ideals of $A$ to the dual coaction invariant ideals of full and reduced crossed products associated to $(\mathcal{H},V,U)$. In particular, these lattice isomorphisms are determined by either the maximality or normality of the coaction $α$. This result directly generalizes the main theorem of Gillespie, Kaliszewski, Quigg, and Williams in arXiv:2406.06780, which in turn generalized an older ideal correspondence result of Gootman and Lazar for locally compact amenable groups. Throughout, we also develop basic conventions and motivate through elementary examples how crossed product $\mathrm{C}^{*}$-algebras by quantum groups generalize the classical crossed product theory.