The Ladder Technique -- Quantum Groups

Abstract

Given a regular $\mathrm{C}^{*}$-algebraic locally compact quantum group $(S_r,\Delta)$ with universal quantum group $(S_f,\Delta_f)$, a $\mathrm{C}^{*}$-algebra $A$, and a sufficiently well-behaved full coaction $S_f \overset{\alpha}{\curvearrowright} A$, we construct natural lattice isomorphisms from the strongly coaction invariant ideals of $A$ to the strongly coaction invariant ideals of full and reduced crossed product $\mathrm{C}^{*}$-algebras as an application of the `ladder technique' developed by the author, S. Kaliszewski, John Quigg and Dana P. Williams. In particular, these lattice isomorphisms are determined by either the maximality or normality of the coaction $\alpha$. This result directly generalizes a recent theorem proven by the aforementioned authors for locally compact groups, which in turn generalized a theorem of Elliot Gootman and Aldo Lazar for amenable groups.