The Ladder Technique -- Quantum Groups

Abstract

Given a regular ${\mathrm{C}}^{\ast }$-algebraic locally compact quantum group $(S_r,\hat{I}”)$ with universal quantum group $(S_f,\hat{I}{”}_f)$, a ${\mathrm{C}}^{\ast }$-algebra $A$, and a sufficiently well-behaved full coaction $S_f\stackrel{\hat{I}}{\pm }\curvearrowright A$, 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 ${\mathrm{C}}^{\ast }$-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 $\hat{I}\pm$. 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.

Published version. Title changed to match the journal version. 24 pages