Full quantum crossed products, invariant measures, and type-I lifting

TL;DR

This paper investigates properties of crossed products and representations in the setting of locally compact quantum groups, establishing conditions for type-I representations and embedding results related to invariant measures and states.

Contribution

It introduces new results on the structure of crossed products, including conditions for type-I lifting and isomorphisms between full and reduced crossed products in LCQGs.

Findings

Type-I property of representations lifts from subgroup to group under invariant measure.

Full group algebra embeds into the crossed product if an invariant state exists.

Full and reduced crossed products are isomorphic for dual-coamenable LCQGs.

Abstract

We show that for a closed embedding $\mathbb{H}\le \mathbb{G}$ of locally compact quantum groups (LCQGs) with $\mathbb{G}/\mathbb{H}$ admitting an invariant probability measure, a unitary $\mathbb{G}$-representation is type-I if its restriction to $\mathbb{H}$ is. On a related note, we also prove that if an action $\mathbb{G}\circlearrowright A$ of an LCQG on a unital $C^*$-algebra admits an invariant state then the full group algebra of $\mathbb{G}$ embeds into the resulting full crossed product (and into the multiplier algebra of that crossed product if the original algebra is not unital). We also prove a few other results on crossed products of LCQG actions, some of which seem to be folklore; among them are (a) the fact that two mutually dual quantum-group morphisms produce isomorphic full crossed products, and (b) the fact that full and reduced crossed products by dual-coamenable LCQGs are isomorphic.