Partial Pontryagin duality for actions of quantum groups on C*-algebras

TL;DR

This paper explores a duality between two constructions of locally compact quantum groups acting on C*-algebras, extending Baaj-Skandalis duality and examining implications for quantum Baum-Connes conjecture.

Contribution

It introduces a partial Pontryagin duality between bicrossed and double crossed products of quantum groups, generalizing existing dualities and preserving monoidal structures.

Findings

Established a duality between bicrossed and double crossed products.

Extended Baaj-Skandalis duality to quantum group actions.

Discussed implications for quantum Baum-Connes conjecture.

Abstract

We compare actions on C*-algebras of two constructions of locally compact quantum groups, the bicrossed product and the double crossed product. We give a duality between them as a generalization of Baaj-Skandalis duality. In the case of quantum doubles, this duality also preserves monoidal structures given by twisted tensor products. We also discuss its consequences for equivariant Kasparov theories in relation to the quantum analogue of the Baum-Connes conjecture.