Landstad-Vaes theory for locally compact quantum groups

TL;DR

This paper extends Landstad-Vaes theory to non-regular locally compact quantum groups by introducing weak actions, enabling the characterization of crossed products and the original algebra's position within them.

Contribution

It generalizes Landstad-Vaes theory to non-regular quantum groups through the concept of weak actions, allowing for the construction and analysis of crossed products.

Findings

Defined weak actions of quantum groups on C*-algebras.

Characterized the position of original algebra inside crossed products.

Proved the uniqueness of the crossed product up to isomorphism.

Abstract

Landstad-Vaes theory deals with the structure of the crossed product of a C$^*$-algebra by an action of locally compact (quantum) group. In particular it describes the position of original algebra inside crossed product. The problem was solved in 1979 by Landstad for locally compact groups and in 2005 by Vaes for regular locally compact quantum groups. To extend the result to non-regular groups we modify the notion of $G$-dynamical system introducing the concept of weak action of quantum groups on C$^*$-algebras. It is still possible to define crossed product (by weak action) and characterise the position of original algebra inside the crossed product. The crossed product is unique up to an isomorphism. At the end we discuss a few applications.