Landstad–Vaes theory for locally compact quantum groups

Abstract

Landstad–Vaes theory deals with the structure of the crossed product of a [Formula: see text]-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 [Formula: see text]-dynamical system introducing the concept of weak action of quantum groups on [Formula: see text]-algebras. It is still possible to define crossed product (by weak action) and characterize 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.