Generalized fixed point algebras for coactions of locally compact quantum groups

TL;DR

This paper extends the concept of generalized fixed point algebras to locally compact quantum groups, establishing a correspondence with Hilbert modules and analyzing their properties in the quantum setting.

Contribution

It introduces a generalized notion of continuous square-integrability for quantum groups and proves a correspondence between Hilbert modules over equivariant C*-algebras and their crossed products.

Findings

Generalized fixed point algebras are characterized as compact operators on associated Hilbert modules.

A new notion of continuous square-integrability for quantum groups is developed.

There is a proven correspondence between Hilbert modules over equivariant C*-algebras and modules over their crossed products.

Abstract

We extend the construction of generalized fixed point algebras to the setting of locally compact quantum groups - in the sense of Kustermans and Vaes - following the treatment of Marc Rieffel, Ruy Exel and Ralf Meyer in the group case. We mainly follow Meyer's approach analyzing the constructions in the realm of equivariant Hilbert modules. We generalize the notion of continuous square-integrability, which is exactly what one needs in order to define generalized fixed point algebras. As in the group case, we prove that there is a correspondence between continuously square-integrable Hilbert modules over an equivariant C*-algebra B and Hilbert modules over the reduced crossed product of B by the underlying quantum group. The generalized fixed point algebra always appears as the algebra of compact operators of the associated Hilbert module over the reduced crossed product.