Imprimitivity theorems for weakly proper actions of locally compact groups

TL;DR

This paper extends the theory of weakly proper group actions on C*-algebras to Hilbert modules, establishing new imprimitivity theorems and Morita equivalences involving exotic crossed products and fixed-point algebras.

Contribution

It introduces the notion of weakly proper actions on Hilbert modules and proves several imprimitivity theorems, including exotic versions of Green's and symmetric imprimitivity theorems.

Findings

Established Morita equivalences between various crossed products and fixed-point algebras.

Proved an exotic version of Green's imprimitivity theorem.

Developed a general symmetric imprimitivity theorem for product groups.

Abstract

In a recent paper the authors introduced universal and exotic generalized fixed-point algebras for weakly proper group actions on C*-algebras. Here we extend the notion of weakly proper actions to actions on Hilbert modules. As a result we obtain several imprimitivity theorems establishing important Morita equivalences between universal, reduced, or exotic crossed products and appropriate universal, reduced, or exotic fixed-point algebras, respectively. In particular, we obtain an exotic version of Green's imprimitivity theorem and a very general version of the symmetric imprimitivity theorem by weakly proper actions of product groups GxH. In addition, we study functorial properties of generalized fixed-point algebras for equivariant categories of C*-algebras based on correspondences.