Imprimitivity theorems for weakly proper actions of locally compact groups

Abstract

In a recent paper the authors introduced universal and exotic generalized fixed-point algebras for weakly proper group actions on$C^{\ast }$-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$G\times H$. In addition, we study functorial properties of generalized fixed-point algebras for equivariant categories of$C^{\ast }$-algebras based on correspondences.