Imprimitivity theorems for weakly proper actions of locally compact groups

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.

The Geometry of Singular Quaternionic Kähler Quotients

Two descriptions of quaternionic Kähler quotients by proper group actions are given: the first as a union of smooth manifolds, some of which come equipped with quaternionic Kähler or locally Kähler structures; the second as a union of quaternionic Kähler orbifolds. In particular the quotient always has an open set which is a smooth quaternionic Kähler manifold. When the original manifold and the group are compact, we describe a length space structure on the quotient. Similar descriptions of singular hyperKähler and 3-Sasakian quotients are given.