Analytical signatures and proper actions

Proper actions and proper invariant metrics

We show that if a locally compact group G acts properly on a locally compact σ-compact space X, then there is a family of G-invariant proper continuous finite-valued pseudometrics which induces the topology of X. If X is, furthermore, metrizable, then G acts properly on X if and only if there exists a G-invariant proper compatible metric on X.

Singular Reduction for Proper Actions

Geometric Quantization for Proper Actions

We first introduce an invariant index for G-equivariant elliptic differential operators on a locally compact manifold M admitting a proper cocompact action of a locally compact group G. It generalizes the Kawasaki index for orbifolds to the case of proper cocompact actions. Our invariant index is used to show that an analog of the Guillemin-Sternberg geometric quantization conjecture holds if M...

Equivariant K-theory, Groupoids and Proper Actions

In this paper we define complex equivariantK-theory for actions of Lie groupoids. For a Bredon-compatible Lie groupoid G, this defines a periodic cohomology theory on the category of finite G-CW-complexes. A suitable groupoid allows us to define complex equivariant K-theory for proper actions of non-compact Lie groups, which is a natural extension of the theory defined in [24]. For the particul...

Proper Actions of Groupoids on C-algebras

This thesis contains some results concerning groupoid dynamical systems and crossed products. We introduce the notion of a proper groupoid dynamical system and of its generalized fixed point algebra. We show that our notion of proper groupoid dynamical system extends both the notion of proper actions of groups on topological spaces and the notion of the proper group dynamical systems introduced...