Groupoid Crossed Products

TL;DR

This paper investigates the structure of groupoid crossed products and algebras, providing topological descriptions of their spectra and primitive ideals, and introduces a theory of principal groupoid bundles and locally unitary actions.

Contribution

It offers new topological identifications of spectra and primitive ideals of groupoid crossed products and develops a duality theory for principal groupoid bundles and actions.

Findings

Spectrum of the crossed product is homeomorphic to a quotient of the spectrum of the stabilizer group bundle.

Primitive ideal space of the groupoid algebra is homeomorphic to a quotient of the dual of the stabilizer group bundle.

Characterization of locally unitary actions via cohomology classes from principal bundles.

Abstract

We present a number of findings concerning groupoid dynamical systems and groupoid crossed products. The primary result is an identification of the spectrum of the groupoid crossed product when the groupoid has continuously varying abelian stabilizers and a well behaved orbit space. In this case, the spectrum of the crossed product is homeomorphic, via an induction map, to a quotient of the spectrum of the crossed product by the stabilizer group bundle. The main theorem is also generalized in the groupoid algebra case to an identification of the primitive ideal space. This generalization replaces the assumption that the orbit space is well behaved with an amenability hypothesis. We then use induction to show that the primitive ideal space of the groupoid algebra is homeomorphic to a quotient of the dual of the stabilizer group bundle. In both cases the identification is topological. We then apply these theorems in a number of examples, and examine when a groupoid algebra has Hausdorff spectrum. As a separate result, we also develop a theory of principal groupoid group bundles and locally unitary groupoid actions. We prove that such actions are characterized, up to exterior equivalence, by a cohomology class which arises from a principal bundle. Furthermore, we also demonstrate how to construct a locally unitary action from a given principal bundle. This last result uses a duality theorem for abelian group bundles which is also included as part of this thesis.