[Paper Review] Groupoid Crossed Products

Geoff Goehle|ArXiv.org|May 28, 2009

Homotopy and Cohomology in Algebraic Topology4 citations

TL;DR

This paper establishes a topological identification of the spectrum and primitive ideal space of groupoid crossed products under specific conditions—particularly when the groupoid has continuously varying abelian stabilizers and a well-behaved orbit space. The key result identifies the spectrum of the crossed product as a quotient of the dual of the stabilizer group bundle via an induction map, with a generalization to groupoid algebras under an amenability hypothesis.

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.

Motivation & Objective

• To characterize the spectrum of groupoid crossed products when the groupoid has continuously varying abelian stabilizers and a well-behaved orbit space.
• To generalize the spectrum identification to the primitive ideal space in the groupoid algebra case using an amenability hypothesis.
• To develop a theory of principal groupoid group bundles and locally unitary groupoid actions.
• To establish a cohomological classification of locally unitary actions up to exterior equivalence.
• To construct locally unitary actions from given principal bundles using a duality theorem for abelian group bundles.

Proposed method

• Utilizes induction maps to relate the spectrum of the crossed product to the dual of the stabilizer group bundle.
• Applies topological quotient constructions to identify the spectrum and primitive ideal space as quotients of the dual of the stabilizer bundle.
• Employs cohomological techniques to classify locally unitary groupoid actions via a characteristic class arising from a principal bundle.
• Introduces a duality theorem for abelian group bundles to construct actions from principal bundles.
• Relies on the structure of groupoid dynamical systems with abelian stabilizers and orbit space regularity.
• Uses the amenability hypothesis to replace the need for a well-behaved orbit space in the primitive ideal space identification.

Experimental results

Research questions

• RQ1 How can the spectrum of a groupoid crossed product be topologically identified when the stabilizers are abelian and vary continuously?
• RQ2 What conditions allow the primitive ideal space of a groupoid algebra to be identified as a quotient of the dual of the stabilizer bundle?
• RQ3 How are locally unitary groupoid actions classified up to exterior equivalence?
• RQ4 Can a locally unitary action be explicitly constructed from a given principal bundle?
• RQ5 What role does the duality theorem for abelian group bundles play in constructing such actions?

Key findings

• The spectrum of the groupoid crossed product is homeomorphic to a quotient of the dual of the stabilizer group bundle via an induction map when stabilizers are abelian and vary continuously.
• In the groupoid algebra case, the primitive ideal space is homeomorphic to a quotient of the dual of the stabilizer bundle under an amenability hypothesis.
• Locally unitary groupoid actions are classified up to exterior equivalence by a cohomology class derived from a principal bundle.
• A duality theorem for abelian group bundles enables the construction of locally unitary actions from given principal bundles.
• The spectrum of the groupoid algebra is Hausdorff if and only if the orbit space is Hausdorff and the stabilizer bundle satisfies certain regularity conditions.
• The results are applied to multiple examples, illustrating conditions under which the spectrum is Hausdorff and the structure of the primitive ideal space.