Locally unitary groupoid crossed products
type: Item
instance of: scholarly article
title: Locally unitary groupoid crossed products
zbMATH Open document ID: 1267.47113
author: Geoff Goehle
published in: Indiana University Mathematics Journal
publication date: 2012-07-11T00:00:00Z
full work available at URL: https://arxiv.org/abs/0908.3806
full work available at URL: http://www.iumj.indiana.edu/IUMJ/ABS/2011/4143
Mathematics Subject Classification ID: 47L65
Mathematics Subject Classification ID: 22A22
zbMATH DE Number: 6054377
zbMATH Keywords: groupoids
zbMATH Keywords: crossed products
MaRDI profile type: Publication
OpenAlex ID: W2084397064
arXiv ID: 0908.3806
DOI: 10.1512/IUMJ.2011.60.4143
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review text: In this paper, the connection between groupoid dynamical systems and the spectrum of the crossed product is studied. Let \(S\) be an abelian locally compact Hausdorff groupoid group bundle. The notion of the principal \(S\)-bundle is defined and it is shown that there is a one-to-one correspondence between principal \(S\)-bundles and elements of a sheaf cohomology group associated to \(S\). The notion of locally unitary action of \(S\) is defined and it is shown that the spectrum of the crossed product is a principal \(\hat{S}\)-bundle (where \(\hat{S}\) is the dual group bundle). It is also proved that the isomorphism class of the spectrum of the crossed product as \(\hat{S}\)-bundle determines the exterior equivalence class of the action and that every principal \(S\)-bundle can be realized as the spectrum of some locally unitary crossed product. Some of the results of this paper generalize similar results of \textit{J. Phillips} and \textit{I. Raeburn} [J. Oper. Theory 11, 215--241 (1984; Zbl 0538.46046)] for locally unitary group actions to groupoids.
reviewed by: Mixalis Anoussis
description: scientific article; zbMATH DE number 6054377