Renault's equivalence theorem for reduced groupoid \(C^\ast\)-algebras
type: Item
instance of: scholarly article
zbMATH Open document ID: 1265.46101
author: Aidan Sims
author: Dana P. Williams
publication date: 2012-10-04T00:00:00Z
full work available at URL: https://arxiv.org/abs/1002.3093
Mathematics Subject Classification ID: 46L55
Mathematics Subject Classification ID: 54H20
zbMATH DE Number: 6090242
zbMATH Keywords: groupoid
zbMATH Keywords: groupoid equivalence
zbMATH Keywords: reduced \(C^\ast\)-algebra
zbMATH Keywords: equivalence theorem
zbMATH Keywords: Hilbert bimodule
zbMATH Keywords: \(C^\ast\)-correspondence
MaRDI profile type: Publication
arXiv ID: 1002.3093
arXiv classification: math.OA
arXiv classification: math.FA
title: Renault's equivalence theorem for reduced groupoid \(C^\ast\)-algebras
published in: Journal of Operator Theory
review text: The purpose of the paper is threefold: firstly to give a precise statement and proof of the equivalence result for reduced groupoid \(C^\ast\)-algebras; secondly to illustrate that the equivalence result for reduced algebras is compatible with the result for the full algebras and Rieffel induction; and thirdly, and possibly most importantly, to highlight the role of the linking groupoid, which is the main tool in proofs.NEWLINENEWLINE The main results of the paper imply that if \(G\) and \(H\) are equivalent groups, then their reduced groupoid \(C^\ast\)-algebras \(C^\ast_r(G)\) and \(C^\ast_r(H)\) are Morita equivalent via a quotient \(X_r\) of \(X\) (\(X\) is Renault's imprimitivity bimodule). This equivalence is compatible in a natural way with the Equivalence Theorem for full groupoid \(C^\ast\)-algebras.
description: scientific article; zbMATH DE number 6090242