Simplicity of finitely aligned \(k\)-graph \(C^\ast\)-algebras
type: Item
instance of: scholarly article
zbMATH Open document ID: 1265.46086
author: Jacob Shotwell
publication date: 2012-06-13T00:00:00Z
full work available at URL: https://arxiv.org/abs/0810.4567
Mathematics Subject Classification ID: 46L05
zbMATH DE Number: 6045971
zbMATH Keywords: \(k\)-graph
zbMATH Keywords: \(C^\ast\)-algebra
zbMATH Keywords: graph algebra
MaRDI profile type: Publication
arXiv ID: 0810.4567
arXiv classification: math.OA
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title: Simplicity of finitely aligned \(k\)-graph \(C^\ast\)-algebras
published in: Journal of Operator Theory
review text: For row-finite \(k\)-graphs without sources in the paper [\textit{A. Kumjian} and \textit{D. Pask}, New York J. Math. 6, 1--20 (2000; Zbl 0946.46044)] an aperiodicity hypothesis is introduced and it is shown that if \(\Lambda\) satisfies this condition, then \(C^\ast(\Lambda)\) is simple if and only if \(\Lambda\) is cofinal. The notion of no local periodicity for row-finite \(k\)-graphs without sources is introduced in the paper [\textit{D. I. Robertson} and \textit{A. Sims}, Bull. Lond. Math. Soc. 39, No. 2, 337--344 (2007; Zbl 1125.46045)]. Then it is proved that \(C^\ast(\Lambda)\) is simple if and only if \(\Lambda\) is cofinal and has no local periodicity. In this paper, the results of Robertson and Sims are generalized to the finitely aligned \(k\)-graphs, and it is proved that for arbitrary finitely aligned \(k\)-graphs ``no local periodicity'' is equivalent to the aperiodicity condition. Then it is proved (Theorem 3.5) that for a finitely aligned \(k\)-graph \(\Lambda\), \(C^\ast(\Lambda)\) is simple if and only if \(\Lambda\) is cofinal and has no local periodicity.
description: scientific article; zbMATH DE number 6045971