Cuntz-Krieger algebras of directed graphs
type: Item
instance of: scholarly article
title: Cuntz-Krieger algebras of directed graphs
zbMATH Open document ID: 0917.46056
author: David Pask
author: Iain Raeburn
author: Alex Kumjian
published in: Pacific Journal of Mathematics
publication date: 1999-07-19T00:00:00Z
review text: In the article [\textit{A. Kumjian, D. Pask, I. Raeburn} and \textit{J. Renault}, J. Funct. Anal. 144, No. 2, 505-541 (1997)] a Cuntz-Krieger \(C^*\)-algebra \(C^*(E)\) for directed graph \(E\) is introduced. A directed graph with a condition (K) is considered and it is obtained a theorem which relates the loop structure in a graph \(E\) to the ideal structure of \(C^*(E)\). In the present paper the authors discuss the following ``exit'' condition (L): a graph \(E\) satisfies the condition (L) if all loops in \(E\) have exits. They study the question how the distribution of loops in \(E\) is reflected by the structure of \(C^*(E)\), and prove that a directed graph \(E\) has no loops if and only if \(C^*(E)\) is an AF algebra. In the paper an exit condition (L) on loops in \(E\) is described which allows the authors to prove an analogue of the Cuntz-Krieger theorem of uniqueness and give a characterization of purely infiniteness of \(C^*(E)\), namely, it is proved that the \(C^*\)-algebra \(C^*(E)\) is purely infinite if and only if every vertex connects to a loop and the graph \(E\) satisfies condition (L). This allowes the authors to prove the following main result: let the graph \(E\) satisfy the condition (L) and be cofinal; if \(E\) has no loops, then \(C^*(E)\) is an AF algebra, if \(E\) has a loop, then \(C^*(E)\) is purely infinite.
Mathematics Subject Classification ID: 46L10
Mathematics Subject Classification ID: 05C20
Mathematics Subject Classification ID: 46L89
Mathematics Subject Classification ID: 94C15
zbMATH DE Number: 1186200
zbMATH Keywords: Cuntz-Krieger algebras
zbMATH Keywords: directed graphs
zbMATH Keywords: loop of graph
zbMATH Keywords: AF algebra
zbMATH Keywords: Cuntz-Krieger theorem of uniqueness
zbMATH Keywords: purely infiniteness
reviewed by: A. A. Rakhimov
MaRDI profile type: Publication
full work available at URL: pjm.1998.184.161
OpenAlex ID: W2162427658
DOI: 10.2140/PJM.1998.184.161
description: scientific article; zbMATH DE number 1186200