\(C^*\)-algebras associated with presentations of subshifts
type: Item
instance of: scholarly article
title: \(C^*\)-algebras associated with presentations of subshifts
zbMATH Open document ID: 1007.46051
author: Kengo Matsumoto
published in: Documenta Mathematica
publication date: 2002-06-03T00:00:00Z
full work available at URL: https://eudml.org/doc/122446
full work available at URL: 01.html
review text: Let \(\Sigma\) be a finite set. A \(\lambda\)-graph system \(L=(V,E,\lambda,\iota)\) consists of a vertex set \(V=V_0\cup V_1\cup V_2\cup\ldots\), an edge set \(E=E_{0,1}\cup E_{1,2}\cup E_{2,3}\cup\ldots\), a labeling map \(\lambda:E\to\Sigma\) and a surjection \(\iota_i:V_i\to V_{i-1}\) for each \(i\in\mathbb N\). The author introduces \(C^*\)-algebras from \(\lambda\)-graph systems. If a \(\lambda\)-graph system is the canonical \(\lambda\)-graph system for a subshift \(\Lambda\), then the corresponding \(C^*\)-algebra coincides with the Cuntz-Krieger algebra \({\mathcal O}_\Lambda\) associated with \(\Lambda\). If a \(\lambda\)-graph system is irreducible (resp. aperiodic), its \(C^*\)-algebra is simple (resp. simple purely infinite). An example of a simple purely infinite \(C^*\)-algebra of a \(\lambda\)-graph system that is not stably isomorphic to any Cuntz-Krieger algebra is produced.
Mathematics Subject Classification ID: 46L35
Mathematics Subject Classification ID: 37B10
zbMATH DE Number: 1748373
zbMATH Keywords: Bratteli diagram
zbMATH Keywords: groupoid \(C^*\)-algebra
zbMATH Keywords: subshift
zbMATH Keywords: Cuntz-Krieger algebra
zbMATH Keywords: \(\lambda\)-graph systems
reviewed by: Vladimir M. Manuilov
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1748373