Cuntz-Krieger algebras and \(C^{*}\)-algebras of topological graphs
type: Item
instance of: scholarly article
title: Cuntz-Krieger algebras and \(C^{*}\)-algebras of topological graphs
zbMATH Open document ID: 1193.46042
author: Takeshi Katsura
published in: Acta Applicandae Mathematicae
publication date: 2010-01-18T00:00:00Z
review text: The author studies Cuntz-Krieger algebras using methods of topological graphs and singly generated dynamical systems. Based on this approach, he presents a new proof of the theorem of \textit{J. Cuntz} and \textit{W. Krieger} which shows that conjugate one-sided Markov chains arising from \(\{0,1\}\) matrices with no zero rows give rise to isomorphic \(C^*\)-algebras, cf. [Invent. Math. 56, 251--268 (1980; Zbl 0434.46045)]. The author introduced topological graphs in [Trans.\ Am.\ Math.\ Soc.\ 356, No.\,~11, 4287--4322 (2004; Zbl 1049.46039)]. To a topological graph \(E\) he associated a \(C^*\)-algebra \(\mathcal{O}(E)\). Given a topological graph \(E\), the infinite path space \(X_E\) together with the one-sided shift map, in this case a local homeomorphism, becomes a singly generated dynamical system \((X_E, \sigma_E)\) in the sense of \textit{J.\,Renault} [in: Operator theoretical methods.\ Proceedings of the 17th international conference on operator theory, Timişoara, Romania, June 23--26, 1998.\ Bucharest:\ The Theta Foundation, 371--386 (2000; Zbl 1032.46535)]. This singly generated dynamical system gives rise to a new topological graph \(E_\infty\), and it is proved here that \(\mathcal{O}(E_\infty)\) is isomorphic to the transformation group algebra \(C^*(X_E, \sigma_E)\). The author showed in [``Topological graphs and singly generated dynamical systems'', preprint (2009)] that \(\mathcal{O}(E)\) and \(\mathcal{O}(E_\infty)\) are canonically isomorphic. Hence by starting from a \(\{0,1\}\)-matrix with no zero row, there is a topological graph whose \(C^*\)-algebra recovers the Cuntz-Krieger algebra associated to \(A\), and the associated \((X_E, \sigma_E)\) is the one-sided Markov chain associated to \(A\), thereby providing the claimed result.
Mathematics Subject Classification ID: 46L55
Mathematics Subject Classification ID: 54H20
Mathematics Subject Classification ID: 22A22
zbMATH DE Number: 5659976
zbMATH Keywords: \(C^{*}\)-algebra
zbMATH Keywords: Cuntz-Krieger algebra
zbMATH Keywords: Markov chain
zbMATH Keywords: groupoid
zbMATH Keywords: topological graph
zbMATH Keywords: factor map
zbMATH Keywords: singly generated dynamical system
reviewed by: Nadia S. Larsen
MaRDI profile type: Publication
full work available at URL: s10440-009-9535-0
OpenAlex ID: W2052839324
cites work: A class of C*-algebras and topological Markov chains
cites work: Q4859247
cites work: A class of \boldmath{$C^*$}-algebras generalizing both graph algebras and homeomorphism \boldmath{$C^*$}-algebras I, fundamental results
cites work: Q4463843
cites work: A CLASS OF C*-ALGEBRAS GENERALIZING BOTH GRAPH ALGEBRAS AND HOMEOMORPHISM C*-ALGEBRAS II, EXAMPLES
cites work: Q3063276
cites work: Q4335109
cites work: A groupoid approach to C*-algebras
cites work: Q4515714
DOI: 10.1007/S10440-009-9535-0
Recommended article: Q4515714
Recommended article: State splitting, strong shift equivalence and stable isomorphism of Cuntz–Krieger algebras
Recommended article: Topological Markov chains, \(C^*\)-algebras, and \(K^ 2\)
Recommended article: Q4529406
Recommended article: $C^*$-algebras arising from Dyck systems of topological Markov chains
Recommended article: Orbit equivalence of topological Markov shifts and Cuntz-Krieger algebras
Recommended article: On one-sided topological conjugacy of topological Markov shifts and gauge actions on Cuntz–Krieger algebras
Recommended article: A short note on Cuntz splice from a viewpoint of continuous orbit equivalence of topological Markov shifts
Recommended article: The uniqueness of Cuntz-Krieger type algebras
Recommended article: Classification of Cuntz-Krieger algebras by orbit equivalence of topological Markov shifts
description: scientific article; zbMATH DE number 5659976