Cuntz-Krieger algebras and \(C^{*}\)-algebras of topological graphs
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.
scientific article; zbMATH DE number 5659976