Cuntz-Krieger algebras of directed graphs
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.
scientific article; zbMATH DE number 1186200