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Purely infinite Cuntz-Krieger algebras of directed graphs. (English) Zbl 1042.46028

A \(C^*\)-algebra \(A\) is said to be purely infinite if it has no characters and if for every pair of positive elements \(x,y \in A\) such that \(y\) lies in the closed two-sided ideal generated by \(x\), there exists a sequence \((a_n) \subset A\) such that \(a_n^*xa_n \rightarrow y\). The purpose of the present note is to give a convenient characterization of generalized Cuntz-Krieger algebras \(C^*(E)\) based on directed graphs \(E\) that are purely infinite. It is also shown that \(C^*(E)\) has real rank zero if and only if the graph \(E\) satisfies the so-called “Condition (K)”.

MSC:

46L05 General theory of \(C^*\)-algebras
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