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Cuntz-Krieger algebras for infinite matrices. (English) Zbl 0932.47053
Authors’ abstract: Given an arbitrary infinite matrix \(A= \{A(i,j)\}_{i,j\in{\mathcal G}}\) with entries in \(\{0,1\}\) and having no identically zero rows, we define an algebra \({\mathcal O}_A\) as the universal \(C^*\)-algebra generated by partial isometries subject to conditions that generalize, to the infinite case, those introduced by Cuntz and Krieger for finite matrices. We realize \({\mathcal O}_A\) as the crossed product algebra for a partial dynamical system and, based on this description, we extend to the infinite case some of the main results known to hold in the finite case, namely the uniqueness theorem, the classification of ideals, and the simplicity criteria. \({\mathcal O}_A\) is always nuclear and we obtain conditions for it to be unital and purely infinite.

MSC:
47L65 Crossed product algebras (analytic crossed products)
46L05 General theory of \(C^*\)-algebras
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