A Cuntz-Krieger uniqueness theorem for semigraph \(C^\ast\)-algebras (Q437721)

The author gives a higher rank generalization of Tomforde's ultragraph \(C^\ast\)-algebras. The class of ultragraph \(C^\ast\)-algebras was introduced by \textit{M. Tomforde} [J. Oper. Theory 50, No. 2, 345--368 (2003; Zbl 1061.46048)], who proved that it includes the classes of graph \(C^\ast\)-algebras and Exel-Laca \(C^\ast\)-algebras. It was proved later [\textit{T. Katsura} et al., J. Reine Angew. Math. 640, 135--165 (2010; Zbl 1207.46049)] that the three classes agree up to Morita equivalence. In the present paper, the author uses some ideas coming from the theory of \(C^\ast\)-algebras of higher rank graphs (as in [\textit{A. Kumjian} and \textit{D. Pask}, New York J. Math. 6, 1--20 (2000; Zbl 0946.46044)]) to construct higher rank versions of ultragraph \(C^\ast\)-algebras. He proves Cuntz-Krieger uniqueness theorems for cancelling semigraph algebras and aperiodic semigraph algebras.