Simplicity of finitely aligned \(k\)-graph \(C^\ast\)-algebras (Q2891148)

For row-finite \(k\)-graphs without sources in the paper [\textit{A. Kumjian} and \textit{D. Pask}, New York J. Math. 6, 1--20 (2000; Zbl 0946.46044)] an aperiodicity hypothesis is introduced and it is shown that if \(\Lambda\) satisfies this condition, then \(C^\ast(\Lambda)\) is simple if and only if \(\Lambda\) is cofinal. The notion of no local periodicity for row-finite \(k\)-graphs without sources is introduced in the paper [\textit{D. I. Robertson} and \textit{A. Sims}, Bull. Lond. Math. Soc. 39, No. 2, 337--344 (2007; Zbl 1125.46045)]. Then it is proved that \(C^\ast(\Lambda)\) is simple if and only if \(\Lambda\) is cofinal and has no local periodicity. In this paper, the results of Robertson and Sims are generalized to the finitely aligned \(k\)-graphs, and it is proved that for arbitrary finitely aligned \(k\)-graphs ``no local periodicity'' is equivalent to the aperiodicity condition. Then it is proved (Theorem 3.5) that for a finitely aligned \(k\)-graph \(\Lambda\), \(C^\ast(\Lambda)\) is simple if and only if \(\Lambda\) is cofinal and has no local periodicity.