Simplicity of \(C^*\)-algebras associated to row-finite locally convex higher-rank graphs. (English) Zbl 1189.46047
Higher-rank graphs \(\Lambda\), or \(k\)-graphs, together with their associated \(C^{*}\)-algebras \(C^{*}(\Lambda)\), were introduced by A.Kumjian and D.Pask in [New York J. Math.6, 1–20 (2000; Zbl 0946.46044)]. Because of the combinatorial complexity of \(k\)-graphs, it is common to restrict attention to row-finite \(k\)-graphs without sources. For example, in their original paper, Kumjian and Pask showed that for such graphs a cofinality and aperiodicity condition on \(\Lambda\) implied that \(C^{*}(\Lambda)\) was simple. In a previous paper [Bull.Lond.Math.Soc.39, No.2, 337–344 (2007; Zbl 1125.46045)], the authors established the converse. In the present paper, the authors extend this characterization to locally convex row-finite \(k\)-graphs which may have sources as studied by I.Raeburn, A.Sims and T.Yeend in [J. Funct.Anal.213, No.1, 206–240 (2004; Zbl 1063.46041)]. An essential technique in their proof is C.Farthing’s “removing sources” construction from [J. Oper.Theory 60, No.1, 165–198 (2008; Zbl 1164.46026)].
There is an unfortunate typo in the statement of Theorem 3.4: the words “with no sources” should be removed from the statement.
There is an unfortunate typo in the statement of Theorem 3.4: the words “with no sources” should be removed from the statement.
Reviewer: Dana P. Williams (Hanover)
MSC:
| 46L05 | General theory of \(C^*\)-algebras |
References:
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