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When is the Cuntz-Krieger algebra of a higher-rank graph approximately finite-dimensional?

Journal Article


Abstract


  • We investigate the question: when is a higher-rank graph C*-algebra approximately finite-dimensional? We prove that the absence of an appropriate higher-rank analogue of a cycle is necessary. We show that it is not in general sufficient, but that it is sufficient for higher-rank graphs with finitely many vertices. We give a detailed description of the structure of the C*-algebra of a row-finite locally convex higher rank graph with finitely many vertices. Our results are also sufficient to establish that if the C*-algebra of a higher-rank graph is AF, then its every ideal must be gauge-invariant. We prove that for a higher-rank graph C*-algebra to be AF it is necessary and sufficient for all the corners determined by vertex projections to be AF. We close with a number of examples which illustrate why our question is so much more difficult for higher-rank graphs than for ordinary graphs.

Publication Date


  • 2012

Citation


  • Evans, D. Gwion. & Sims, A. (2012). When is the Cuntz-Krieger algebra of a higher-rank graph approximately finite-dimensional?. Journal of Functional Analysis, 263 (1), 183-215.

Scopus Eid


  • 2-s2.0-84860450641

Ro Metadata Url


  • http://ro.uow.edu.au/infopapers/1917

Number Of Pages


  • 32

Start Page


  • 183

End Page


  • 215

Volume


  • 263

Issue


  • 1

Place Of Publication


  • http://dx.doi.org/10.1016/j.jfa.2012.03.024

Abstract


  • We investigate the question: when is a higher-rank graph C*-algebra approximately finite-dimensional? We prove that the absence of an appropriate higher-rank analogue of a cycle is necessary. We show that it is not in general sufficient, but that it is sufficient for higher-rank graphs with finitely many vertices. We give a detailed description of the structure of the C*-algebra of a row-finite locally convex higher rank graph with finitely many vertices. Our results are also sufficient to establish that if the C*-algebra of a higher-rank graph is AF, then its every ideal must be gauge-invariant. We prove that for a higher-rank graph C*-algebra to be AF it is necessary and sufficient for all the corners determined by vertex projections to be AF. We close with a number of examples which illustrate why our question is so much more difficult for higher-rank graphs than for ordinary graphs.

Publication Date


  • 2012

Citation


  • Evans, D. Gwion. & Sims, A. (2012). When is the Cuntz-Krieger algebra of a higher-rank graph approximately finite-dimensional?. Journal of Functional Analysis, 263 (1), 183-215.

Scopus Eid


  • 2-s2.0-84860450641

Ro Metadata Url


  • http://ro.uow.edu.au/infopapers/1917

Number Of Pages


  • 32

Start Page


  • 183

End Page


  • 215

Volume


  • 263

Issue


  • 1

Place Of Publication


  • http://dx.doi.org/10.1016/j.jfa.2012.03.024