Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-25T04:37:51.407Z Has data issue: false hasContentIssue false
  • English
  • Français

Remarks on some fundamental results about higher-rank graphs and their C*-algebras

Part of: Selfadjoint operator algebras Graph theory

Published online by Cambridge University Press:  30 April 2013

Robert Hazlewood ,
Iain Raeburn ,
Aidan Sims  and
Samuel B. G. Webster
Robert Hazlewood
Affiliation:
School of Mathematics and Statistics, University of New South Wales, Sydney, NSW 2052, Australia ([email protected])
Iain Raeburn
Affiliation:
Department of Mathematics and Statistics, University of Otago, PO Box 56, Dunedin 9054, New Zealand ([email protected])
Aidan Sims
Affiliation:
School of Mathematics and Applied Statistics, University of Wollongong, NSW 2522, Australia ([email protected]; [email protected])
Samuel B. G. Webster
Affiliation:
School of Mathematics and Applied Statistics, University of Wollongong, NSW 2522, Australia ([email protected]; [email protected])
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Results of Fowler and Sims show that every k-graph is completely determined by its k-coloured skeleton and collection of commuting squares. Here we give an explicit description of the k-graph associated with a given skeleton and collection of squares and show that two k-graphs are isomorphic if and only if there is an isomorphism of their skeletons which preserves commuting squares. We use this to prove directly that each k-graph Λ is isomorphic to the quotient of the path category of its skeleton by the equivalence relation determined by the commuting squares, and show that this extends to a homeomorphism of infinite-path spaces when the k-graph is row finite with no sources. We conclude with a short direct proof of the characterization, originally due to Robertson and Sims, of simplicity of the C*-algebra of a row-finite k-graph with no sources.

Keywords

higher-rank graphC*-algebraCuntz–Krieger algebra

MSC classification

Secondary: 05C20: Directed graphs (digraphs), tournaments 46L05: General theory of $C^*$-algebras
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 56 , Issue 2 , June 2013 , pp. 575 - 597
Copyright
Copyright © Edinburgh Mathematical Society 2013

References

1.Pino, G. Aranda, Clark, J., Huef, A. and Raeburn, I., Kumjian–Pask algebras of higher-rank graphs, Trans. Am. Math. Soc., in press.Google Scholar
2.Baum, P. F., Hajac, P. M., Matthes, R. and Szymański, W., The K-theory of Heegaard-type quantum 3-spheres, K-Theory 35 (2005), 159–186.CrossRefGoogle Scholar
3.Davidson, K. R. and Yang, D., Periodicity in rank 2 graph algebras, Can. J. Math. 61 (2009), 1239–1261.CrossRefGoogle Scholar
4.Davidson, K. R. and Yang, D., Representations of higher rank graph algebras, New York J. Math. 15 (2009), 169–198.Google Scholar
5.Deicke, K., Hong, J. H. and Szymański, W., Stable rank of graph algebras: type I graph algebras and their limits, Indiana Univ. Math. J. 52 (2003), 963–979.CrossRefGoogle Scholar
6.Drinen, D., Viewing AF-algebras as graph algebras, Proc. Am. Math. Soc. 128 (2000), 1991–2000.CrossRefGoogle Scholar
7.Evans, D. G., On the K-theory of higher-rank graph C*-algebras, New York J. Math. 14 (2008), 1–31.Google Scholar
8.Farthing, C., Muhly, P. S. and Yeend, T., Higher-rank graph C*-algebras: an inverse semigroup and groupoid approach, Semigroup Forum 71 (2005), 159–187.CrossRefGoogle Scholar
9.Fowler, N. J. and Sims, A., Product systems over right-angled Artin semigroups, Trans. Am. Math. Soc. 354 (2002), 1487–1509.CrossRefGoogle Scholar
10.Green, E. R., Graph products of groups, PhD Thesis, University of Leeds (1990) (available at http://etheses.whiterose.ac.uk/236/1/uk_bl_ethos_254954.pdf).Google Scholar
11.Hong, J. H. and Szymański, W., Quantum spheres and projective spaces as graph algebras, Commun. Math. Phys. 232 (2002), 157–188.CrossRefGoogle Scholar
12.Hong, J. H. and Szymański, W., The primitive ideal space of the C*-algebras of infinite graphs, J. Math. Soc. Jpn 56 (2004), 45–64.Google Scholar
13.Jeong, J. A. and Park, G. H., Graph C*-algebras with real rank zero, J. Funct. Analysis 188 (2002), 216–226.CrossRefGoogle Scholar
14.Kumjian, A. and Pask, D., Higher rank graph C*-algebras, New York J. Math. 6 (2000), 1–20.Google Scholar
15.Lewin, P. and Sims, A., Aperiodicity and co.nality for finitely aligned higher-rank graphs, Math. Proc. Camb. Phil. Soc. 149 (2010), 333–350.CrossRefGoogle Scholar
16.Pask, D., Quigg, J. and Raeburn, I., Fundamental groupoids of k-graphs, New York J. Math. 10 (2004), 195–207.Google Scholar
17.Pask, D., Raeburn, I., Rørdam, M. and Sims, A., Rank-two graphs whose C*-algebras are direct limits of circle algebras, J. Funct. Analysis 239 (2006), 137–178.CrossRefGoogle Scholar
18.Pask, D., Raeburn, I. and Weaver, N. A., A family of 2-graphs arising from two-dimensional subshifts, Ergod. Theory Dynam. Syst. 29 (2009), 1613–1639.CrossRefGoogle Scholar
19.Raeburn, I., Sims, A. and Yeend, T., Higher-rank graphs and their C*-algebras, Proc. Edinb. Math. Soc. 46(2) (2003), 99–115.CrossRefGoogle Scholar
20.Raeburn, I., Sims, A. and Yeend, T., The C*-algebras of finitely aligned higher-rank graphs, J. Funct. Analysis 213 (2004), 206–240.CrossRefGoogle Scholar
21.Robertson, D. I. and Sims, A., Simplicity of C*-algebras associated to higher-rank graphs, Bull. Lond. Math. Soc. 39 (2007), 337–344.CrossRefGoogle Scholar
22.Robertson, D. I. and Sims, A., Simplicity of C*-algebras associated to row-finite locally convex higher-rank graphs, Israel J. Math. 172 (2009), 171–192.CrossRefGoogle Scholar
23.Schubert, H., Categories (transl. by Gray, E.) (Springer, 1972).CrossRefGoogle Scholar
24.Shotwell, J., Simplicity of finitely aligned k-graph C*-algebras. J. Operat. Theory 67 (2012), 335–347.Google Scholar
25.Spielberg, J., Graph-based models for Kirchberg algebras, J. Operat. Theory 57 (2007), 347–374.Google Scholar
26.Webster, S. B. G., The path space of a higher-rank graph, Studia Math. 204 (2011), 155–185.CrossRefGoogle Scholar