Skip to main content
×
×
Home

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

  • Robert Hazlewood (a1), Iain Raeburn (a2), Aidan Sims (a3) and Samuel B. G. Webster (a3)
Abstract

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.

Copyright
References
Hide All
1.Pino, G. Aranda, Clark, J., Huef, A. and Raeburn, I., Kumjian–Pask algebras of higher-rank graphs, Trans. Am. Math. Soc., in press.
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), 159186.
3.Davidson, K. R. and Yang, D., Periodicity in rank 2 graph algebras, Can. J. Math. 61 (2009), 12391261.
4.Davidson, K. R. and Yang, D., Representations of higher rank graph algebras, New York J. Math. 15 (2009), 169198.
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), 963979.
6.Drinen, D., Viewing AF-algebras as graph algebras, Proc. Am. Math. Soc. 128 (2000), 19912000.
7.Evans, D. G., On the K-theory of higher-rank graph C*-algebras, New York J. Math. 14 (2008), 131.
8.Farthing, C., Muhly, P. S. and Yeend, T., Higher-rank graph C*-algebras: an inverse semigroup and groupoid approach, Semigroup Forum 71 (2005), 159187.
9.Fowler, N. J. and Sims, A., Product systems over right-angled Artin semigroups, Trans. Am. Math. Soc. 354 (2002), 14871509.
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).
11.Hong, J. H. and Szymański, W., Quantum spheres and projective spaces as graph algebras, Commun. Math. Phys. 232 (2002), 157188.
12.Hong, J. H. and Szymański, W., The primitive ideal space of the C*-algebras of infinite graphs, J. Math. Soc. Jpn 56 (2004), 4564.
13.Jeong, J. A. and Park, G. H., Graph C*-algebras with real rank zero, J. Funct. Analysis 188 (2002), 216226.
14.Kumjian, A. and Pask, D., Higher rank graph C*-algebras, New York J. Math. 6 (2000), 120.
15.Lewin, P. and Sims, A., Aperiodicity and co.nality for finitely aligned higher-rank graphs, Math. Proc. Camb. Phil. Soc. 149 (2010), 333350.
16.Pask, D., Quigg, J. and Raeburn, I., Fundamental groupoids of k-graphs, New York J. Math. 10 (2004), 195207.
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), 137178.
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), 16131639.
19.Raeburn, I., Sims, A. and Yeend, T., Higher-rank graphs and their C*-algebras, Proc. Edinb. Math. Soc. 46(2) (2003), 99115.
20.Raeburn, I., Sims, A. and Yeend, T., The C*-algebras of finitely aligned higher-rank graphs, J. Funct. Analysis 213 (2004), 206240.
21.Robertson, D. I. and Sims, A., Simplicity of C*-algebras associated to higher-rank graphs, Bull. Lond. Math. Soc. 39 (2007), 337344.
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), 171192.
23.Schubert, H., Categories (transl. by Gray, E.) (Springer, 1972).
24.Shotwell, J., Simplicity of finitely aligned k-graph C*-algebras. J. Operat. Theory 67 (2012), 335347.
25.Spielberg, J., Graph-based models for Kirchberg algebras, J. Operat. Theory 57 (2007), 347374.
26.Webster, S. B. G., The path space of a higher-rank graph, Studia Math. 204 (2011), 155185.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Proceedings of the Edinburgh Mathematical Society
  • ISSN: 0013-0915
  • EISSN: 1464-3839
  • URL: /core/journals/proceedings-of-the-edinburgh-mathematical-society
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Keywords

MSC classification

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed