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Aperiodicity and cofinality for finitely aligned higher-rank graphs

Published online by Cambridge University Press:  10 May 2010

PETER LEWIN
Affiliation:
School of Mathematics and Applied Statistics, Austin Keane Building (15), University of Wollongong, Wollongong, NSW, Australia. e-mail: asims@uow.edu.au
AIDAN SIMS
Affiliation:
School of Mathematics and Applied Statistics, Austin Keane Building (15), University of Wollongong, Wollongong, NSW, Australia. e-mail: asims@uow.edu.au

Abstract

We introduce new formulations of aperiodicity and cofinality for finitely aligned higher-rank graphs Λ, and prove that C*(Λ) is simple if and only if Λ is aperiodic and cofinal. The main advantage of our versions of aperiodicity and cofinality over existing ones is that ours are stated in terms of finite paths. To prove our main result, we first characterise each of aperiodicity and cofinality of Λ in terms of the ideal structure of C*(Λ). In an appendix we show how our new cofinality condition simplifies in a number of special cases which have been treated previously in the literature; even in these settings our results are new.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 2010

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References

REFERENCES

[1]Bates, T., Hong, J. H., Raeburn, I. and Szymaǹski, W.The ideal structure of the C*-algebras of infinite graphs. Illinois J. Math. 46 (2002), no. 4, 11591176.CrossRefGoogle Scholar
[2]Bates, T., Pask, D., Raeburn, I. and Szymaǹski, W.The C*-algebras of row-finite graphs. New York J. Math. 6 (2000), 307324.Google Scholar
[3]Burgstaller, B.The uniqueness of Cuntz–Krieger type algebras. J. reine angew. Math. 594 (2006), 207236.Google Scholar
[4]Cuntz, J. and Krieger, W.A class of C*-algebras and topological Markov chains. Invent. Math. 56 (1980), 251268.Google Scholar
[5]Davidson, K. R., Power, S. and Yang, D.Atomic representations of rank 2 graph algebras. J. Funct. Anal. 255 (2008), no. 4, 819853.CrossRefGoogle Scholar
[6]Dritschel, M. A., Marcantognini, S. and McCullough, S.Interpolation in semigroupoid algebras. J. reine angew. Math. 606 (2007), 140.Google Scholar
[7]Enomoto, M. and Watatani, Y.A graph theory for C*-algebras. Math. Japon 25 (1980), 435442.Google Scholar
[8]Exel, R.Inverse semigroups and combinatorial C*-algebras. Bull. Braz. Math. Soc. (N.S.) 39 (2008), no. 2, 191313.CrossRefGoogle Scholar
[9]Exel, R. and Laca, M.Cuntz–Krieger algebras for infinite matrices. J. reine angew. Math. 512 (1999), 119172.CrossRefGoogle Scholar
[10]Farthing, C.Removing sources from higher-rank graphs. J. Operator Theory, 60 (2008), 165198.Google Scholar
[11]Farthing, C., Muhly, P. S. and Yeend, T.Higher-rank graph C*-algebras: an inverse semigroup and groipoid approach. Semigroup Forum 71 (2005), no. 2, 159187.Google Scholar
[12]Fowler, N. J., Laca, M. and Raeburn, I.The C*-algebras of infinite graphs. Proc. Amer. Math. Soc. 128 (2000), no. 8, 23192327.CrossRefGoogle Scholar
[13]Hajac, P. M., Matthes, R. and Szymaǹski, W.A locally trivial quantum Hopf fibration. Algebr. Represent. Theory 9 (2006), no. 2, 121146.CrossRefGoogle Scholar
[14]Katsoulis, E. and Kribs, D.W.The C*-envelope of the tensor algebra of a directed graph. Integral Equations Operator Theory 56 (2006), no. 3, 401414.Google Scholar
[15]Katsura, T.A class of C*-algebras generalising both graph algebras and homeomorphism C*-algebras I. Fundamental results. Trans. Amer. Math. Soc. 356 (2004), no. 11, 42874322.CrossRefGoogle Scholar
[16]Kumjian, A. and Pask, D.Higher rank graph C*-algebras. New York J. Math. 6 (2000), 120.Google Scholar
[17]Kumjian, A., Pask, D., and Raeburn, I.Cuntz–Krieger algebras of directed graphs. Pacific J. Math. 184 (1998), no. 1, 161174.CrossRefGoogle Scholar
[18]Kumjian, A., Pask, D., Raeburn, I. and Renault, J.Graphs, groupoids and Cuntz–Krieger algebras. J. Funct. Anal. 144 (1997), no. 2, 505541.Google Scholar
[19]Muhly, P. S. and Tomforde, M.Topological quivers. Internat. J. Math. 16 (2005), 693756.CrossRefGoogle Scholar
[20]Pask, D., Quigg, J. and Raeburn, I.Coverings of k-graphs. J. Algebra 289 (2005), no. 1, 161191.Google Scholar
[21]Raeburn, I., Sims, A. and Yeend, T.Higher-rank graphs and their C*-algebras. Proc. Edinburgh Math. Soc. 46 (2003), 99115.Google Scholar
[22]Raeburn, I., Sims, A. and Yeend, T.The C*-algebras of finitely aligned higher-rank graphs. J. Funct. Anal. 213 (2004), 206240.Google Scholar
[23]Robertson, D. and Sims, A.Simplicity of C*-algebras associated to higher-rank graphs. Bull. London Math. Soc. 39 (2007), 337344.CrossRefGoogle Scholar
[24]Robertson, D. and Sims, A.Simplicity of C*-algebras associated to row finite locally convex higher-rank graphs. Israel J. Math. 172 (2009), 171192.CrossRefGoogle Scholar
[25]Robertson, G. and Steger, T.Affine buildings, tiling systems and higher rank Cuntz–Krieger algebras. J. reine. angew. Math. 513 (1999), 115144.Google Scholar
[26]Shotwell, J. Simplicity of finitely-aligned k-graph C*-algebras, preprint (arXiv:0810.4567 [Math OA]).Google Scholar
[27]Sims, A.C*-algebras associated to higher-rank graphs. PhD Thesis, University of Newcastle, 2003.Google Scholar
[28]Sims, A.Relative Cuntz–Krieger algebras of finitely aligned higher-rank graphs. Indiana Univ. Math. J. 55 (2006), 849868.CrossRefGoogle Scholar
[29]Sims, A.Gauge-invariant ideals in the C*-algebras of finitely aligned higher-rank graphs. Canad. J. Math. 58 (2006), 12681290.CrossRefGoogle Scholar
[30]Skalski, A. and Zacharias, J.Entropy of shifts on higher-rank graph C*-algebras. Houston J. Math. 34 (2008), no. 1, 269282.Google Scholar
[31]Spielberg, J.Graph-based models for Kirchberg algebras. J. Operator Theory 57 (2007), no. 2, 347374.Google Scholar
[32]Tomforde, M.A unified approach to Exel-Laca algebras and C*-algebras associated to graphs. J. Operator Theory 50 (2003), no. 2, 345368.Google Scholar