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PERIODIC 2-GRAPHS ARISING FROM SUBSHIFTS

Published online by Cambridge University Press:  23 March 2010

DAVID PASK
Affiliation:
School of Mathematics and Applied Statistics, University of Wollongong, NSW 2522, Australia (email: [email protected])
IAIN RAEBURN*
Affiliation:
School of Mathematics and Applied Statistics, University of Wollongong, NSW 2522, Australia (email: [email protected])
NATASHA A. WEAVER
Affiliation:
School of Mathematical and Physical Sciences, University of Newcastle, NSW 2308, Australia (email: [email protected])
*
For correspondence; e-mail: [email protected]
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Abstract

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Higher-rank graphs were introduced by Kumjian and Pask to provide models for higher-rank Cuntz–Krieger algebras. In a previous paper, we constructed 2-graphs whose path spaces are rank-two subshifts of finite type, and showed that this construction yields aperiodic 2-graphs whoseC*-algebras are simple and are not ordinary graph algebras. Here we show that the construction also gives a family of periodic 2-graphs which we call domino graphs. We investigate the combinatorial structure of domino graphs, finding interesting points of contact with the existing combinatorial literature, and prove a structure theorem for the C*-algebras of domino graphs.

Type
Research Article
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2010

Footnotes

This research was supported by the Australian Research Council, and Natasha Weaver was supported by an Australian Postgraduate Award.

References

[1]Adji, S., Raeburn, I. and Rosjanuardi, R., ‘Group extensions and the primitive ideal spaces of group algebras’, Glasg. Math. J. 49 (2007), 8192.CrossRefGoogle Scholar
[2]Dang Ngoc, N., ‘Produits croisés restreintes et extensions de groupes’, mimeographed notes, Paris 1977.Google Scholar
[3]Davidson, K. R., Power, S. C. and Yang, D., ‘Atomic representations of rank 2 graph algebras’, J. Funct. Anal. 255 (2008), 819853.CrossRefGoogle Scholar
[4]Davidson, K. R. and Yang, D., ‘Periodicity in rank 2 graph algebras’, Canad. J. Math. 61 (2009), 12391261.CrossRefGoogle Scholar
[5]Echterhoff, S., ‘On induced covariant systems’, Proc. Amer. Math. Soc. 108 (1990), 703706.Google Scholar
[6]Evans, D. G., ‘On the K-theory of higher rank graph C *-algebras’, New York J. Math. 14 (2008), 131.Google Scholar
[7]Farthing, C., Pask, D. and Sims, A., ‘Crossed products of k-graph algebras by ℤl’, Houston J. Math. 35 (2009), 903933.Google Scholar
[8]Fowler, N. J. and Raeburn, I., ‘The Toeplitz algebra of a Hilbert bimodule’, Indiana Univ. Math. J. 48 (1999), 155181.CrossRefGoogle Scholar
[9]Green, P., ‘The local structure of twisted covariance algebras’, Acta Math. 140 (1978), 191250.Google Scholar
[10]Katayama, Y. and Takehana, H., ‘On automorphisms of generalized Cuntz algebras’, Int. J. Math. 9 (1998), 493512.CrossRefGoogle Scholar
[11]Kishimoto, A., ‘Outer automorphisms and reduced crossed products of simple C *-algebras’, Commun. Math. Phys. 81 (1981), 429435.CrossRefGoogle Scholar
[12]Kitchens, B. and Schmidt, K., ‘Mixing sets and relative entropies for higher dimensional Markov shifts’, Ergod. Th. & Dynam. Sys. 13 (1993), 705735.CrossRefGoogle Scholar
[13]Kumjian, A. and Pask, D., ‘Higher rank graph C *-algebras’, New York J. Math. 6 (2000), 120.Google Scholar
[14]Lyndon, R. C., ‘On Burnside problem I’, Trans. Amer. Math. Soc. 77 (1954), 202215.Google Scholar
[15]Marelli, D. and Raeburn, I., ‘Proper actions which are not saturated’, Proc. Amer. Math. Soc. 137 (2009), 22732283.Google Scholar
[16]Olesen, D. and Pedersen, G. K., ‘Partially inner C *-dynamical systems’, J. Funct. Anal. 66 (1986), 263281.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. Anal. 239 (2006), 137178.Google Scholar
[18]Pask, D., Raeburn, I. and Weaver, N. A., ‘A family of 2-graphs arising from two-dimensional subshifts’, Ergod. Th. & Dynam. Sys. 29 (2009), 16131639.CrossRefGoogle Scholar
[19]Pask, D. and Rho, S.-J., ‘Some intrinsic properties of simple graph C *-algebras’, in: Proceedings of the Conference on Operator Algebras and Mathematical Physics, Constanţa, 2001 (Theta Foundation, Bucharest, 2003), pp. 325340.Google Scholar
[20]Pimsner, M. V. and Voiculescu, D., ‘Exact sequences for K-groups and Ext-groups of certain cross-products of C *-algebras’, J. Operator Theory 4 (1980), 93118.Google Scholar
[21]Raeburn, I., Graph Algebras, CBMS Regional Conference Series in Mathematics, 103 (American Mathematical Society, Providence, RI, 2005).CrossRefGoogle Scholar
[22]Raeburn, I., Sims, A. and Yeend, T., ‘Higher-rank graphs and their C *-algebras’, Proc. Edinb. Math. Soc. 46 (2003), 99115.CrossRefGoogle Scholar
[23]Raeburn, I. and Williams, D. P., Morita Equivalence and Continuous-Trace C *-Algebras, Mathematical Surveys and Monographs, 60 (American Mathematical Society, Providence, RI, 1998).CrossRefGoogle Scholar
[24]Rieffel, M. A., ‘Proper actions of groups on C*-algebras’, in: Mappings of Operator Algebras, Progress in Mathematics, 84 (Birkhäuser, Boston, MA, 1990), pp. 141182.Google Scholar
[25]Robertson, D. I. and Sims, A., ‘Simplicity of C *-algebras associated to higher rank graphs’, Bull. London Math. Soc. 39 (2007), 337344.CrossRefGoogle Scholar
[26]Robertson, G. and Steger, T., ‘Affine buildings, tiling systems and higher rank Cuntz–Krieger algebras’, J. reine angew. Math. 513 (1999), 115144.CrossRefGoogle Scholar
[27]Rørdam, M., Larsen, F. and Lausten, N. J., An Introduction to K-Theory for C *-Algebras, London Mathematical Society Student Texts, 49 (Cambridge University Press, Cambridge, 2000).Google Scholar
[28]Rosenberg, J. and Schochet, C., ‘The Künneth theorem and the universal coefficent theorem for Kasparov’s generalised K-functor’, Duke Math. J. 55 (1987), 431474.CrossRefGoogle Scholar
[29]Ruskey, F., Miers, C. R. and Sawada, J., ‘The number of irreducible polynomials and Lyndon words with given trace’, SIAM J. Discrete Math. 41 (2001), 240245.Google Scholar
[30]Schmidt, K., Dynamical Systems of Algebraic Origin, Progress in Mathematics, 128 (Birkhäuser, Basel, 1995).Google Scholar
[31]Skalski, A. and Zacharias, J., ‘Entropy of shifts on higher-rank graph C *-algebras’, Houston J. Math. 34 (2008), 269282.Google Scholar
[32]Williams, D. P., Crossed Products of C *-Algebras, Mathematical Surveys and Monographs, 134 (American Mathematical Society, Providence, RI, 2007).CrossRefGoogle Scholar