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A dual graph construction for higher-rank graphs, and $K$-theory for finite 2-graphs


Authors: Stephen Allen, David Pask and Aidan Sims
Journal: Proc. Amer. Math. Soc. 134 (2006), 455-464
MSC (2000): Primary 46L05
DOI: https://doi.org/10.1090/S0002-9939-05-07994-3
Published electronically: June 29, 2005
MathSciNet review: 2176014
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Abstract: Given a $k$-graph $ \Lambda $ and an element $p$ of $\mathbb{N} ^k$, we define the dual $k$-graph, $p\Lambda $. We show that when $\Lambda $ is row-finite and has no sources, the $C^*$-algebras $C^*(\Lambda )$ and $C^*(p\Lambda )$ coincide. We use this isomorphism to apply Robertson and Steger's results to calculate the $K$-theory of $C^*(\Lambda )$ when $\Lambda $ is finite and strongly connected and satisfies the aperiodicity condition.


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  • 1. T. Bates, Applications of the gauge-invariant uniqueness theorem for graph algebras, Bull. Austral. Math. Soc. 65 (2002), 55-67. MR 1922607 (2003g:46064)
  • 2. T. Bates and D. Pask, Flow equivalence of graph algebras, Ergodic Theory Dynam. Systems 24 (2004), 367-382. MR 2054048 (2004m:37019)
  • 3. T. Bates, D. Pask, I. Raeburn, and W. Szymanski, The $C^*$-algebras of row-finite graphs, New York J. Math. 6 (2000), 307-324. MR 1777234 (2001k:46084)
  • 4. J. Cuntz and W. Krieger, A class of $C^*$-algebras and topological Markov chains, Invent. Math. 56 (1980), 251-268. MR 0561974 (82f:46073a)
  • 5. M. Enomoto and Y. Watatani, A graph theory for $C^* $-algebras, Math. Japon. 25 (1980), 435-442. MR 0594544 (83d:46069a)
  • 6. D. G. Evans, On higher-rank graph $C^*$-algebras, Ph.D. Thesis, Univ. Wales, 2002.
  • 7. N. J. Fowler, M. Laca, and I. Raeburn, The $C^*$-algebras of infinite graphs, Proc. Amer. Math. Soc. 128 (2000), 2319-2327. MR 1670363 (2000k:46079)
  • 8. A. Kumjian and D. Pask, Higher rank graph $C^*$-algebras, New York J. Math. 6 (2000) 1-20. MR 1745529 (2001b:46102)
  • 9. A. Kumjian, D. Pask, and I. Raeburn, Cuntz-Krieger algebras of directed graphs, Pacific J. Math. 184 (1998), 161-174. MR 1626528 (99i:46049)
  • 10. A. Kumjian, D. Pask, I. Raeburn, and J. Renault, Graphs, groupoids and Cuntz-Krieger algebras, J. Funct. Anal. 144 (1997), 505-541. MR 1432596 (98g:46083)
  • 11. M.H. Mann, I. Raeburn, and C.E. Sutherland, Representations of finite groups and Cuntz-Krieger algebras, Bull. Austral. Math. Soc. 46 (1992), 225-243. MR 1183780 (93k:46046)
  • 12. N.C. Phillips, A classification theorem for nuclear purely infinite simple $C^*$-algebras, Documenta Math. 5 (2000), 49-114. MR 1745197 (2001d:46086b)
  • 13. I. Raeburn and W. Szymanski, Cuntz-Krieger algebras of infinite graphs and matrices, Trans. Amer. Math. Soc. 356 (2004), 39-59. MR 2020023 (2004i:46087)
  • 14. I. Raeburn, A. Sims, and T. Yeend, Higher-rank graphs and their $C^*$-algebras, Proc. Edinb. Math. Soc 46 (2003), 99-115. MR 1961175 (2004f:46068)
  • 15. G. Robertson and T. Steger, Affine buildings, tiling systems and higher rank Cuntz-Krieger algebras, J. Reine Angew. Math. 513 (1999), 115-144. MR 1713322 (2000j:46109)
  • 16. G. Robertson and T. Steger, Asymptotic $K$-theory for groups acting on $\tilde A_2$ buildings, Can. J. Math. 53 (2001), 809-833. MR 1848508 (2002f:46141)
  • 17. W. Szymanski, The range of $K$-invariants for $C^*$-algebras of infinite graphs, Indiana Univ. Math. J. 51 (2002), 239-249. MR 1896162 (2003b:46077)

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Additional Information

Stephen Allen
Affiliation: Department of Mathematics, University of Newcastle, New South Wales 2308, Australia
Email: stephen.allen@studentmail.newcastle.edu.au

David Pask
Affiliation: Department of Mathematics, University of Newcastle, New South Wales 2308, Australia
Email: david.pask@newcastle.edu.au

Aidan Sims
Affiliation: Department of Mathematics, University of Newcastle, New South Wales 2308, Australia
Email: aidan.sims@newcastle.edu.au

DOI: https://doi.org/10.1090/S0002-9939-05-07994-3
Keywords: Graphs as categories, graph algebra, $C^*$-algebra, $K$-theory
Received by editor(s): March 22, 2004
Received by editor(s) in revised form: September 20, 2004
Published electronically: June 29, 2005
Additional Notes: This research was supported by the Australian Research Council.
Communicated by: David R. Larson
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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