A dual graph construction for higher-rank graphs, and $K$-theory for finite 2-graphs
HTML articles powered by AMS MathViewer
- by Stephen Allen, David Pask and Aidan Sims
- Proc. Amer. Math. Soc. 134 (2006), 455-464
- DOI: https://doi.org/10.1090/S0002-9939-05-07994-3
- Published electronically: June 29, 2005
- PDF | Request permission
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.References
- Teresa Bates, Applications of the gauge-invariant uniqueness theorem for graph algebras, Bull. Austral. Math. Soc. 66 (2002), no. 1, 57–67. MR 1922607, DOI 10.1017/S0004972700020670
- Teresa Bates and David Pask, Flow equivalence of graph algebras, Ergodic Theory Dynam. Systems 24 (2004), no. 2, 367–382. MR 2054048, DOI 10.1017/S0143385703000348
- Teresa Bates, David Pask, Iain Raeburn, and Wojciech Szymański, The $C^*$-algebras of row-finite graphs, New York J. Math. 6 (2000), 307–324. MR 1777234
- Joachim Cuntz and Wolfgang Krieger, A class of $C^{\ast }$-algebras and topological Markov chains, Invent. Math. 56 (1980), no. 3, 251–268. MR 561974, DOI 10.1007/BF01390048
- Masatoshi Enomoto and Yasuo Watatani, A graph theory for $C^{\ast }$-algebras, Math. Japon. 25 (1980), no. 4, 435–442. MR 594544
- D. G. Evans, On higher-rank graph $C^*$-algebras, Ph.D. Thesis, Univ. Wales, 2002.
- Neal J. Fowler, Marcelo Laca, and Iain Raeburn, The $C^*$-algebras of infinite graphs, Proc. Amer. Math. Soc. 128 (2000), no. 8, 2319–2327. MR 1670363, DOI 10.1090/S0002-9939-99-05378-2
- Alex Kumjian and David Pask, Higher rank graph $C^\ast$-algebras, New York J. Math. 6 (2000), 1–20. MR 1745529
- Alex Kumjian, David Pask, and Iain Raeburn, Cuntz-Krieger algebras of directed graphs, Pacific J. Math. 184 (1998), no. 1, 161–174. MR 1626528, DOI 10.2140/pjm.1998.184.161
- Alex Kumjian, David Pask, Iain Raeburn, and Jean Renault, Graphs, groupoids, and Cuntz-Krieger algebras, J. Funct. Anal. 144 (1997), no. 2, 505–541. MR 1432596, DOI 10.1006/jfan.1996.3001
- M. H. Mann, Iain Raeburn, and C. E. Sutherland, Representations of finite groups and Cuntz-Krieger algebras, Bull. Austral. Math. Soc. 46 (1992), no. 2, 225–243. MR 1183780, DOI 10.1017/S0004972700011862
- Eberhard Kirchberg and N. Christopher Phillips, Embedding of exact $C^*$-algebras in the Cuntz algebra $\scr O_2$, J. Reine Angew. Math. 525 (2000), 17–53. MR 1780426, DOI 10.1515/crll.2000.065
- Iain Raeburn and Wojciech Szymański, Cuntz-Krieger algebras of infinite graphs and matrices, Trans. Amer. Math. Soc. 356 (2004), no. 1, 39–59. MR 2020023, DOI 10.1090/S0002-9947-03-03341-5
- Iain Raeburn, Aidan Sims, and Trent Yeend, Higher-rank graphs and their $C^*$-algebras, Proc. Edinb. Math. Soc. (2) 46 (2003), no. 1, 99–115. MR 1961175, DOI 10.1017/S0013091501000645
- Guyan Robertson and Tim Steger, Affine buildings, tiling systems and higher rank Cuntz-Krieger algebras, J. Reine Angew. Math. 513 (1999), 115–144. MR 1713322, DOI 10.1515/crll.1999.057
- Guyan Robertson and Tim Steger, Asymptotic $K$-theory for groups acting on $\~A_2$ buildings, Canad. J. Math. 53 (2001), no. 4, 809–833. MR 1848508, DOI 10.4153/CJM-2001-033-4
- Wojciech Szymański, The range of $K$-invariants for $C^*$-algebras of infinite graphs, Indiana Univ. Math. J. 51 (2002), no. 1, 239–249. MR 1896162, DOI 10.1512/iumj.2002.51.1920
Bibliographic 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
- MR Author ID: 671497
- Email: aidan.sims@newcastle.edu.au
- 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
- © Copyright 2005
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 134 (2006), 455-464
- MSC (2000): Primary 46L05
- DOI: https://doi.org/10.1090/S0002-9939-05-07994-3
- MathSciNet review: 2176014