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Generalised morphisms of $ k$-graphs: $ k$-morphs


Authors: Alex Kumjian, David Pask and Aidan Sims
Journal: Trans. Amer. Math. Soc. 363 (2011), 2599-2626
MSC (2000): Primary 46L05
DOI: https://doi.org/10.1090/S0002-9947-2010-05152-9
Published electronically: December 20, 2010
MathSciNet review: 2763728
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Abstract: In a number of recent papers, $ (k+l)$-graphs have been constructed from $ k$-graphs by inserting new edges in the last $ l$ dimensions. These constructions have been motivated by $ C^*$-algebraic considerations, so they have not been treated systematically at the level of higher-rank graphs themselves. Here we introduce $ k$-morphs, which provide a systematic unifying framework for these various constructions. We think of $ k$-morphs as the analogue, at the level of $ k$-graphs, of $ C^*$-correspondences between $ C^*$-algebras. To make this analogy explicit, we introduce a category whose objects are $ k$-graphs and whose morphisms are isomorphism classes of $ k$-morphs. We show how to extend the assignment $ \Lambda \mapsto C^*(\Lambda)$ to a functor from this category to the category whose objects are $ C^*$-algebras and whose morphisms are isomorphism classes of $ C^*$-correspondences.


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  • 1. S. Allen, D. Pask, and A. Sims, A dual graph construction for higher-rank graphs, and $ K$-theory for finite $ 2$-graphs, Proc. Amer. Math. Soc. 134 (2006), 455-464. MR 2176014 (2006e:46078)
  • 2. T. Bates, Applications of the gauge-invariant uniqueness theorem for graph algebras, Bull. Austral. Math. Soc. 66 (2002), 57-67. MR 1922607 (2003g:46064)
  • 3. T. Bates, D. Pask, I. Raeburn, and W. Szymański, The $ C^*$-algebras of row-finite graphs, New York J. Math. 6 (2000), 307-324. MR 1777234 (2001k:46084)
  • 4. B. Blackadar, Operator algebras. Theory of $ C\sp *$-algebras and von Neumann algebras. Encyclopaedia of Mathematical Sciences, 122. Operator Algebras and Non-commutative Geometry, III. Springer-Verlag, Berlin, 2006. MR 2188261 (2006k:46082)
  • 5. V. Deaconu, Continuous graphs and $ C\sp *$-algebras, Operator theoretical methods (Timişoara, 1998), 137-149, Theta Found., Bucharest, 2000. MR 1770320 (2001g:46123)
  • 6. V. Deaconu, Iterating the Pimsner construction, New York J. Math. 13 (2007), 199-213. MR 2336239 (2008g:46093)
  • 7. K. Deicke, J.H. Hong, and W. Szymański, Stable rank of graph algebras. Type I graph algebras and their limits, Indiana Univ. Math. J. 52 (2003), 963-979. MR 2001940 (2004e:46065)
  • 8. D. Drinen and M. Tomforde, Computing K-theory and Ext for graph C*-algebras, Illinois J. Math. 46 (2002), 81-91. MR 1936076 (2003k:46103)
  • 9. S. Echterhoff, S. Kaliszewski, J. Quigg and I. Raeburn, A categorical approach to imprimitivity theorems for $ C\sp *$-dynamical systems, Mem. Amer. Math. Soc. 180 (2006), viii+169 pp. MR 2203930 (2007m:46107)
  • 10. D. G. Evans, On the $ K$-theory of higher-rank graph $ C^*$-algebras, New York J. Math. 14 (2008), 1-31. MR 2383584 (2009e:46065)
  • 11. C. Farthing, Removing sources from higher-rank graphs, J. Operator Theory, 60 (2008), no. 1, 165-198. MR 2415563 (2009k:46097)
  • 12. C. Farthing, P. S. Muhly, and T. Yeend, Higher-rank graph $ C^*$-algebras: An inverse semigroup approach, Semigroup Forum 71 (2005), 159-187. MR 2184052 (2006h:46052)
  • 13. 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)
  • 14. N. J. Fowler, P. S. Muhly and I. Raeburn, Representations of Cuntz-Pimsner algebras, Indiana Univ. Math. J. 52 (2003), 569-605. MR 1986889 (2005d:46114)
  • 15. C. Farthing, D. Pask and A. Sims, Crossed products of $ k$-graph $ C^*$-algebras by $ \mathbb{Z}^l$, Houston J. Math. 35 (2009), no. 3, 903-933. MR 2534288 (2010h:46079)
  • 16. T. Katsura, A class of $ C^*$-algebras generalizing both graph algebras and homeomorphism $ C^*$-algebras. I. Fundamental results, Trans. Amer. Math. Soc. 356 (2004), 4287-4322. MR 2067120 (2005b:46119)
  • 17. T. Katsura, On $ C^*$-algebras associated with $ C^*$-correspondences, J. Funct. Anal. 217 (2004), 366-401. MR 2102572 (2005e:46099)
  • 18. A. Kumjian and D. Pask, Higher rank graph $ C^*$-algebras, New York J. Math. 6 (2000), 1-20. MR 1745529 (2001b:46102)
  • 19. A. Kumjian and D. Pask, Actions of $ \mathbb{Z}^k$ associated to higher-rank graphs, Ergod. Th. & Dynam. Sys. 23 (2003), 1153-1172. MR 1997971 (2004f:46066)
  • 20. 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)
  • 21. A. Kumjian, D. Pask and A. Sims, $ C^*$-algebras associated to coverings of k-graphs, Doc. Math. 13 (2008), 161-205. MR 2520478 (2010f:46110)
  • 22. E. C. Lance, Hilbert $ C^*$-modules: A toolkit for operator algebraists, London Math. Soc. Lecture Note Series, vol. 210, Cambridge Univ. Press, Cambridge, 1994. MR 1325694 (96k:46100)
  • 23. N.P. Landsman, Bicategories of operator algebras and Poisson manifolds (English summary), Mathematical physics in mathematics and physics (Siena, 2000), 271-286, Fields Inst. Commun., 30, Amer. Math. Soc., Providence, RI, 2001. MR 1867561 (2002h:46099)
  • 24. P.S. Muhly, D. Pask and M. Tomforde, Strong shift equivalence of $ C^*$-correspondences, Israel J. Math. 167 (2008), 315-346. MR 2448028
  • 25. P.S. Muhly and M. Tomforde, Topological quivers, Internat. J. Math. 16 (2005), 693-755. MR 2158956 (2006i:46099)
  • 26. D. Pask, J. Quigg and I. Raeburn, Coverings of $ k$-graphs, J. Algebra 289 (2005), 161-191. MR 2139097 (2006b:46077)
  • 27. D. Pask, J. Quigg and A. Sims, Coverings of skew-products and crossed products by coactions, J. Austral. Math. Soc., 86 (2009), no. 3, 379-398. MR 2529331 (2010f:46103)
  • 28. D. Pask, I. Raeburn, M. Rørdam and A. Sims, Rank-2 graphs whose $ C^*$-algebras are direct limits of circle algebras, J. Funct. Anal. 239 (2006), 137-178. MR 2258220 (2007e:46053)
  • 29. M. V. Pimsner, A class of $ C^*$-algebras generalizing both Cuntz-Krieger algebras and crossed products by $ \mathbb{Z}$, Fields Institute Communications 12 (1997), 189-212. MR 1426840 (97k:46069)
  • 30. I. Raeburn, Graph algebras, CBMS Regional Conference Series in Mathematics, Vol. 103, Amer. Math. Soc., 2005. MR 2135030 (2005k:46141)
  • 31. 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)
  • 32. I. Raeburn, A. Sims, and T. Yeend, The $ C^*$-algebras of finitely aligned higher-rank graphs, J. Funct. Anal. 213 (2004), 206-240. MR 2069786 (2005e:46103)
  • 33. I. Raeburn and D. P. Williams, Morita equivalence and continuous-trace $ C^*$-algebras, Math. Surveys and Monographs, vol. 60, Amer. Math. Soc., Providence, 1998. MR 1634408 (2000c:46108)
  • 34. D.I. Robertson, $ C^*$-algebras generated by $ 2$-graphs and variations under factorisation properties, unpublished notes, University of Newcastle, 2006. http://www.uow.edu.au/$ \sim$asims/index_files/page0003.html
  • 35. 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)
  • 36. 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)
  • 37. J. Schweizer, Crossed products by $ C^*$-correspondences and Cuntz-Pimsner algebras, $ C^*$-algebras (Springer, Berlin, 2000), pp. 203-226. MR 1798598 (2002f:46133)
  • 38. J. Spielberg, A functorial approach to the $ C^*$-algebras of a graph, Internat. J. Math. 13 (2002), 245-277. MR 1911104 (2004e:46084)
  • 39. J. Spielberg, A commuting three-coloured graph which is not a $ 3$-graph, private communication.

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Additional Information

Alex Kumjian
Affiliation: Department of Mathematics (084), University of Nevada, Reno, Nevada 89557-0084
Email: alex@unr.edu

David Pask
Affiliation: School of Mathematics and Applied Statistics, University of Wollongong, NSW 2522, Australia
Email: dpask@uow.edu.au

Aidan Sims
Affiliation: School of Mathematics and Applied Statistics, University of Wollongong, NSW 2522, Australia
Email: asims@uow.edu.au

DOI: https://doi.org/10.1090/S0002-9947-2010-05152-9
Keywords: $C^{*}$-algebra, graph algebra, $k$-graph, $C^{*}$-correspondence.
Received by editor(s): December 6, 2007
Received by editor(s) in revised form: June 30, 2009
Published electronically: December 20, 2010
Additional Notes: This research was supported by the Australian Research Council.
Article copyright: © Copyright 2010 American Mathematical Society

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