A dual graph construction for higher-rank graphs, and -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

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Given a -graph and an element of , we define the dual -graph, . We show that when is row-finite and has no sources, the -algebras and coincide. We use this isomorphism to apply Robertson and Steger's results to calculate the -theory of when is finite and strongly connected and satisfies the aperiodicity condition.

**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 -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 -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 -algebras*, Math. Japon.**25**(1980), 435-442. MR**0594544 (83d:46069a)****6.**D. G. Evans,*On higher-rank graph -algebras*, Ph.D. Thesis, Univ. Wales, 2002.**7.**N. J. Fowler, M. Laca, and I. Raeburn,*The**-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 -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 -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 -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 -theory for groups acting on buildings*, Can. J. Math.**53**(2001), 809-833. MR**1848508 (2002f:46141)****17.**W. Szymanski,*The range of -invariants for -algebras of infinite graphs*, Indiana Univ. Math. J.**51**(2002), 239-249. MR**1896162 (2003b:46077)**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
46L05

Retrieve articles in all journals with MSC (2000): 46L05

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.