The 𝐶*-algebras of infinite graphs

Fowler, Neal; Laca, Marcelo; Raeburn, Iain

doi:10.1090/S0002-9939-99-05378-2

The $C^\ast$-algebras of infinite graphs
HTML articles powered by AMS MathViewer

by Neal J. Fowler, Marcelo Laca and Iain Raeburn

Proc. Amer. Math. Soc. 128 (2000), 2319-2327

DOI: https://doi.org/10.1090/S0002-9939-99-05378-2

Published electronically: December 8, 1999

PDF | Request permission

Abstract:

We associate $C^*$-algebras to infinite directed graphs that are not necessarily locally finite. By realizing these algebras as Cuntz-Krieger algebras in the sense of Exel and Laca, we are able to give criteria for their uniqueness and simplicity, generalizing results of Kumjian, Pask, Raeburn, and Renault for locally finite directed graphs.

References

T. Bates, D. Pask, I. Raeburn and W. Szymański, The $C^*$-algebras of row-finite graphs, The University of Newcastle, preprint, 1999.
Bruce Blackadar, Shape theory for $C^\ast$-algebras, Math. Scand. 56 (1985), no. 2, 249–275. MR 813640, DOI 10.7146/math.scand.a-12100
Joachim Cuntz, Simple $C^*$-algebras generated by isometries, Comm. Math. Phys. 57 (1977), no. 2, 173–185. MR 467330, DOI 10.1007/BF01625776
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
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
R. Exel, M. Laca and J. C. Quigg, Partial dynamical systems and $C^*$-algebras generated by partial isometries, The University of Newcastle, preprint, 1997.
R. Exel and M. Laca, Cuntz-Krieger algebras for infinite matrices, J. reine angew. Math., to appear.
N. J. Fowler and I. Raeburn, The Toeplitz algebra of a Hilbert bimodule, Indiana Univ. Math. J., to appear.
Masatoshi Enomoto and Yasuo Watatani, A graph theory for $C^{\ast }$-algebras, Math. Japon. 25 (1980), no. 4, 435–442. MR 594544
Astrid an Huef and Iain Raeburn, The ideal structure of Cuntz-Krieger algebras, Ergodic Theory Dynam. Systems 17 (1997), no. 3, 611–624. MR 1452183, DOI 10.1017/S0143385797079200
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
A. Kumjian, D. Pask and I. Raeburn, Cuntz-Krieger algebras of directed graphs, Pacific J. Math. 184 (1998), 161–174.
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
Michael V. Pimsner, A class of $C^*$-algebras generalizing both Cuntz-Krieger algebras and crossed products by $\textbf {Z}$, Free probability theory (Waterloo, ON, 1995) Fields Inst. Commun., vol. 12, Amer. Math. Soc., Providence, RI, 1997, pp. 189–212. MR 1426840