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$C^{\ast}$-algebras associated with branched coverings


Authors: Valentin Deaconu and Paul S. Muhly
Journal: Proc. Amer. Math. Soc. 129 (2001), 1077-1086
MSC (1991): Primary 46L55, 43A35; Secondary 43A07, 43A15, 43A22
DOI: https://doi.org/10.1090/S0002-9939-00-05697-5
Published electronically: October 4, 2000
MathSciNet review: 1814145
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Abstract: In this note we analyze the $C^{\ast}$-algebra associated with a branched covering both as a groupoid $C^{\ast}$-algebra and as a Cuntz-Pimsner algebra. We determine conditions when the algebra is simple and purely infinite. We indicate how to compute the K-theory of several examples, including one related to rational maps on the Riemann sphere.


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  • 1. Abadie, B., Eilers S., Exel, R. Morita equivalence for crossed products by Hilbert $C^{\ast}$-bimodules, Trans. Amer. Math. Soc. 350 (1998), 3043-3054. MR 98k:46109
  • 2. Anantharaman-Delaroche, C. Purely infinite $C^{\ast} $-algebras arising from dynamical systems, Bull. Soc. Math. France 125 (1997), 199-225. MR 99i:46051
  • 3. Beardon, A.F. Iteration of rational functions, Graduate Texts in Mathematics 132, Springer-Verlag 1991. MR 92j:30026
  • 4. Deaconu, V. Groupoids associated with endomorphisms, Trans. of the AMS 347 (1995), 1779-1786.
  • 5. Deaconu, V. A path model for circle algebras, J. Operator Theory 34 (1995), 57-89. MR 96m:46113
  • 6. Deaconu, V. Generalized Cuntz-Krieger algebras, Proc. Amer. Math. Soc. 124 (1996), 3427-3435. MR 97a:46081
  • 7. Deaconu, V. Generalized solenoids and $C^{\ast} $-algebras, to appear in Pacific J. Math.
  • 8. Deaconu, V. Embeddings of toral algebras and the corresponding Cuntz-Krieger algebras, preprint.
  • 9. Exel, R. Circle actions on $C^{\ast}$-algebras, partial automorphisms and a generalized Pimsner-Voiculescu exact sequence, J. Funct. Anal. 122 (1994), no. 2, 361-401. MR 95g:46122
  • 10. Fox, R. H. Covering spaces with singularities, Algebraic geometry and topology, Princeton 1957, 243-257. MR 23:A626
  • 11. Jiang, X., and Su, H. A classification of simple inductive limits of splitting interval algebras, J. Funct. Anal. 151 (1997), no. 1, 50-76. MR 99d:46076
  • 12. Kumjian, A. Preliminary algebras arising from local homeomorphisms, Math. Scand 52 (1983), 269-278. MR 85b:46078
  • 13. Kumjian, A. Fell bundles over groupoids, Proc. Amer. Math. Soc. 126 (1998), 1115-1125. MR 98i:46055
  • 14. Kumjian, A., Muhly, P. S., Renault, J. N., and Williams, D. P. The Brauer group of a locally compact groupoid, Amer. J. Math. 120 (1998), 901-954. CMP 99:01
  • 15. Muhly, P. S. Coordinates in Operator Algebra, to appear.
  • 16. Muhly, P. S., Renault, J. and Williams, D. P. Equivalence and isomorphism for groupoid $C^{\ast}$-algebras, J. Operator Theory 17 (1987), 3-22. MR 88h:46123
  • 17. Muhly, P. S., and Williams, D. P. Groupoid cohomology and the Dixmier-Douady class, Proc. London Math. Soc. 71 (1995), 109-134. MR 97d:46082
  • 18. Pimsner, M. A class of $C^{\ast}$-algebras generalizing both Cuntz-Krieger algebras and crossed products by $\mathbb{Z} $, in Free probability theory (D. Voiculescu, Ed.), 189-212, Fields Institute Communications 12, Amer. Math. Soc., Providence, 1997. MR 97k:46069
  • 19. Renault, J. A groupoid approach to $C^{\ast}$-algebras, Springer Lecture Notes, no. 793, 1980. MR 82h:46075
  • 20. Renault, J. Répresentations des produits croisés d'algèbres de groupoïdes, J. Operator Theory 18 (1987), 67-97. MR 89g:46108
  • 21. Renault, J. Cuntz-like Algebras, preprint 1998.
  • 22. Watatani, Y. Index for $C^{\ast}$-algebras, Memoirs of the AMS 83 (1990), No. 424, vi-117. MR 90i:46104

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Additional Information

Valentin Deaconu
Affiliation: Department of Mathematics, University of Nevada, Reno, Nevada 89557
Email: vdeaconu@math.unr.edu

Paul S. Muhly
Affiliation: Department of Mathematics, University of Iowa, Iowa City, Iowa 52242
Email: muhly@math.uiowa.edu

DOI: https://doi.org/10.1090/S0002-9939-00-05697-5
Keywords: $C^{\ast}$-algebras, branched coverings, dynamical systems
Received by editor(s): June 30, 1999
Published electronically: October 4, 2000
Additional Notes: This research was supported in part by grants from the National Science Foundation.
Communicated by: David R. Larson
Article copyright: © Copyright 2000 American Mathematical Society

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