𝐶*-algebras associated with branched coverings

Deaconu, Valentin; Muhly, Paul

doi:10.1090/S0002-9939-00-05697-5

$C^{\ast }$-algebras associated with branched coverings
HTML articles powered by AMS MathViewer

by Valentin Deaconu and Paul S. Muhly PDF

Proc. Amer. Math. Soc. 129 (2001), 1077-1086 Request permission

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.

References

Beatriz Abadie, Søren Eilers, and Ruy Exel, Morita equivalence for crossed products by Hilbert $C^*$-bimodules, Trans. Amer. Math. Soc. 350 (1998), no. 8, 3043–3054. MR 1467459, DOI 10.1090/S0002-9947-98-02133-3
Claire Anantharaman-Delaroche, Purely infinite $C^*$-algebras arising from dynamical systems, Bull. Soc. Math. France 125 (1997), no. 2, 199–225 (English, with English and French summaries). MR 1478030, DOI 10.24033/bsmf.2304
Alan F. Beardon, Iteration of rational functions, Graduate Texts in Mathematics, vol. 132, Springer-Verlag, New York, 1991. Complex analytic dynamical systems. MR 1128089, DOI 10.1007/978-1-4612-4422-6
Deaconu, V. Groupoids associated with endomorphisms, Trans. of the AMS 347 (1995), 1779-1786.
Valentin Deaconu, A path model for circle algebras, J. Operator Theory 34 (1995), no. 1, 57–89. MR 1361567
Valentin Deaconu, Generalized Cuntz-Krieger algebras, Proc. Amer. Math. Soc. 124 (1996), no. 11, 3427–3435. MR 1343688, DOI 10.1090/S0002-9939-96-03484-3
Deaconu, V. Generalized solenoids and $C^*$-algebras, to appear in Pacific J. Math.
Deaconu, V. Embeddings of toral algebras and the corresponding Cuntz-Krieger algebras, preprint.
Ruy Exel, Circle actions on $C^*$-algebras, partial automorphisms, and a generalized Pimsner-Voiculescu exact sequence, J. Funct. Anal. 122 (1994), no. 2, 361–401. MR 1276163, DOI 10.1006/jfan.1994.1073
Ralph H. Fox, Covering spaces with singularities, A symposium in honor of S. Lefschetz, Princeton University Press, Princeton, N.J., 1957, pp. 243–257. MR 0123298
Xinhui Jiang and Hongbing Su, A classification of simple limits of splitting interval algebras, J. Funct. Anal. 151 (1997), no. 1, 50–76. MR 1487770, DOI 10.1006/jfan.1997.3120
Alexander Kumjian, Preliminary algebras arising from local homeomorphisms, Math. Scand. 52 (1983), no. 2, 269–278. MR 702958, DOI 10.7146/math.scand.a-12006
Alex Kumjian, Fell bundles over groupoids, Proc. Amer. Math. Soc. 126 (1998), no. 4, 1115–1125. MR 1443836, DOI 10.1090/S0002-9939-98-04240-3
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.
Muhly, P. S. Coordinates in Operator Algebra, to appear.
Paul S. Muhly, Jean N. Renault, and Dana P. Williams, Equivalence and isomorphism for groupoid $C^\ast$-algebras, J. Operator Theory 17 (1987), no. 1, 3–22. MR 873460
Paul S. Muhly and Dana P. Williams, Groupoid cohomology and the Dixmier-Douady class, Proc. London Math. Soc. (3) 71 (1995), no. 1, 109–134. MR 1327935, DOI 10.1112/plms/s3-71.1.109
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
Jean Renault, A groupoid approach to $C^{\ast }$-algebras, Lecture Notes in Mathematics, vol. 793, Springer, Berlin, 1980. MR 584266, DOI 10.1007/BFb0091072
Jean Renault, Représentation des produits croisés d’algèbres de groupoïdes, J. Operator Theory 18 (1987), no. 1, 67–97 (French). MR 912813
Renault, J. Cuntz-like Algebras, preprint 1998.
Yasuo Watatani, Index for $C^*$-subalgebras, Mem. Amer. Math. Soc. 83 (1990), no. 424, vi+117. MR 996807, DOI 10.1090/memo/0424