On partial actions and groupoids
HTML articles powered by AMS MathViewer
- by Fernando Abadie PDF
- Proc. Amer. Math. Soc. 132 (2004), 1037-1047 Request permission
Abstract:
We prove that, as in the case of global actions, any partial action gives rise to a groupoid provided with a Haar system, whose $C^*$-algebra agrees with the crossed product by the partial action.References
- F. Abadie, Enveloping actions and Takai duality for partial actions, J. Funct. Anal. 197 (2003), 14–67.
- 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
- Ruy Exel, The Bunce-Deddens algebras as crossed products by partial automorphisms, Bol. Soc. Brasil. Mat. (N.S.) 25 (1994), no. 2, 173–179. MR 1306559, DOI 10.1007/BF01321306
- Ruy Exel, Approximately finite $C^\ast$-algebras and partial automorphisms, Math. Scand. 77 (1995), no. 2, 281–288. MR 1379271, DOI 10.7146/math.scand.a-12566
- Ruy Exel, Twisted partial actions: a classification of regular $C^*$-algebraic bundles, Proc. London Math. Soc. (3) 74 (1997), no. 2, 417–443. MR 1425329, DOI 10.1112/S0024611597000154
- Ruy Exel, Partial actions of groups and actions of inverse semigroups, Proc. Amer. Math. Soc. 126 (1998), no. 12, 3481–3494. MR 1469405, DOI 10.1090/S0002-9939-98-04575-4
- Ruy Exel and Marcelo Laca, Cuntz-Krieger algebras for infinite matrices, J. Reine Angew. Math. 512 (1999), 119–172. MR 1703078, DOI 10.1515/crll.1999.051
- Ruy Exel, Marcelo Laca, and John Quigg, Partial dynamical systems and $C^*$-algebras generated by partial isometries, J. Operator Theory 47 (2002), no. 1, 169–186. MR 1905819
- John W. Green, Harmonic functions in domains with multiple boundary points, Amer. J. Math. 61 (1939), 609–632. MR 90, DOI 10.2307/2371316
- André Haefliger, Structures feuilletées et cohomologie à valeur dans un faisceau de groupoïdes, Comment. Math. Helv. 32 (1958), 248–329 (French). MR 100269, DOI 10.1007/BF02564582
- Kevin McClanahan, $K$-theory for partial crossed products by discrete groups, J. Funct. Anal. 130 (1995), no. 1, 77–117. MR 1331978, DOI 10.1006/jfan.1995.1064
- Alexandru Nica, On a groupoid construction for actions of certain inverse semigroups, Internat. J. Math. 5 (1994), no. 3, 349–372. MR 1274123, DOI 10.1142/S0129167X94000206
- Alan L. T. Paterson, Groupoids, inverse semigroups, and their operator algebras, Progress in Mathematics, vol. 170, Birkhäuser Boston, Inc., Boston, MA, 1999. MR 1724106, DOI 10.1007/978-1-4612-1774-9
- John Quigg and Iain Raeburn, Characterisations of crossed products by partial actions, J. Operator Theory 37 (1997), no. 2, 311–340. MR 1452280
- John Quigg and Nándor Sieben, $C^*$-actions of $r$-discrete groupoids and inverse semigroups, J. Austral. Math. Soc. Ser. A 66 (1999), no. 2, 143–167. MR 1671944, DOI 10.1017/S1446788700039288
- Jean Renault, A groupoid approach to $C^{\ast }$-algebras, Lecture Notes in Mathematics, vol. 793, Springer, Berlin, 1980. MR 584266, DOI 10.1007/BFb0091072
- Nándor Sieben, $C^\ast$-crossed products by partial actions and actions of inverse semigroups, J. Austral. Math. Soc. Ser. A 63 (1997), no. 1, 32–46. MR 1456588, DOI 10.1017/S1446788700000306
Additional Information
- Fernando Abadie
- Affiliation: Centro de Matemática, Facultad de Ciencias, Universidad de la República, Iguá 4225, 11400, Montevideo, Uruguay
- Email: fabadie@cmat.edu.uy
- Received by editor(s): April 25, 2001
- Received by editor(s) in revised form: October 24, 2002
- Published electronically: November 7, 2003
- Additional Notes: This work was partially financied by Fapesp, Brazil, Processo No. 95/04097-9
- Communicated by: David R. Larson
- © Copyright 2003 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 132 (2004), 1037-1047
- MSC (2000): Primary 46L05
- DOI: https://doi.org/10.1090/S0002-9939-03-07300-3
- MathSciNet review: 2045419