Spectra for compact quantum group coactions and crossed products
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- by Raluca Dumitru and Costel Peligrad
- Trans. Amer. Math. Soc. 364 (2012), 3699-3713
- DOI: https://doi.org/10.1090/S0002-9947-2012-05562-0
- Published electronically: March 9, 2012
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Abstract:
We present definitions of both the Connes spectrum and the strong Connes spectrum for actions of compact quantum groups on $C^*$-algebras and obtain necessary and sufficient conditions for a crossed product to be a prime or a simple $C^*$-algebra. Our results extend to the case of compact quantum actions, namely the results of Gootman, Lazar and Peligrad, which in turn, generalize results by Connes, Olesen, Pedersen and Kishimoto for abelian group actions. We prove in addition that the Connes spectra are closed under tensor products. These results are new for compact nonabelian groups as well.References
- Saad Baaj and Georges Skandalis, Unitaires multiplicatifs et dualité pour les produits croisés de $C^*$-algèbres, Ann. Sci. École Norm. Sup. (4) 26 (1993), no. 4, 425–488 (French, with English summary). MR 1235438
- Florin P. Boca, Ergodic actions of compact matrix pseudogroups on $C^*$-algebras, Astérisque 232 (1995), 93–109. Recent advances in operator algebras (Orléans, 1992). MR 1372527
- Lawrence G. Brown, Stable isomorphism of hereditary subalgebras of $C^*$-algebras, Pacific J. Math. 71 (1977), no. 2, 335–348. MR 454645
- Alain Connes, Une classification des facteurs de type $\textrm {III}$, Ann. Sci. École Norm. Sup. (4) 6 (1973), 133–252 (French). MR 341115
- Raluca Dumitru, Simple and prime crossed products of $C^*$-algebras by compact quantum group coactions, J. Funct. Anal. 257 (2009), no. 5, 1480–1492. MR 2541277, DOI 10.1016/j.jfa.2009.05.023
- Raluca Dumitru, Unitary representations of compact quantum groups, Perspectives in operator algebras and mathematical physics, Theta Ser. Adv. Math., vol. 8, Theta, Bucharest, 2008, pp. 85–90. MR 2433028
- Raluca Dumitru and Costel Peligrad, Compact quantum group actions on $C^*$-algebras and invariant derivations, Proc. Amer. Math. Soc. 135 (2007), no. 12, 3977–3984. MR 2341948, DOI 10.1090/S0002-9939-07-09011-9
- Elliot C. Gootman, Aldo J. Lazar, and Costel Peligrad, Spectra for compact group actions, J. Operator Theory 31 (1994), no. 2, 381–399. MR 1331784
- Akitaka Kishimoto, Simple crossed products of $C^{\ast }$-algebras by locally compact abelian groups, Yokohama Math. J. 28 (1980), no. 1-2, 69–85. MR 623751
- M. B. Landstad, Operator algebras and compact groups, Operator algebras and group representations, Vol. II (Neptun, 1980) Monogr. Stud. Math., vol. 18, Pitman, Boston, MA, 1984, pp. 33–47. MR 733301
- Gert K. Pedersen, $C^{\ast }$-algebras and their automorphism groups, London Mathematical Society Monographs, vol. 14, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1979. MR 548006
- C. Peligrad, Compact actions commuting with ergodic actions and applications to crossed products, Trans. Amer. Math. Soc. 331 (1992), no. 2, 825–836. MR 1044964, DOI 10.1090/S0002-9947-1992-1044964-4
- Costel Peligrad, Locally compact group actions on $C^*$-algebras and compact subgroups, J. Funct. Anal. 76 (1988), no. 1, 126–139. MR 923048, DOI 10.1016/0022-1236(88)90052-3
- Piotr Podleś, Symmetries of quantum spaces. Subgroups and quotient spaces of quantum $\textrm {SU}(2)$ and $\textrm {SO}(3)$ groups, Comm. Math. Phys. 170 (1995), no. 1, 1–20. MR 1331688
- Marc A. Rieffel, Actions of finite groups on $C^{\ast }$-algebras, Math. Scand. 47 (1980), no. 1, 157–176. MR 600086, DOI 10.7146/math.scand.a-11882
- S. L. Woronowicz, Compact matrix pseudogroups, Comm. Math. Phys. 111 (1987), no. 4, 613–665. MR 901157
- S. L. Woronowicz, Compact quantum groups, Symétries quantiques (Les Houches, 1995) North-Holland, Amsterdam, 1998, pp. 845–884. MR 1616348
Bibliographic Information
- Raluca Dumitru
- Affiliation: Department of Mathematics and Statistics, University of North Florida, 1 UNF Drive, Jacksonville, Florida 32224 – and – Institute of Mathematics of the Romanian Academy, Bucharest, Romania
- Email: raluca.dumitru@unf.edu
- Costel Peligrad
- Affiliation: Department of Mathematical Sciences, University of Cincinnati, 5514 French Hall West, 2815 Commons Way, Cincinnati, Ohio 45221-0025
- Email: costel.peligrad@uc.edu
- Received by editor(s): June 6, 2010
- Received by editor(s) in revised form: December 28, 2010
- Published electronically: March 9, 2012
- Additional Notes: The first author was supported by a UNF Summer Research Grant
- © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 364 (2012), 3699-3713
- MSC (2010): Primary 47L65, 20G42
- DOI: https://doi.org/10.1090/S0002-9947-2012-05562-0
- MathSciNet review: 2901231