Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Spectra for compact quantum group coactions and crossed products


Authors: Raluca Dumitru and Costel Peligrad
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
Published electronically: March 9, 2012
MathSciNet review: 2901231
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • 1. S. Baaj, G. Skandalis, Unitaires multiplicatifs et dualité pour les produits croisés de $ C^*$-
    algè bres, Ann. Sci. École Norm. Sup. (4) 26 (4)1993, 425-488. MR 1235438 (94e:46127)
  • 2. F. Boca, Ergodic actions of compact matrix pseudogroups on $ C^*$-algebras, in: Recent Advances in Operator Algebras, Orléans, 1992, Astérisque 232 (1995), 93-109. MR 1372527 (97d:46075)
  • 3. L.G. Brown, Stable isomorphisms of hereditary subalgebras of $ C^*$-algebras, Pacific J. Math. 71 (1977), 335-349. MR 0454645 (56:12894)
  • 4. A. Connes, Une classification des facteurs de type III, Ann. Sci. Ecole Norm. Sup., Paris 6 (1973), 133-252. MR 0341115 (49:5865)
  • 5. R. Dumitru, Simple and prime crossed products of $ C^*$-algebras by compact quantum group coactions, J. Funct. Anal., 257 (2009), 1480-1492. MR 2541277 (2010g:46111)
  • 6. R. Dumitru, Unitary representations of compact quantum groups, in: Perspectives in Operator Algebras and Math. Physics, Bucharest, 2005, 85-90, arXiv:math/0609676v1. MR 2433028 (2009k:46095)
  • 7. R. Dumitru, C. Peligrad, Compact quantum group actions on $ C^*$-algebras and invariant derivations, Proc. Amer. Math. Soc., 135 (12) (2007), 3977-3984. MR 2341948 (2009d:46127)
  • 8. E. C. Gootman, A. J. Lazar, C. Peligrad, Spectra for compact group actions, J. Operator Theory, 31 (1994), 381-399. MR 1331784 (96f:46117)
  • 9. A. Kishimoto, Simple crossed products of $ C^*$-algebras by locally compact abelian actions, Yokohama Math. J., 28 (1980), 69-85. MR 623751 (82g:46110)
  • 10. M.B. Landstad, Operator algebras and compact groups, Proceedings of International Conference held in Neptun, Romania, 1980, vol. II, Pitman, Boston-London-Melbourne, 1981, 33-47. MR 733301 (85f:46125)
  • 11. G. K. Pedersen, $ C^*$-algebras and their automorphism groups, Academic Press, 1979. MR 548006 (81e:46037)
  • 12. C. Peligrad, Compact actions commuting with ergodic actions and applications to crossed products, Transactions of the Amer. Math. Soc. 331 (1992) 825-837. MR 1044964 (92h:46097)
  • 13. C. Peligrad, Locally compact group actions on $ C^*$-algebras and compact subgroups, J. Funct. Anal., Vol. 76, No.1, January 1988, 126-139. MR 923048 (89h:46097)
  • 14. P. Podleś, Symmetries of quantum spaces. Subgroups and quotient spaces of quantum SU(2) and SO(3) groups, Comm. Math. Phys. 170(1995), no.1, 1-20. MR 1331688 (96j:58013)
  • 15. M. Rieffel, Actions of finite groups on $ C^*$-algebras, Math. Scand., 47 (1980), 157-176. MR 600086 (83c:46062)
  • 16. S. L. Woronowicz, Compact matrix pseudogroups, Comm. Math. Phys. 111 (4) 1987, 613-665. MR 901157 (88m:46079)
  • 17. S. L. Woronowicz, Compact quantum groups, in: Quantum Symmetries, Proceedings of the Les Houches Summer School, 1995, North-Holland, Amsterdam, 1998, 845-884. MR 1616348 (99m:46164)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 47L65, 20G42

Retrieve articles in all journals with MSC (2010): 47L65, 20G42


Additional 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

DOI: https://doi.org/10.1090/S0002-9947-2012-05562-0
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
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society