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 Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(e) ISSN 0002-9939(p)

     

Quantum spaces without group structure

Author(s): Piotr M. Sołtan
Journal: Proc. Amer. Math. Soc. 138 (2010), 2079-2086.
MSC (2000): Primary 46L89, 46L85, 17B37, 81R60, 20G42
Posted: February 17, 2010
MathSciNet review: 2596045
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Abstract | References | Similar articles | Additional information

Abstract: We prove that some well known compact quantum spaces such as quantum tori and some quantum two-spheres do not admit a compact quantum group structure. This is achieved by considering existence of traces, characters and nuclearity of the corresponding $ \mathrm{C}^*$-algebras.

References:

1.
E. Bedos, G. Murphy and L. Tuset, Co-amenability of compact quantum groups. J. Geom. Phys. 40 (2001), 129-153. MR 1862084 (2002m:46100)

2.
E. Bedos, G. Murphy and L. Tuset, Amenability and co-amenability of algebraic quantum groups. II. J. Funct. Anal. 201 (2003), 303-340. MR 1986692 (2004e:46085)

3.
O. Bratteli, G.A. Elliott, D.E. Evans and A. Kishimoto, Non-commutative spheres. I. Int. J. Math. 3 (1991), 139-166. MR 1094701 (92d:58008)

4.
O. Bratteli, G.A. Elliott, D.E. Evans and A. Kishimoto, Non-commutative spheres. II. Rational rotations. J. Op. Th. 27 (1992), 53-85. MR 1241115 (94j:46068)

5.
O. Bratteli and A. Kishimoto, Non-commutative spheres. III. Irrational rotations. Comm. Math. Phys. 147 (1992), 605-624. MR 1175495 (93g:58008)

6.
A. Connes, $ \mathit{C}^*$-algebrès et géométrie différentielle. C. R. Acad. Sci. Paris Sér. A-B, 290 (1980), 599-604. MR 572645 (81c:46053)

7.
A. Connes, Noncommutative Geometry. Academic Press, 1994. MR 1303779 (95j:46063)

8.
L. Dabrowski, The garden of quantum spheres. In Noncommutative Geometry and Quantum Groups, Banach Center Publications, 61, Polish Academy of Sciences, 2003, 37-48. MR 2024420 (2004k:58009)

9.
J. Dixmier, $ \mathit{C}^*$-algebras. North-Holland, 1977. MR 0458185 (56:16388)

10.
P.M. Hajac and T. Masuda, Quantum double torus. C. R. Acad. Sci. Paris Ser. I Math. 327, no. 6 (1998), 553-558. MR 1650603 (99h:16060)

11.
P. Kruszyśki and S.L. Woronowicz, A non-commutative Gelfand-Naimark theorem. J. Op. Th. 8 (1982), 361-389. MR 677419 (84b:46068)

12.
A. Maes and A. Van Daele, Notes on compact quantum groups. Nieuw Arch. Wiskd. IV 16, no. 1-2 (1998), 73-112. MR 1645264 (99g:46105)

13.
T. Natsume and C.L. Olsen, A new family of noncommutative $ 2$-spheres. J. Func. Anal. 202 (2003), 363-391. MR 1990530 (2005f:46126)

14.
C. Pearcy and D. Topping, On commutators in ideals of compact operators. Michigan Math. J. 18, no. 3 (1971), 247-252. MR 0284853 (44:2077)

15.
P. Podleś, Quantum spheres. Lett. Math. Phys. 14 (1987), 193-202. MR 919322 (89b:46081)

16.
M.A. Rieffel, Non-commutative tori -- a case study of non-commutative differentiable manifolds. Contemporary Mathematics, 105, Amer. Math. Soc., Providence, RI, 1990, 191-211. MR 1047281 (91d:58012)

17.
M.A. Rieffel, $ \mathrm{C}^*$-algebras associated to irrational rotations. Pac. J. Math. 93 (1981), 415-429. MR 623572 (83b:46087)

18.
Z.-J. Ruan, Amenability of Hopf von Neumann algebras and Kac algebras. J. Funct. Anal. 139 (1996), 466-499. MR 1402773 (98e:46077)

19.
P.M. Sołtan, Quantum Bohr compactification. Ill. J. Math. 49, no. 4 (2005), 1245-1270. MR 2210362 (2007a:46079)

20.
P.M. Sołtan, Quantum families of maps and quantum semigroups on finite quantum spaces. J. Geom. Phys. 59 (2009), 354-368. MR 2501746

21.
P.M. Sołtan, Examples of quantum commutants. Arab. J. Sci. Eng. 33, no. 2C (2008), 447-457. MR 2500052

22.
P.M. Sołtan, Quantum $ \mathrm{SO}(3)$ groups and quantum group actions on $ M_2$. J. Noncommut. Geom. 4 (2010), 2-28.

23.
R. Tomatsu, Amenable discrete quantum groups. J. Math. Soc. Japan 58 (2006), 949-964. MR 2276175 (2008g:46122)

24.
A. Van Daele, The Haar measure on a compact quantum group. Proc. Amer. Math. Soc. 123 (1995), 3125-3128. MR 1277138 (95m:46097)

25.
S.L. Woronowicz, Compact matrix pseudogroups. Comm. Math. Phys. 111 (1987), 613-665. MR 901157 (88m:46079)

26.
S.L. Woronowicz, Compact quantum groups. In Symétries Quantiques, les Houches, Session LXIV 1995, Elsevier, 1998, 845-884. MR 1616348 (99m:46164)

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Additional Information:

Piotr M. Sołtan
Affiliation: Institute of Mathematics, Polish Academy of Sciences - and - Department of Mathematical Methods in Physics, Faculty of Physics, University of Warsaw, Warsaw, Poland
Email: piotr.soltan@fuw.edu.pl

DOI: 10.1090/S0002-9939-10-10265-2
PII: S 0002-9939(10)10265-2
Keywords: Compact quantum group, quantum space, non-commutative topology
Received by editor(s): May 11, 2009,
Received by editor(s) in revised form: September 21, 2009
Posted: February 17, 2010
Additional Notes: This research was partially supported by a Polish government grant, no. N201 1770 33.
Communicated by: Marius Junge
Copyright of article: Copyright 2010, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.




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