Quantum spaces without group structure

Sołtan, Piotr

doi:10.1090/S0002-9939-10-10265-2

Quantum spaces without group structure
HTML articles powered by AMS MathViewer

by Piotr M. Sołtan

Proc. Amer. Math. Soc. 138 (2010), 2079-2086

DOI: https://doi.org/10.1090/S0002-9939-10-10265-2

Published electronically: February 17, 2010

PDF | Request permission

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

E. Bédos, G. J. Murphy, and L. Tuset, Co-amenability of compact quantum groups, J. Geom. Phys. 40 (2001), no. 2, 130–153. MR 1862084, DOI 10.1016/S0393-0440(01)00024-9
E. Bédos, G. J. Murphy, and L. Tuset, Amenability and co-amenability of algebraic quantum groups. II, J. Funct. Anal. 201 (2003), no. 2, 303–340. MR 1986692, DOI 10.1016/S0022-1236(03)00021-1
Ola Bratteli, George A. Elliott, David E. Evans, and Akitaka Kishimoto, Noncommutative spheres. I, Internat. J. Math. 2 (1991), no. 2, 139–166. MR 1094701, DOI 10.1142/S0129167X91000090
O. Bratteli, G. A. Elliott, D. E. Evans, and A. Kishimoto, Noncommutative spheres. II. Rational rotations, J. Operator Theory 27 (1992), no. 1, 53–85. MR 1241115
Ola Bratteli and Akitaka Kishimoto, Noncommutative spheres. III. Irrational rotations, Comm. Math. Phys. 147 (1992), no. 3, 605–624. MR 1175495
Alain Connes, $C^{\ast }$ algèbres et géométrie différentielle, C. R. Acad. Sci. Paris Sér. A-B 290 (1980), no. 13, A599–A604 (French, with English summary). MR 572645
Alain Connes, Noncommutative geometry, Academic Press, Inc., San Diego, CA, 1994. MR 1303779
Ludwik Dąbrowski, The garden of quantum spheres, Noncommutative geometry and quantum groups (Warsaw, 2001) Banach Center Publ., vol. 61, Polish Acad. Sci. Inst. Math., Warsaw, 2003, pp. 37–48. MR 2024420, DOI 10.4064/bc61-0-3
Jacques Dixmier, $C^*$-algebras, North-Holland Mathematical Library, Vol. 15, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977. Translated from the French by Francis Jellett. MR 0458185
Piotr M. Hajac and Tetsuya Masuda, Quantum double-torus, C. R. Acad. Sci. Paris Sér. I Math. 327 (1998), no. 6, 553–558 (English, with English and French summaries). MR 1650603, DOI 10.1016/S0764-4442(98)89162-3
PawełKruszyński and Stanisław L. Woronowicz, A noncommutative Gel′fand-Naĭmark theorem, J. Operator Theory 8 (1982), no. 2, 361–389. MR 677419
Ann Maes and Alfons Van Daele, Notes on compact quantum groups, Nieuw Arch. Wisk. (4) 16 (1998), no. 1-2, 73–112. MR 1645264
Toshikazu Natsume and Catherine L. Olsen, A new family of noncommutative 2-spheres, J. Funct. Anal. 202 (2003), no. 2, 363–391. MR 1990530, DOI 10.1016/S0022-1236(03)00023-5
Carl Pearcy and David Topping, On commutators in ideals of compact operators, Michigan Math. J. 18 (1971), 247–252. MR 284853
P. Podleś, Quantum spheres, Lett. Math. Phys. 14 (1987), no. 3, 193–202. MR 919322, DOI 10.1007/BF00416848
Marc A. Rieffel, Noncommutative tori—a case study of noncommutative differentiable manifolds, Geometric and topological invariants of elliptic operators (Brunswick, ME, 1988) Contemp. Math., vol. 105, Amer. Math. Soc., Providence, RI, 1990, pp. 191–211. MR 1047281, DOI 10.1090/conm/105/1047281
Marc A. Rieffel, $C^{\ast }$-algebras associated with irrational rotations, Pacific J. Math. 93 (1981), no. 2, 415–429. MR 623572
Zhong-Jin Ruan, Amenability of Hopf von Neumann algebras and Kac algebras, J. Funct. Anal. 139 (1996), no. 2, 466–499. MR 1402773, DOI 10.1006/jfan.1996.0093
Piotr M. Sołtan, Quantum Bohr compactification, Illinois J. Math. 49 (2005), no. 4, 1245–1270. MR 2210362
Piotr M. Sołtan, Quantum families of maps and quantum semigroups on finite quantum spaces, J. Geom. Phys. 59 (2009), no. 3, 354–368. MR 2501746, DOI 10.1016/j.geomphys.2008.11.007
Piotr M. Sołtan, Examples of quantum commutants, Arab. J. Sci. Eng. Sect. C Theme Issues 33 (2008), no. 2, 447–457 (English, with English and Arabic summaries). MR 2500052
P.M. Sołtan, Quantum $\mathrm {SO}(3)$ groups and quantum group actions on $M_2$. J. Noncommut. Geom. 4 (2010), 2–28.
Reiji Tomatsu, Amenable discrete quantum groups, J. Math. Soc. Japan 58 (2006), no. 4, 949–964. MR 2276175
A. Van Daele, The Haar measure on a compact quantum group, Proc. Amer. Math. Soc. 123 (1995), no. 10, 3125–3128. MR 1277138, DOI 10.1090/S0002-9939-1995-1277138-0
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