Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 8 (2012), 091, 12 pages      arXiv:1101.2751      http://dx.doi.org/10.3842/SIGMA.2012.091

Covariant Fields of C*-Algebras under Rieffel Deformation

Fabian Belmonte a and Marius Măntoiu b
a) D728, building A, SISSA-ISAS, Via Bonomea 265, 34136 Trieste, Italy
b) Departamento de Matemáticas, Universidad de Chile, Las Palmeras 3425, Casilla 653, Santiago, Chile

Received August 26, 2012, in final form November 22, 2012; Published online November 28, 2012

Abstract
We show that Rieffel's deformation sends covariant C(T)-algebras into C(T)-algebras. We also treat the lower semi-continuity issue, proving that Rieffel's deformation transforms covariant continuous fields of C*-algebras into continuous fields of C*-algebras. Some examples are indicated, including certain quantum groups.

Key words: pseudodifferential operator; Rieffel deformation; C*-algebra; continuous field; noncommutative dynamical system.

pdf (397 kb)   tex (22 kb)

References

  1. Belmonte F., Măntoiu M., Continuity of spectra in Rieffel's pseudodifferential calculus, in Spectral Analysis of Quantum Hamiltonians, Oper. Theory Adv. Appl., Vol. 224, Birkhäuser, Basel, 2012, 11-24.
  2. Beltiţă I., Măntoiu M., Rieffel deformation and twisted crossed products, Int. Math. Res. Not., to appear, arXiv:1208.6548.
  3. Blanchard E., A few remarks on exact C(X)-algebras, Rev. Roumaine Math. Pures Appl. 45 (2000), 565-576, math.OA/0012127.
  4. Blanchard É., Déformations de C*-algèbres de Hopf, Bull. Soc. Math. France 124 (1996), 141-215.
  5. Dixmier J., Malliavin P., Factorisations de fonctions et de vecteurs indéfiniment différentiables, Bull. Sci. Math. (2) 102 (1978), 307-330.
  6. Doplicher S., Fredenhagen K., Roberts J.E., The quantum structure of spacetime at the Planck scale and quantum fields, Comm. Math. Phys. 172 (1995), 187-220, hep-th/0303037.
  7. Fell J.M.G., The structure of algebras of operator fields, Acta Math. 106 (1961), 233-280.
  8. Folland G.B., Harmonic analysis in phase space, Annals of Mathematics Studies, Vol. 122, Princeton University Press, Princeton, NJ, 1989.
  9. Hannabuss K.C., Mathai V., Noncommutative principal torus bundles via parametrised strict deformation quantization, in Superstrings, Geometry, Topology, and C*-Algebras, Proc. Sympos. Pure Math., Vol. 81, Amer. Math. Soc., Providence, RI, 2010, 133-147, arXiv:0911.1886.
  10. Hannabuss K.C., Mathai V., Parametrized strict deformation quantization of C*-bundles and Hilbert C*-modules, J. Aust. Math. Soc. 90 (2011), 25-38, arXiv:1007.4696.
  11. Kasprzak P., Rieffel deformation via crossed products, J. Funct. Anal. 257 (2009), 1288-1332, math.OA/0606333.
  12. Landsman N.P., Mathematical topics between classical and quantum mechanics, Springer Monographs in Mathematics, Springer-Verlag, New York, 1998.
  13. Lee R.Y., On the C*-algebras of operator fields, Indiana Univ. Math. J. 25 (1976), 303-314.
  14. Măntoiu M., Rieffel's pseudodifferential calculus and spectral analysis for quantum Hamiltonians, Ann. Inst. Fourier (Grenoble) 62 (2012), 1551-1580, arXiv:1003.3149.
  15. Nilsen M., C*-bundles and C0(X)-algebras, Indiana Univ. Math. J. 45 (1996), 463-477.
  16. Packer J.A., Raeburn I., Twisted crossed products of C*-algebras. II, Math. Ann. 287 (1990), 595-612.
  17. Piacitelli G., Quantum spacetime: a disambiguation, SIGMA 6 (2010), 073, 43 pages, arXiv:1004.5261.
  18. Raeburn I., Williams D.P., Morita equivalence and continuous-trace C*-algebras, Mathematical Surveys and Monographs, Vol. 60, American Mathematical Society, Providence, RI, 1998.
  19. Rieffel M.A., Compact quantum groups associated with toral subgroups, in Representation Theory of Groups and Algebras, Contemp. Math., Vol. 145, Amer. Math. Soc., Providence, RI, 1993, 465-491.
  20. Rieffel M.A., Continuous fields of C*-algebras coming from group cocycles and actions, Math. Ann. 283 (1989), 631-643.
  21. Rieffel M.A., Deformation quantization for actions of Rd, Mem. Amer. Math. Soc. 106 (1993), no. 506, x+93 pages.
  22. Rieffel M.A., Quantization and C*-algebras, in C*-Algebras: 1943-1993 (San Antonio, TX, 1993), Contemp. Math., Vol. 167, Amer. Math. Soc., Providence, RI, 1994, 66-97.
  23. Tomiyama J., Topological representation of C*-algebras, Tôhoku Math. J. 14 (1962), 187-204.
  24. Williams D.P., Crossed products of C*-algebras, Mathematical Surveys and Monographs, Vol. 134, American Mathematical Society, Providence, RI, 2007.

Previous article  Next article   Contents of Volume 8 (2012)