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Year 2018, , 54 - 60, 11.03.2018
https://doi.org/10.32323/ujma.382008

Abstract

References

  • [1] A. Connes, Compact metric spaces, Fredholm modules and hyperfiniteness, Ergo. Th. Dyn. Sys. 9 (1989), 207–220.
  • [2] A. Connes, Noncommutative Geometry, Academic Press, 1994.
  • [3] G. Kuperberg, N. Weaver, A von Neumann algebra approach to quantum metrics/quantum relations, Vol. 215, no. 1010. American Mathematical Society, 2012.
  • [4] S. T. Rachev, Probability Metrics and the Stability of Stochastic Models, John Wiley and Sons, 1991.
  • [5] M. A. Rieffel, Metrics on states from actions of compact groups, Doc. Math. 3 (1998), 215–229.
  • [6] M. A. Rieffel, Metrics on state spaces, Doc. Math. 4 (1999), 559–600.
  • [7] M. A. Rieffel, Gromov-Hausdorff distance for quantum metric spaces, Mem. Amer. Math. Soc. 168 (2004), 1–65.
  • [8] M. A. Rieffel, Compact quantum metric spaces, Contemp. Math. 365 (2004), 315–330.
  • [9] M. A. Rieffel, Leibniz seminorms for Matrix algebras converge to the sphere, Quanta of maths, Clay Math. Proc., vol. 11, Amer. Math. Soc., Providence, RI, 2010, 543–578.
  • [10] M. M. Sadr, Quantum functor Mor, Math. Pannonica 21 no. 1 (2010), 77–88.
  • [11] M. M. Sadr, A kind of compact quantum semigroups, Int. J. Math. Math. Sci. 2012 (2012), Article ID 725270, 10 pages.
  • [12] M. M. Sadr, On the quantum groups and semigroups of maps between noncommutative spaces, Czechoslovak Math. J. 67 no. 1 (2017), 97–121.
  • [13] M. M. Sadr, Quantum metrics on noncommutative spaces, available at https://arxiv.org/pdf/1606.00661.pdf
  • [14] M. M. Sadr, Metric operator fields, available at https://arxiv.org/pdf/1705.03378.pdf
  • [15] P. M. Sołtan, Quantum families of maps and quantum semigroups on finite quantum spaces, J. Geom. Phys. 59 (2009), 354–368.
  • [16] S. L. Woronowicz, Pseudogroups, pseudospaces and Pontryagin duality, Proceedings of the International Conference on Mathematical Physics, Lausanne 1979 , Lecture Notes in Physics 116, 407–412.

Quantum metric spaces of quantum maps

Year 2018, , 54 - 60, 11.03.2018
https://doi.org/10.32323/ujma.382008

Abstract

We show that any quantum family of quantum maps from a noncommutative space to a compact quantum metric space has a canonical quantum pseudo-metric structure. Here by a 'compact quantum metric space' we mean a unital C*-algebra together with a Lipschitz seminorm, in the sense of Rieffel, which induces the weak* topology on the state space of the C*-algebra. Our main result generalizes a classical result to noncommutative world.

References

  • [1] A. Connes, Compact metric spaces, Fredholm modules and hyperfiniteness, Ergo. Th. Dyn. Sys. 9 (1989), 207–220.
  • [2] A. Connes, Noncommutative Geometry, Academic Press, 1994.
  • [3] G. Kuperberg, N. Weaver, A von Neumann algebra approach to quantum metrics/quantum relations, Vol. 215, no. 1010. American Mathematical Society, 2012.
  • [4] S. T. Rachev, Probability Metrics and the Stability of Stochastic Models, John Wiley and Sons, 1991.
  • [5] M. A. Rieffel, Metrics on states from actions of compact groups, Doc. Math. 3 (1998), 215–229.
  • [6] M. A. Rieffel, Metrics on state spaces, Doc. Math. 4 (1999), 559–600.
  • [7] M. A. Rieffel, Gromov-Hausdorff distance for quantum metric spaces, Mem. Amer. Math. Soc. 168 (2004), 1–65.
  • [8] M. A. Rieffel, Compact quantum metric spaces, Contemp. Math. 365 (2004), 315–330.
  • [9] M. A. Rieffel, Leibniz seminorms for Matrix algebras converge to the sphere, Quanta of maths, Clay Math. Proc., vol. 11, Amer. Math. Soc., Providence, RI, 2010, 543–578.
  • [10] M. M. Sadr, Quantum functor Mor, Math. Pannonica 21 no. 1 (2010), 77–88.
  • [11] M. M. Sadr, A kind of compact quantum semigroups, Int. J. Math. Math. Sci. 2012 (2012), Article ID 725270, 10 pages.
  • [12] M. M. Sadr, On the quantum groups and semigroups of maps between noncommutative spaces, Czechoslovak Math. J. 67 no. 1 (2017), 97–121.
  • [13] M. M. Sadr, Quantum metrics on noncommutative spaces, available at https://arxiv.org/pdf/1606.00661.pdf
  • [14] M. M. Sadr, Metric operator fields, available at https://arxiv.org/pdf/1705.03378.pdf
  • [15] P. M. Sołtan, Quantum families of maps and quantum semigroups on finite quantum spaces, J. Geom. Phys. 59 (2009), 354–368.
  • [16] S. L. Woronowicz, Pseudogroups, pseudospaces and Pontryagin duality, Proceedings of the International Conference on Mathematical Physics, Lausanne 1979 , Lecture Notes in Physics 116, 407–412.
There are 16 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Maysam Maysami Sadr 0000-0003-0747-4180

Publication Date March 11, 2018
Submission Date January 21, 2018
Acceptance Date February 28, 2018
Published in Issue Year 2018

Cite

APA Maysami Sadr, M. (2018). Quantum metric spaces of quantum maps. Universal Journal of Mathematics and Applications, 1(1), 54-60. https://doi.org/10.32323/ujma.382008
AMA Maysami Sadr M. Quantum metric spaces of quantum maps. Univ. J. Math. Appl. March 2018;1(1):54-60. doi:10.32323/ujma.382008
Chicago Maysami Sadr, Maysam. “Quantum Metric Spaces of Quantum Maps”. Universal Journal of Mathematics and Applications 1, no. 1 (March 2018): 54-60. https://doi.org/10.32323/ujma.382008.
EndNote Maysami Sadr M (March 1, 2018) Quantum metric spaces of quantum maps. Universal Journal of Mathematics and Applications 1 1 54–60.
IEEE M. Maysami Sadr, “Quantum metric spaces of quantum maps”, Univ. J. Math. Appl., vol. 1, no. 1, pp. 54–60, 2018, doi: 10.32323/ujma.382008.
ISNAD Maysami Sadr, Maysam. “Quantum Metric Spaces of Quantum Maps”. Universal Journal of Mathematics and Applications 1/1 (March 2018), 54-60. https://doi.org/10.32323/ujma.382008.
JAMA Maysami Sadr M. Quantum metric spaces of quantum maps. Univ. J. Math. Appl. 2018;1:54–60.
MLA Maysami Sadr, Maysam. “Quantum Metric Spaces of Quantum Maps”. Universal Journal of Mathematics and Applications, vol. 1, no. 1, 2018, pp. 54-60, doi:10.32323/ujma.382008.
Vancouver Maysami Sadr M. Quantum metric spaces of quantum maps. Univ. J. Math. Appl. 2018;1(1):54-60.

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