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Year 2018, Volume: 1 Issue: 1, 57 - 60, 30.06.2018
https://doi.org/10.33401/fujma.401097

Abstract

References

  • [1] A. Connes, Noncommutative geometry, Academic Press, 1994.
  • [2] A. Connes, On the spectral characterization of manifolds, J. Noncommutative Geometry 7 (1) (2013): 1–82.
  • [3] R.V. Kadison, J.R. Ringrose, Fundamentals of the theory of operator algebras: Advanced theory, Vol. 2, American Mathematical Soc., 1997.
  • [4] G. Kuperberg, N. Weaver, A von Neumann algebra approach to quantum metrics/quantum relations, Vol. 215, no. 1010. American Mathematical Society, 2012.
  • [5] P. Martinetti, From Monge to Higgs: a survey of distance computations in noncommutative geometry, Noncommutative Geometry and Optimal Transport 676, 2016.
  • [6] M.A. Rieffel, Metrics on state spaces, Doc. Math. 4 (1999): 559-600.
  • [7] M.A. Rieffel, Group C*-algebras as compact quantum metric spaces, Doc. Math. 7 (2002): 605–651.
  • [8] M.A. Rieffel, Gromov-Hausdorff distance for quantum metric spaces/Matrix algebras converge to the sphere for quantum Gromov-Hausdorff distance, Vol. 168, no. 796. American Mathematical Soc., 2004.
  • [9] M.M. Sadr, Metric operator fields. (arXiv:1705.03378 [math.OA])
  • [10] M.M. Sadr, Vietoris topology on hyperspaces associated to a noncommutative compact space, Mathematica, 60 (83) (1) (2018): 72–82.
  • [11] M.M. Sadr, Quantum metric spaces of quantum maps, Universal Journal of Mathematics and Applications, 1 (1) (2018): 54–60.
  • [12] M. Takesaki, Theory of operator algebras I, Reprint of the first (1979) edition, Encyclopaedia of Mathematical Sciences, 124, Operator Algebras and Noncommutative Geometry, 5. (2002).

Quantum metrics on noncommutative spaces

Year 2018, Volume: 1 Issue: 1, 57 - 60, 30.06.2018
https://doi.org/10.33401/fujma.401097

Abstract

We introduce two new algebraic formulations for the notion of 'quantum metric on noncommutative space'. For a compact noncommutative space associated to a unital C*-algebra, our quantum metrics are elements of the spatial tensor product of the C*-algebra with itself. We consider some basic properties of these new objects, and state some connections with the Rieffel theory of compact quantum metric spaces.

References

  • [1] A. Connes, Noncommutative geometry, Academic Press, 1994.
  • [2] A. Connes, On the spectral characterization of manifolds, J. Noncommutative Geometry 7 (1) (2013): 1–82.
  • [3] R.V. Kadison, J.R. Ringrose, Fundamentals of the theory of operator algebras: Advanced theory, Vol. 2, American Mathematical Soc., 1997.
  • [4] G. Kuperberg, N. Weaver, A von Neumann algebra approach to quantum metrics/quantum relations, Vol. 215, no. 1010. American Mathematical Society, 2012.
  • [5] P. Martinetti, From Monge to Higgs: a survey of distance computations in noncommutative geometry, Noncommutative Geometry and Optimal Transport 676, 2016.
  • [6] M.A. Rieffel, Metrics on state spaces, Doc. Math. 4 (1999): 559-600.
  • [7] M.A. Rieffel, Group C*-algebras as compact quantum metric spaces, Doc. Math. 7 (2002): 605–651.
  • [8] M.A. Rieffel, Gromov-Hausdorff distance for quantum metric spaces/Matrix algebras converge to the sphere for quantum Gromov-Hausdorff distance, Vol. 168, no. 796. American Mathematical Soc., 2004.
  • [9] M.M. Sadr, Metric operator fields. (arXiv:1705.03378 [math.OA])
  • [10] M.M. Sadr, Vietoris topology on hyperspaces associated to a noncommutative compact space, Mathematica, 60 (83) (1) (2018): 72–82.
  • [11] M.M. Sadr, Quantum metric spaces of quantum maps, Universal Journal of Mathematics and Applications, 1 (1) (2018): 54–60.
  • [12] M. Takesaki, Theory of operator algebras I, Reprint of the first (1979) edition, Encyclopaedia of Mathematical Sciences, 124, Operator Algebras and Noncommutative Geometry, 5. (2002).
There are 12 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Maysam Maysami Sadr 0000-0003-0747-4180

Publication Date June 30, 2018
Submission Date March 3, 2018
Acceptance Date March 15, 2018
Published in Issue Year 2018 Volume: 1 Issue: 1

Cite

APA Maysami Sadr, M. (2018). Quantum metrics on noncommutative spaces. Fundamental Journal of Mathematics and Applications, 1(1), 57-60. https://doi.org/10.33401/fujma.401097
AMA Maysami Sadr M. Quantum metrics on noncommutative spaces. Fundam. J. Math. Appl. June 2018;1(1):57-60. doi:10.33401/fujma.401097
Chicago Maysami Sadr, Maysam. “Quantum Metrics on Noncommutative Spaces”. Fundamental Journal of Mathematics and Applications 1, no. 1 (June 2018): 57-60. https://doi.org/10.33401/fujma.401097.
EndNote Maysami Sadr M (June 1, 2018) Quantum metrics on noncommutative spaces. Fundamental Journal of Mathematics and Applications 1 1 57–60.
IEEE M. Maysami Sadr, “Quantum metrics on noncommutative spaces”, Fundam. J. Math. Appl., vol. 1, no. 1, pp. 57–60, 2018, doi: 10.33401/fujma.401097.
ISNAD Maysami Sadr, Maysam. “Quantum Metrics on Noncommutative Spaces”. Fundamental Journal of Mathematics and Applications 1/1 (June 2018), 57-60. https://doi.org/10.33401/fujma.401097.
JAMA Maysami Sadr M. Quantum metrics on noncommutative spaces. Fundam. J. Math. Appl. 2018;1:57–60.
MLA Maysami Sadr, Maysam. “Quantum Metrics on Noncommutative Spaces”. Fundamental Journal of Mathematics and Applications, vol. 1, no. 1, 2018, pp. 57-60, doi:10.33401/fujma.401097.
Vancouver Maysami Sadr M. Quantum metrics on noncommutative spaces. Fundam. J. Math. Appl. 2018;1(1):57-60.

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