Involve: A Journal of Mathematics

  • Involve
  • Volume 12, Number 2 (2019), 329-342.

Quantum metrics from traces on full matrix algebras

Konrad Aguilar and Samantha Brooker

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We prove that, in the sense of the Gromov–Hausdorff propinquity, certain natural quantum metrics on the algebras of (n×n)-matrices are separated by a positive distance when n is not prime.

Article information

Involve, Volume 12, Number 2 (2019), 329-342.

Received: 5 December 2017
Accepted: 7 March 2018
First available in Project Euclid: 25 October 2018

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 46L89: Other "noncommutative" mathematics based on C-algebra theory [See also 58B32, 58B34, 58J22] 46L30: States 58B34: Noncommutative geometry (à la Connes)

noncommutative metric geometry Gromov–Hausdorff propinquity quantum metric spaces Lip-norms C*-algebras full matrix algebras


Aguilar, Konrad; Brooker, Samantha. Quantum metrics from traces on full matrix algebras. Involve 12 (2019), no. 2, 329--342. doi:10.2140/involve.2019.12.329.

Export citation


  • K. Aguilar and F. Latrémolière, “Quantum ultrametrics on AF algebras and the Gromov–Hausdorff propinquity”, Studia Math. 231:2 (2015), 149–193.
  • O. Bratteli, “Inductive limits of finite dimensional $C\sp{\ast} $-algebras”, Trans. Amer. Math. Soc. 171 (1972), 195–234.
  • D. Burago, Y. Burago, and S. Ivanov, A course in metric geometry, Graduate Studies in Mathematics 33, Amer. Math. Soc., Providence, RI, 2001.
  • K. R. Davidson, $C^*$-algebras by example, Fields Institute Monographs 6, Amer. Math. Soc., Providence, RI, 1996.
  • F. Latrémolière, “Convergence of fuzzy tori and quantum tori for the quantum Gromov–Hausdorff propinquity: an explicit approach”, Münster J. Math. 8:1 (2015), 57–98.
  • F. Latrémolière, “The quantum Gromov–Hausdorff propinquity”, Trans. Amer. Math. Soc. 368:1 (2016), 365–411.
  • F. Latrémolière, “Quantum metric spaces and the Gromov–Hausdorff propinquity”, pp. 47–133 in Noncommutative geometry and optimal transport (Besançon, 2014), edited by P. Martinetti and J.-C. Wallet, Contemp. Math. 676, Amer. Math. Soc., Providence, RI, 2016.
  • F. Latrémolière, “A compactness theorem for the dual Gromov–Hausdorff propinquity”, Indiana Univ. Math. J. 66:5 (2017), 1707–1753.
  • G. J. Murphy, $C^*$-algebras and operator theory, Academic Press, Boston, 1990.
  • N. Ozawa and M. A. Rieffel, “Hyperbolic group $C^*$-algebras and free-product $C^*$-algebras as compact quantum metric spaces”, Canad. J. Math. 57:5 (2005), 1056–1079.
  • M. A. Rieffel, “Metrics on states from actions of compact groups”, Doc. Math. 3 (1998), 215–229.
  • M. A. Rieffel, “Metrics on state spaces”, Doc. Math. 4 (1999), 559–600.
  • M. A. Rieffel, Gromov-Hausdorff distance for quantum metric spaces; Matrix algebras converge to the sphere for quantum Gromov–Hausdorff distance, Mem. Amer. Math. Soc. 796, Amer. Math. Soc., Providence, RI, 2004.
  • M. A. Rieffel, “Matricial bridges for `matrix algebras converge to the sphere”', pp. 209–233 in Operator algebras and their applications (San Antonio, 2015), edited by R. S. Doran, Contemp. Math. 671, Amer. Math. Soc., 2016.