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

SIGMA 14 (2018), 079, 21 pages      arXiv:1803.04036

Metrized Quantum Vector Bundles over Quantum Tori Built from Riemannian Metrics and Rosenberg's Levi-Civita Connections

Leonard Huang
Department of Mathematics, University of Colorado at Boulder, Campus Box 395, 2300 Colorado Avenue, Boulder, CO 80309-0395, USA

Received March 13, 2018, in final form July 21, 2018; Published online July 29, 2018

We build metrized quantum vector bundles, over a generically transcendental quantum torus, from Riemannian metrics, using Rosenberg's Levi-Civita connections for these metrics. We also prove that two metrized quantum vector bundles, corresponding to positive scalar multiples of a Riemannian metric, have distance zero between them with respect to the modular Gromov-Hausdorff propinquity.

Key words: quantum torus; generically transcendental; quantum metric space; metrized quantum vector bundle; Riemannian metric; Levi-Civita connection.

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  1. Aguilar K., Latrémolière F., Quantum ultrametrics on AF algebras and the Gromov-Hausdorff propinquity, Studia Math. 231 (2015), 149-193, arXiv:1511.07114.
  2. Bratteli O., Elliott G.A., Jorgensen P.E.T., Decomposition of unbounded derivations into invariant and approximately inner parts, J. Reine Angew. Math. 346 (1984), 166-193.
  3. Latrémolière F., Curved noncommutative tori as Leibniz quantum compact metric spaces, J. Math. Phys. 56 (2015), 123503, 16 pages, arXiv:1507.08771.
  4. Latrémolière F., The quantum Gromov-Hausdorff propinquity, Trans. Amer. Math. Soc. 368 (2016), 365-411, arXiv:1302.4058.
  5. Latrémolière F., Quantum metric spaces and the Gromov-Hausdorff propinquity, in Noncommutative Geometry and Optimal Transport, Contemp. Math., Vol. 676, Amer. Math. Soc., Providence, RI, 2016, 47-133, arXiv:1506.04341.
  6. Latrémolière F., The modular Gromov-Hausdorff propinquity, arXiv:1608.04881.
  7. Latrémolière F., Convergence of the Heisenberg modules over quantum two-tori for the modular Gromov-Hausdorff propinquity, arXiv:1703.07073.
  8. Latrémolière F., Packer J., Noncommutative solenoids and the Gromov-Hausdorff propinquity, Proc. Amer. Math. Soc. 145 (2017), 2043-2057, arXiv:1601.02707.
  9. Morrison T.J., Functional analysis: an introduction to Banach space theory, Pure and Applied Mathematics (New York), Wiley-Interscience, New York, 2001.
  10. Rieffel M.A., Metrics on states from actions of compact groups, Doc. Math. 3 (1998), 215-229, math.OA/9807084.
  11. Rieffel M.A., Metrics on state spaces, Doc. Math. 4 (1999), 559-600, math.OA/9906151.
  12. Rieffel M.A., Gromov-Hausdorff distance for quantum metric spaces, Mem. Amer. Math. Soc. 168 (2004), 1-65, math.OA/0011063.
  13. Rieffel M.A., Leibniz seminorms for ''matrix algebras converge to the sphere'', in Quanta of Maths, Clay Math. Proc., Vol. 11, Amer. Math. Soc., Providence, RI, 2010, 543-578, arXiv:0707.3229.
  14. Rosenberg J., Levi-Civita's theorem for noncommutative tori, SIGMA 9 (2013), 071, 9 pages, arXiv:1307.3775.

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