Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-26T18:06:53.319Z Has data issue: false hasContentIssue false
  • English
  • Français

Heisenberg Modules over Quantum 2-tori are Metrized Quantum Vector Bundles

Part of: Selfadjoint operator algebras Infinite-dimensional manifolds

Published online by Cambridge University Press:  28 March 2019

Frédéric Latrémolière
Frédéric Latrémolière*
Affiliation:
Department of Mathematics, University of Denver, Denver CO 80208 Email: frederic@math.du.eduhttp://www.math.du.edu/∼frederic
Get access Rights & Permissions [Opens in a new window]

Abstract

The modular Gromov–Hausdorff propinquity is a distance on classes of modules endowed with quantum metric information, in the form of a metric form of a connection and a left Hilbert module structure. This paper proves that the family of Heisenberg modules over quantum two tori, when endowed with their canonical connections, form a family of metrized quantum vector bundles, as a first step in proving that Heisenberg modules form a continuous family for the modular Gromov–Hausdorff propinquity.

Keywords

noncommutative metric geometryGromov–Hausdorff convergenceMonge-Kantorovich distancequantum metric spaceLip-normD-normHilbert modulenoncommutative connectionnoncommutative Riemannian geometryunstable K-theory

MSC classification

Primary: 46L89: Other “noncommutative” mathematics based on $C^*$-algebra theory
Secondary: 46L30: States 58B34: Noncommutative geometry (à la Connes)
Type
Article
Information
Canadian Journal of Mathematics , Volume 72 , Issue 4 , August 2020 , pp. 1044 - 1081
Check for updates
Copyright
© Canadian Mathematical Society 2019

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

This work is part of the project supported by the grant H2020-MSCA-RISE-2015-691246-QUANTUM DYNAMICS.

References

Hawkins, A., Skalski, A., White, S., and Zacharias, J., On spectral triples on crossed products arising from equicontinuous actions. Math. Scand. 113(2013), 262–291. https://doi.org/10.7146/math.scand.a-15572Google Scholar
Aguilar, K. and Kaad, J., The Podlès sphere as a spectral metric space. J. Geom. Phys. 133(2018), 260–278. https://doi.org/10.1016/j.geomphys.2018.07.015Google Scholar
Aguilar, K. and Latrémolière, F., Quantum ultrametrics on AF algebras and the Gromov–Hausdorff propinquity. Studia Math. 231(2015), no. 2, 149–194.Google Scholar
Bratteli, O. and Robinson, D., Operator algebras and quantum statistical mechanics i. Springer-Verlag, 1979.10.1007/978-3-662-02313-6Google Scholar
Connes, A., C ∗-algèbres et géométrie differentielle. C. R. Acad. Sci. Paris 290(1980), no. 13, A599–A604.Google Scholar
Connes, A., Compact metric spaces, Fredholm modules and hyperfiniteness. Ergodic Theory Dynam. Systems 9(1989), no. 2, 207–220. https://doi.org/10.1017/S0143385700004934Google Scholar
Connes, A. and Rieffel, M. A., Yang-Mills for noncommutative two-tori. In: Operator algebras and mathematical physics (Iowa City, Iowa, 1985). Contemp. Math., 62, American Mathematical Society, Providence, RI, 1987, pp. 237–266. https://doi.org/10.1090/conm/062/878383Google Scholar
Folland, G., Harmonic analysis in phase space. Annals of Mathematics Studies, Princteon University Press, Princeton, NJ, 1989. https://doi.org/10.1515/9781400882427Google Scholar
Hoegh-Krohn, R., Landstad, M. B., and Stormer, E., Compact ergodic groups of automorphisms. Ann. of Math. 114(1981), 75–86. https://doi.org/10.2307/1971377Google Scholar
Kleppner, A., Multipliers on Abelian groups. Math. Ann. 158(1965), 11–34. https://doi.org/10.1007/BF01370393Google Scholar
Latrémolière, F., Curved noncommutative tori as Leibniz compact quantum metric spaces. J. Math. Phys. 56(2015), no. 12, 123503.10.1063/1.4937444Google Scholar
Latrémolière, F., The dual Gromov–Hausdorff propinquity. J. Math. Pures Appl. 103(2015), no. 2, 303–351. https://doi.org/10.1016/j.matpur.2014.04.006Google Scholar
Latrémolière, F., Quantum metric spaces and the Gromov–Hausdorff propinquity. In: Noncommutative geometry and optimal transport. Contemp. Math., 676, American Mathematical Society, 2015, pp. 47–133.Google Scholar
Latrémolière, F., Equivalence of quantum metrics with a common domain. J. Math. Anal. Appl. 443(2016), 1179–1195. https://doi.org/10.1016/j.jmaa.2016.05.045Google Scholar
Latrémolière, F., The modular Gromov–Hausdorff propinquity. Accepted in Dissertationes Mathematicae (2019), 67 pages, arxiv:1608.04881Google Scholar
Latrémolière, F., The quantum Gromov–Hausdorff propinquity. Trans. Amer. Math. Soc. 368(2016), no. 1, 365–411. https://doi.org/10.1090/tran/6334Google Scholar
Latrémolière, F., A compactness theorem for the dual Gromov–Hausdorff propinquity. Indiana Univ. Math. J. 66(2017), no. 5, 1707–1753. https://doi.org/10.1512/iumj.2017.66.6151Google Scholar
Latrémolière, F., The triangle inequality and the dual Gromov–Hausdorff propinquity. Indiana Univ. Math. J. 66(2017), no. 1, 297–313. https://doi.org/10.1512/iumj.2017.66.5954Google Scholar
Latrémolière, F., Convergence of Heisenberg modules for the modular Gromov–Hausdorff propinquity. 2018. arxiv:1803.06601Google Scholar
Latrémolière, F. and Packer, J., Noncommutative solenoids and the Gromov–Hausdorff propinquity. Proc. Amer. Math. Soc. 145(2017), no. 5, 1179–1195. https://doi.org/10.1090/proc/13229Google Scholar
Li, H., C ∗-algebraic quantum Gromov–Hausdorff distance. 2003. arxiv:math.OA/0312003Google Scholar
Ozawa, N. and Rieffel, M. A., Hyperbolic group C ∗-algebras and free product C ∗-algebras as compact quantum metric spaces. Canad. J. Math. 57(2005), 1056–1079. https://doi.org/10.4153/CJM-2005-040-0Google Scholar
Rieffel, M. A., The cancellation theorem for the projective modules over irrational rotation C ∗-algebras. Proc. London Math. Soc. 47(1983), 285–302. https://doi.org/10.1112/plms/s3-47.2.285Google Scholar
Rieffel, M. A., Projective modules over higher-dimensional non-commutative tori. Canad. J. Math. 40(1988), no. 2, 257–338. https://doi.org/10.4153/CJM-1988-012-9Google Scholar
Rieffel, M. A., Continuous fields of C ∗-algebras coming from group cocycles and actions. Math. Ann. 283(1989), 631–643. https://doi.org/10.1007/BF01442857Google Scholar
Rieffel, M. A., Deformation-quantization for actions of ℝd. Mem. Amer. Math. Soc. 106(1993). https://doi.org/10.1090/memo/0506Google Scholar
Rieffel, M. A., Metrics on states from actions of compact groups. Doc. Math. 3(1998), 215–229.Google Scholar
Rieffel, M. A., Group C ∗-algebras as compact quantum metric spaces. Doc. Math. 7(2002), 605–651.Google Scholar
Rieffel, M. A., Gromov–Hausdorff distance for quantum metric spaces. Mem. Amer. Math. Soc. 168(2004), no. 796, 1–65. https://doi.org/10.1090/memo/0796Google Scholar
Thangavelu, S., Lectures on hermite and laguerre expansions. Math. Notes, 42, Princeton University Press, Princeton, NJ, 1993.Google Scholar
Zeller-Meier, G., Produits croisés d’une C ∗-algèbre par un groupe d’ automorphismes. J. Math. Pures Appl. 47(1968), no. 2, 101–239.Google Scholar