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

Part of: Infinite-dimensional manifolds Selfadjoint operator algebras

Published online by Cambridge University Press:  28 March 2019

Frédéric Latrémolière
Frédéric Latrémolière*
Affiliation:
Department of Mathematics, University of Denver, Denver CO 80208 Email: [email protected]://www.math.du.edu/∼frederic
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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
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Copyright
© Canadian Mathematical Society 2019

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Footnotes

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

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