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Journal of Operator Theory

Volume 84, Issue 1, Summer 2020  pp. 211-237.

Convergence of Heisenberg modules over quantum $2$-tori for the modular Gromov-Hausdorff propinquity

Authors: Frederic Latremoliere
Author institution:Department of Mathematics, University of Denver, Denver CO 80208, U.S.A.

Summary: The modular Gromov-Hausdorff propinquity is a distance on clas\-ses 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 continuous family for the modular propinquity.

Keywords: noncommutative metric geometry, Monge--Kantorovich distance,\break Gromov--Hausdorff convergence, quantum metric spaces, lip-norms, $D$-norms, Hilbert modules, noncommutative connections, noncommutative Riemannian geometry, unstable $K$-theory

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