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Approximation of Quantum Tori by Finite Quantum Tori for the Quantum Gromov-Hausdorff Distance

Frederic Latremoliere
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Abstract

The statement that one can approximate a quantum torus by some twisted convolution C*-algebra of a (finite) quotient of Z^d can be found in the physics literature dealing with quantum field theory and M-theory. In this paper, we show that indeed the quantum tori are limits of such finite dimensional C*-algebras for the quantum Gromov-Hausdorff distance for compact quantum metric spaces introduced by Rieffel, when one chooses the right metric structures. The proof uses a mild extension of a result by Ramazan on continuous fields of C*-algebras obtained from groupoid C*-algebras, and extends on Rieffel's work on the continuity of the quantum tori for the quantum Gromov-Hausdorff distance. We also show that one can collapse a quantum torus to a lower dimensional quantum torus.

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Frederic Latremoliere. 2005-10-18. Approximation of Quantum Tori by Finite Quantum Tori for the Quantum Gromov-Hausdorff Distance. https://arxiv.org/abs/math/0310214

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