Fuzzy tori converge to the flat torus Dirac triple via an extension of spectral propinquity to twisted spectral triples with unbounded twists.
A Compactness Theorem for The Dual Gromov-Hausdorff Propinquity
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abstract
We prove a compactness theorem for the dual Gromov-Hausdorff propinquity as a noncommutative analogue of the Gromov compactness theorem for the Gromov-Hausdorff distance. Our theorem is valid for subclasses of quasi-Leibniz compact quantum metric spaces of the closure of finite dimensional quasi-Leibniz compact quantum metric spaces for the dual propinquity. While finding characterizations of this class proves delicate, we show that all nuclear, quasi-diagonal quasi-Leibniz compact quantum metric spaces are limits of finite dimensional quasi-Leibniz compact quantum metric spaces. This result involves a mild extension of the definition of the dual propinquity to quasi-Leibniz compact quantum metric spaces, which is presented in the first part of this paper.
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UNVERDICTED 2representative citing papers
Certain natural quantum metrics on matrix algebras M_n are separated by positive Gromov-Hausdorff propinquity distance when n is not prime.
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How to approximate the flat spectral triple of a quantum torus by fuzzy tori : a twisted tale
Fuzzy tori converge to the flat torus Dirac triple via an extension of spectral propinquity to twisted spectral triples with unbounded twists.
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Quantum metrics from the trace on full matrix algebras
Certain natural quantum metrics on matrix algebras M_n are separated by positive Gromov-Hausdorff propinquity distance when n is not prime.