Gromov-Hausdorff convergence of state spaces for spectral truncations

Abstract

We study the convergence aspects of the metric on spectral truncations of geometry. We find general conditions on sequences of operator system spectral triples that allows one to prove a result on Gromov-Hausdorff convergence of the corresponding state spaces when equipped with Connes' distance formula. We exemplify this result for spectral truncations of the circle, Fourier series on the circle with a finite number of Fourier modes, and matrix algebras that converge to the sphere.

arXiv comment15 pages, 3 figures (published version)

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• Truncated Geometry on the Circle
• Convergence of Fourier truncations for compact quantum groups and finitely generated groups
• Tolerance Relations and Quantization
• Reconstructing manifolds from truncated spectral triples
• A comparison of two quantum distances