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.

中文翻译：

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用于频谱截断的状态空间的 Gromov-Hausdorff 收敛

我们研究了几何谱截断度量的收敛方面。我们找到了算子系统谱三元组序列的一般条件，当配备 Connes 距离公式时，可以证明相应状态空间的 Gromov-Hausdorff 收敛的结果。我们举例说明了圆的谱截断、具有有限数量傅立叶模式的圆上的傅立叶级数以及收敛到球体的矩阵代数的结果。