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.
Highlights
We continue our study of spectral truncations of geometry that we started in [10] and here focus on the metric convergence aspect of so-called operator system spectral triples
The usual spectral approach to geometry [9] in terms of a ∗-algebra A of operators on H and a self-adjoint operator D on H has been adapted in [10,11] to deal with such spectral truncations
We consider sequences of spectral triples on operator systems and formulate general conditions under which we prove the state spaces equipped with the above distance functions to converge to a limiting state space
Summary
We continue our study of spectral truncations of (noncommutative) geometry that we started in [10] and here focus on the metric convergence aspect of so-called operator system spectral triples. If E = A is a ∗-algebra this reduces to the usual distance function [8,9] on the state space of the C ∗-algebra A = A It agrees with the definition of quantum metric spaces based on order-unit spaces given in [17,18,21,22,23,27,28]. We consider sequences of spectral triples on operator systems and formulate general conditions under which we prove the state spaces equipped with the above distance functions to converge to a limiting state space. The latter is described by an operator system spectral triple. In [13] certain sets of states have been identified for which the Connes’ distance formula has good convergence properties with respect to a given metric on a Riemannian manifold
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