In this letter, we prove that the pure state space on the $$n \times n$$ n × n complex Toeplitz matrices converges in the Gromov–Hausdorff sense to the state space on $$C(S^1)$$ C ( S 1 ) as n grows to infinity, if we equip these sets with the metrics defined by the Connes distance formula for their respective natural Dirac operators. A direct consequence of this fact is that the set of measures on $$S^1$$ S 1 with density functions $$c \prod _{j=1}^n (1-\cos (t-\theta _j))$$ c ∏ j = 1 n ( 1 - cos ( t - θ j ) ) is dense in the set of all positive Borel measures on $$S^1$$ S 1 in the weak$$^*$$ ∗ topology.
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