Truncated Geometry on the Circle

Abstract

In this letter we prove that the pure state space on the $n \times n$ complex Toeplitz matrices converges in Gromov-Hausdorff sense to the state space on $C(S^1)$ 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 $S^1$ with density functions $c \prod_{j=1}^n (1-\cos(t-\theta_j))$ is dense in the set of all positive Borel measures on $S^1$ in the weak$^*$ topology.