Truncated Geometry on the Circle

TL;DR

This paper proves that the state spaces of finite Toeplitz matrices converge to the state space of continuous functions on the circle, with implications for measure approximation on the circle.

Contribution

It establishes the Gromov-Hausdorff convergence of Toeplitz matrix state spaces to the circle's function space and shows density of certain measure sets.

Findings

Convergence of Toeplitz state spaces to C(S^1) in Gromov-Hausdorff sense.

Density of measures with specific densities in all positive measures on S^1.

Connection between matrix state spaces and measure approximation 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.