Pure UCP Maps on Finite Toeplitz Systems and Quantum Gromov--Hausdorff Convergence

Abstract

We study pure unital completely positive maps on the finite Toeplitz operator system $ T_{d}$ of $d \times d$ Toeplitz matrices. Our first main result gives an explicit characterization of pure UCP maps from $T_{d}$ to $M_n$ in terms of positive $n\times n$ matrix-valued trigonometric polynomials of degree at most $d-1$. This characterization provides a checkable criterion for deciding when a given UCP map is pure. As a first application, we show that every pure UCP map from $ T_{d}$ to $M_n$ admits a unique UCP extension to the generated $C^*$-algebra. As a second application, we prove that, for each fixed $n$, the space of pure UCP maps from $T_{d}$ to $M_n$, equipped with the matricial Connes distance, converges in the Gromov--Hausdorff sense to the space of normalized positive $n\times n$ matrix-valued Borel measures on the unit circle, equipped with the matricial Monge--Kantorovich distance.