Pure UCP Maps on Finite Toeplitz Systems and Quantum Gromov--Hausdorff Convergence
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866914623780814848 |
|---|---|
| author | Duhan, Ritul Jindal, Abhay |
| author_facet | Duhan, Ritul Jindal, Abhay |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2606_02561 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Pure UCP Maps on Finite Toeplitz Systems and Quantum Gromov--Hausdorff Convergence Duhan, Ritul Jindal, Abhay Operator Algebras Functional Analysis Quantum Algebra 46L07, 58B34, 15B05, 42A05, 42A82, 54E35 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. |
| title | Pure UCP Maps on Finite Toeplitz Systems and Quantum Gromov--Hausdorff Convergence |
| topic | Operator Algebras Functional Analysis Quantum Algebra 46L07, 58B34, 15B05, 42A05, 42A82, 54E35 |
| url | https://arxiv.org/abs/2606.02561 |