The operator system of Toeplitz matrices
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- by Douglas Farenick;
- Trans. Amer. Math. Soc. Ser. B 8 (2021), 999-1023
- DOI: https://doi.org/10.1090/btran/83
- Published electronically: November 17, 2021
- HTML | PDF
Abstract:
A recent paper of A. Connes and W.D. van Suijlekom [Comm. Math. Phys. 383 (2021), pp. 2021–2067] identifies the operator system of $n\times n$ Toeplitz matrices with the dual of the space of all trigonometric polynomials of degree less than $n$. The present paper examines this identification in somewhat more detail by showing explicitly that the Connes–van Suijlekom isomorphism is a unital complete order isomorphism of operator systems. Applications include two special results in matrix analysis: (i) that every positive linear map of the $n\times n$ complex matrices is completely positive when restricted to the operator subsystem of Toeplitz matrices and (ii) that every linear unital isometry of the $n\times n$ Toeplitz matrices into the algebra of all $n\times n$ complex matrices is a unitary similarity transformation.
An operator systems approach to Toeplitz matrices yields new insights into the positivity of block Toeplitz matrices, which are viewed herein as elements of tensor product spaces of an arbitrary operator system with the operator system of $n\times n$ complex Toeplitz matrices. In particular, it is shown that min and max positivity are distinct if the blocks themselves are Toeplitz matrices, and that the maximally entangled Toeplitz matrix $\xi _n$ generates an extremal ray in the cone of all continuous $n\times n$ Toeplitz-matrix valued functions $f$ on the unit circle $S^1$ whose Fourier coefficients $\hat f(k)$ vanish for $|k|\geq n$. Lastly, it is noted that all positive Toeplitz matrices over nuclear C$^*$-algebras are approximately separable.
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Bibliographic Information
- Douglas Farenick
- Affiliation: Department of Mathematics & Statistics, University of Regina, Regina, Saskatchewan S4S 0A2, Canada
- MR Author ID: 243969
- ORCID: 0000-0002-7151-213X
- Email: douglas.farenick@uregina.ca
- Received by editor(s): April 3, 2021
- Received by editor(s) in revised form: May 22, 2021, and June 7, 2021
- Published electronically: November 17, 2021
- Additional Notes: This work was supported in part by the NSERC Discovery Grant program
- © Copyright 2021 by the author under Creative Commons Attribution 3.0 License (CC BY 3.0)
- Journal: Trans. Amer. Math. Soc. Ser. B 8 (2021), 999-1023
- MSC (2020): Primary 46L07, 47L05
- DOI: https://doi.org/10.1090/btran/83
- MathSciNet review: 4340832