The operator system of Toeplitz matrices

TL;DR

This paper explores the operator system structure of Toeplitz matrices, establishing a complete order isomorphism with trigonometric polynomial duals, and deriving new positivity and isometry results with implications for block Toeplitz matrices.

Contribution

It explicitly proves the Connes--van Suijlekom isomorphism as a unital complete order isomorphism and explores its consequences for positivity and isometries of Toeplitz matrices.

Findings

Every positive linear map on matrices is completely positive on Toeplitz subspace.

Unital isometries of Toeplitz matrices are unitary similarities.

Distinct min and max positivity for Toeplitz block matrices.

Abstract

A recent paper of A.~Connes and W.D.~van Suijlekom 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. Consequences of this complete order isomorphism are also examined, yielding two special results of note: (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. This latter result gives a new proof of a theorem established earlier. 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 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 coffecients $\hat f(k)$ vanish for $|k|\geq n$ .