The operator system of Toeplitz matrices

A recent paper of A. Connes and W.D. van Suijlekom [Comm. Math. Phys. 383 (2021), pp. 2021–2067] identifies the operator system n × n\times n Toeplitz matrices with dual space all trigonometric polynomials degree less than alttext="n"> encoding="application/x-tex">n . The present examines this identification in somewhat more detail by showing explicitly that Connes–van isomorphism is a unital complete order systems. Applications include two special results matrix analysis: (i) every positive linear map complex completely when restricted to subsystem (ii) isometry into algebra unitary similarity transformation. An systems approach yields new insights positivity block matrices, which are viewed herein as elements tensor product spaces an arbitrary matrices. In particular, it shown min max distinct if blocks themselves maximally entangled alttext="xi Subscript ? encoding="application/x-tex">\xi _n generates extremal ray cone continuous Toeplitz-matrix valued functions alttext="f"> f encoding="application/x-tex">f on unit circle alttext="upper S Superscript 1"> S 1 encoding="application/x-tex">S^1 whose Fourier coefficients alttext="ModifyingAbove f With caret left-parenthesis k right-parenthesis"> ^<!-- ^ </mml:mover> stretchy="false">( k stretchy="false">) encoding="application/x-tex">\hat f(k) vanish for alttext="StartAbsoluteValue EndAbsoluteValue greater-than-or-equal-to stretchy="false">| ?<!-- ? encoding="application/x-tex">|k|\geq Lastly, noted over nuclear C ?<!-- ? encoding="application/x-tex">^* -algebras approximately separable.