The operator system of Toeplitz matrices

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 $|k|\geq n$ .


INTRODUCTION
Toeplitz operators and matrices are among the most intensively studied and best understood of all classes of Hilbert space operators; in this paper, they are considered from the perspective of the unital selfadjoint linear subspaces they generate. These subspaces of matrices and operators are concrete instances of operator systems, which in the abstract refer to matrix-ordered involutive complex vector spaces possessing an Archimedean order unit [8].
In addition to classical Toeplitz matrices, this paper considers block Toeplitz matrices, which are matrices x of the form for some s −n+1 , . . . , s n−1 in an operator system S. The present paper considers such matrices x when S is an operator system, and addresses the issue of positivity 2020 Mathematics Subject Classification. 46L07, 47L05. Supported in part by the NSERC Discovery Grant program.
for these Toeplitz matrices, particularly in the cases S = M m (C), the C * -algebra of m×m complex matrices, and S = C(S 1 ) (m) , the operator system of m×m complex Toeplitz matrices. The work in this paper is strongly motivated by recent results of Connes and van Suijlekom [9] which, among other things, identify the operator system of Toeplitz matrices with the dual space of a function system of trigonometric polynomials. To explain the contributions of the present paper and set the notation, let C(S 1 ) denote the unital abelian C * -algebra of all continuous functions f : S 1 → C, where S 1 ⊂ C is the unit circle. For each n ∈ N, let C(S 1 ) (n) denote the vector space of those f ∈ C(S 1 ) for which the Fourier coefficientsf(k) of f satisfyf(k) = 0 for every k ∈ Z such that |k| n. Thus, every f ∈ C(S 1 ) (n) is given by as a function of z ∈ S 1 , where each α k = 1 2π 2π 0 f(e iθ )e −ikθ dθ. The vector space C(S 1 ) (n) is an operator system via the matrix ordering that arises from the identification of M p C(S 1 ) (n) , the space of p × p matrices with entries from C(S 1 ) (n) , with the space of continuous functions F : S 1 → M p (C), and where the Archimedean order unit is the canonical one (namely, the constant function χ 0 : S 1 → C given by χ 0 (z) = 1, for z ∈ S 1 ).
The operator system of all n×n Toeplitz matrices over C is denoted by C(S 1 ) (n) ; the identity matrix in M n (C) is the canonical Archimedean order unit for C(S 1 ) (n) .
As explained in [8], if R is an operator system and R d denotes its dual space, then R d is a matrix-ordered * -vector space. Specifically, a matrix Φ = [ϕ ij ] p i,j=1 of linear functionals ϕ ij : R → C is considered positive whenever the linear map r → [ϕ ij (r)] p i,j=1 is a completely positive linear map of R into the algebra M p (C). Furthermore, if φ : R → S is a linear map of operator systems and φ d : S d → R d denotes the adjoint transformation as linear mapping of matrix-ordered * -vector spaces, then φ is positive if and only if φ d is positive. Likewise, φ ⊗ id M p (C) is positive if and only if φ d ⊗ id M p (C) is positive, for p ∈ N.
If an operator system R has finite dimension, then any faithful state ϕ on R serves an Archimedean order unit for the matrix-ordered * -vector space R d , thereby giving R d the structure of an operator system. Because the linear functional e (n) : C(S 1 ) (n) → C given by is a faithful state, we shall henceforth designate e (n) as the Archimedean order unit for the operator system dual of C(S 1 ) (n) . The category O 1 has as its objects operator systems, and as its morphisms unital completely positive linear maps. Therefore, an isomorphism in this category is a unital completely positive linear map φ : R → S between operator systems R and S such that φ is a linear bijection and both φ and φ −1 are completely positive. (The complete positivity of a linear bijection φ is not sufficient to imply the complete positivity of its inverse φ −1 .) Such a linear isomorphism is called a unital complete order isomorphism and we denote the existence of such an isomorphism between operator systems R and S with the notation R ≃ S.
This notation above has its own ambiguity, as explicit reference to the Archimedean order units of R and S is not made. In this paper, whenever we are speaking of specific operator systems R and S that have had Archimedean order units e R and e S explicitly designated, then R ≃ S implies that there is a complete order isomorphism between these operator systems that sends e R to e S .
The main result of this paper is the following isomorphism theorem in the operator system category; the remaining results in this paper are derived from this isomorphism.
Theorem 1.1, as stated above, was proved in [14,Theorem 4.5] for n = 2 using an approach rather different from the approach of the present paper. In that paper, the operator system C(S 1 ) (2) arises as S 1 , the operator system generated by a universal unitary operator. The approach in the present paper is inspired by and based upon an elegant argument of Connes and van Suijlekom [9, Proposition 4.6], which proves that there is a linear unital order isomorphism φ between the operator system C(S 1 ) (n) and the operator system dual C(S 1 ) (n) d . The new contribution in Theorem 1.1 is the proof that this linear unital order isomorphism φ is a complete order isomorphism. Theorem 1.1 has a curious consequence for positive linear maps ψ : M n (C) → M m (C). Recall that a positive linear map ψ : M n (C) → M m (C) is decomposable if it is the sum of completely positive and completely co-positive linear maps; that is, if there are n × m matrices a 1 , . . . , a k and b 1 , . . . , b ℓ such that where x t denotes the transpose map on M n (C) (which is positive but not completely positive). If x is a Toeplitz matrix, then there is a unitary u independent of x for which x t = u * xu (namely, u = n i=1 e i,n−i+1 , where e ij is a matrix unit of M n (C)). Thus, the restriction of a decomposable positive linear map to the Toeplitz operator system C(S 1 ) (n) is completely positive. However, for all n 3 there exist indecomposable positive linear maps on M n (C) [26], which makes the following assertion somewhat unexpected. Theorems 1.1 and 1.2 also lead to a new proof of the following theorem from [13] concerning linear isometries of operator systems of Toeplitz matrices.
is a unital linear isometry, then there exists a unitary matrix v such that φ(x) = v * xv, for every x ∈ C(S 1 ) (n) .
In §6, to address questions of matrix positivity for Toeplitz matrices x over operator systems S, as in equation (1), operator system tensor products will have a role in giving meaning to various notions of positivity. All of these notions coincide if the s k are assumed to be elements of M m (C) (or, indeed, of any nuclear C *algebra), but the notions become distinct if the elements s k in the Toeplitz matrix x are themselves Toeplitz matrices (Corollary 6.6).
Lastly, there is a well-known relationship between Toeplitz operators on the Hardy space of S 1 and essentially-bounded functions (symbols) on S 1 ; this relationship is addressed in §4 for the symbol spaces C(S 1 ) (n) , giving rise to the identification in the operator system category of certain Toeplitz operators as dual elements to finite Toeplitz matrices.
2. PRELIMINARIES 2.1. Terminology. Throughout this paper, when referring to a positive matrix or operator, what is meant is a selfadjoint operator with spectrum contained in the halfline [0, ∞); thus, "positive" is the same as "positive semidefinite" in this terminology.
The Schur-Hadamard product (or entry-wise product) of d × d matrices a and b is denoted by a • b and is the matrix in which each (i, j)-entry of a • b is given by the product of the corresponding (i, j)-entries of a and b. The Schur-Hadamard isometry is the linear map v : C d → C d ⊗ C d that sends the k-th canonical orthonormal basis vector e k of C d to e k ⊗ e k ∈ C d ⊗ C d ; this isometry has the property that The canonical matrix units for a full matrix algebra M n (C) shall be denoted, henceforth, by e ij , and the analytic shift matrix in M n (C) is the lower-triangular Toeplitz matrix s given The algebra of bounded linear operators acting on a Hilbert space H is denoted by B(H). An operator x ∈ B(H) is irreducible if the commutant of {x, x * } is trivial (i.e., consists only of scalar multiples of the identity operator). Likewise, an operator subsystem S ⊆ B(H) is irreducible if S ′ (the commutant of S) is trivial. Thus, the analytic Toeplitz matrix s is irreducible, along with any operator subsystem of M n (C) that contains s.

Operator systems.
Formally, an operator system is a triple (S, {C n } n∈N , e S ) consisting of: (1) a complex * -vector space S; (2) a family {C n } n∈N of proper cones C n ⊆ M n (S) sa of selfadjoint matrices with the properties that C n ⊕ C m ⊆ C n+m and γ * C n γ ⊆ C m for all n, m ∈ N and all linear transformations γ : C m → C n ; and (3) an Archimedean order unit e S for the ordered real vector space S sa . In concrete situations, operator systems arise as unital * -closed subspaces of unital C * -algebras.
If S and R are operator systems with Archimedean order units e S and e R , then S is said to be an operator subsystem of R if S ⊆ R and e S = e R . A unital completely positive linear map φ : S → T is a unital complete order embedding if φ is a unital complete order isomorphism between S and the operator subsystem φ(S) of T.
The embedding theorem of Choi and Effros [8] states that every operator system R is unitally completely order isomorphic to an operator subsystem of B(H), for some Hilbert space H. Therefore, every operator system R is capable of generating a C * -algebra. Two such C * -algebras of note are the C * -envelope, C * e (R), and the universal C * -algebra, C * u (R), which satisfy, respectively, minimal and maximal universal properties [16,24].
The following result from the literature shows that C * -envelope of an operator system can be nuclear, while the universal C * -algebra of the same operator system need not be exact. (1) C * e (C(S 1 ) (n) ) = M n (C), is an exact C * -algebra. Proof. The first two assertions are given by Propositions 4.2 and 4.3 of [9], while the third assertion is derived from Proposition 6.3 of [20].

Definition 2.2.
An operator subsystem R of a unital C * -algebra A is hyperrigid in A if, for every representation π : A → B(H π ) of A, the ucp map π |R : R → B(H π ) has a unique extension to a completely positive linear map on A.
In the definition above, the unique completely positive extension of the restriction π |R of π to R is of course π itself. As the algebra generated by s and s * is M n (C), this shows that the unitary Toeplitz matrices generate the algebra M n (C). The hyperrigidity of C(S 1 ) (n) now follows from [17,Lemma 3.11], which states that if the unitary elements of an operator subsystem R of a unital C * -algebra A generate A, then R is hyperrigid in A.
The following lemmas on automatic complete positivity will be useful. Proof. The unital complete order embedding ι e : S → C * e (S) has the property that, for any X ∈ M p (S), the matrix ι As every positive linear map of an operator system into a unital abelian C * -algebra is completely positive [25,Theorem 3.9], we deduce that ι e • φ is completely positive. Hence, φ is necessarily completely positive. Proof. The range of φ is a * -closed subspace of B(H). Let T denote the operator subsystem of B(H) spanned by φ(R) and the identity operator on H; thus, T is a finite-dimensional operator system and φ is a positive linear map R → T. Consider the dual linear map φ d : 2.3. Tensor representations. Suppose that an operator system R has a linear basis {r ℓ } ℓ . Fix p ∈ N and let {e ij } p i,j=1 denote the canonical basis of M p (C) given by the standard matrix units. The algebraic tensor product R ⊗ M p (C) may be identified with the set of p×p matrices with entries from R or, alternatively, as "R with entries from M p (C)." As it is this second viewpoint that is required, a brief explanation of what this phrase means is given below.
If x ∈ R ⊗ M p (C), then there exists scalars α ℓij ∈ C, finitely many of which are nonzero, such that as p×p matrices with entries from R, which is the usual point of view in the theory of operator systems and operator spaces. On the other hand, we may express where a ℓ ∈ M p (C) is the matrix i j α ℓij e ij . This give us the conceptual identi- Similarly, if φ : R → S is a linear map of operator systems, then the linear map is, in our two views of x above, given by

Canonical linear bases.
The canonical linear basis for the operator system C(S 1 ) (n) of Toeplitz matrices is given by powers of the shift and their adjoints. That is, if {r −n+1 , . . . , r 0 , . . . , r n−1 } is the set of matrices defined by then {r −n+1 , . . . , r 0 , . . . , r n−1 } is a linear basis of C(S 1 ) (n) . The identity matrix r 0 serves as the Archimedean order unit for the operator system C(S 1 ) (n) .
The operator system C(S 1 ) (n) of trigonometric polynomials of degree less than n has a canonical linear basis consisting of functions χ k : S 1 → C defined by χ k (z) = z k , for k = −n + 1, . . . , n − 1. Recall that χ 0 is the Archimedean order unit for the operator system C(S 1 ) (n) .
Let {e k | −n+1 k n−1} denote the dual basis of the {χ k | −n+1 k n−1}; thus, for each k, for every f ∈ C(S 1 ) (n) . The faithful state e 0 is the designated Archimedean order unit of the operator system dual C(S 1 ) (n) d .

THE CONNES-VAN SUIJLEKOM THEOREM
The Connes-van Suijlekom theorem [9, Proposition 4.6] is formulated and proved below in the context of the operator system category. The proof is modeled on the arguments of Connes and van Suijlekom.
Proof. The case n = 1 is trivial; therefore, it is assumed that n 2.
Consider the linear map φ : a k z k . Note that φ is injective (and hence surjective) and that φ sends the identity matrix in C(S 1 ) (n) to the linear functional f →f(0), which we have identified as the Archimedean order unit e 0 of the operator system dual C(S 1 ) (n) d . Therefore, φ is a unital linear isomorphism, and it remains to show that φ and φ −1 are completely positive.
To show that φ is completely positive, fix p ∈ N and consider the linear iso- which as a matrix is represented as To understand the action of , it is enough to understand the action of e ℓ ⊗ g, for some fixed g ∈ M p (C). Writing e ℓ ⊗ g as a p × p matrix of linear functionals on C(S 1 ) (n) , we obtain implying that the Schur-Hadamard product of g with a ℓ . Therefore, if T is the block Toeplitz matrix in (3), then the evaluation of We now show that if T is positive, then φ [p] (T )[F] is positive for every positive F ∈ C(S 1 ) (n) ⊗M p (C). To this end, assume that the matrix T in (3) is positive. Thus, the matrixT whose (k, ℓ)-entry is τ k−ℓ ⊗ 1 p , where 1 p denotes the identity matrix of M p (C), is positive in M n (M p (C) ⊗ M p (C)). That is, the matrixT is a positive operator on the Hilbert space H constructed from the direct sum of n copies of where the set I j is given by for some a k ∈ M p (C), then, by the operator-valued Riesz-Fejér theorem [10,18], In computing the product H(z) * H(z), we obtain where I j is the set in (5). Hence, Consider the element q = we deduce that which proves that φ [p] is a positive linear map. Hence, φ is completely positive.
Turning now to the proof that φ −1 is completely positive, we begin by showing φ −1 is positive. To this end, let ψ be a pure state on C(S 1 ) (n) , and consider weak*-closed convex set S ψ of all states on C(S 1 ) that extend ψ. By the Krein-Milman theorem S ψ has an extreme point Ψ, and this extreme point is necessarily an extreme point of the state space of the abelian C * -algebra C(S 1 ); hence Ψ is a point evaluation at some point λ ∈ S 1 , implying that ψ(f) = a k z k . Let Λ be the positive Toeplitz matrix given by and note that φ(Λ) = ψ. Hence, φ −1 (ψ) is positive and, therefore, so is φ −1 (ϕ), for every linear functional ϕ that is a limit of positive scalar multiples of convex combinations of pure states on C(S 1 ) (n) . This proves that φ −1 is positive.
is also positive. Furthermore, since the operator system C(S 1 ) (n) is an operator subsystem of the abelian C * -algebra C(S 1 ) and because every positive linear map of an operator system into a unital abelian C *algebra is completely positive by Lemma 2.4, the positive linear map φ −1 d is completely positive. Thus, φ −1 is completely positive.
Hence, φ is a unital complete order isomorphism.
Proof. With any finite-dimensional operator system R, the Archimedean order unit e R serves as an Archimedean order unit for the bidual R dd , implying that the operator system R dd is unitally completely order isomorphic to R. Hence, in passing to operator system duals, Theorem 3.1 yields the conclusion.

OPERATOR SYSTEMS OF TOEPLITZ OPERATORS
The canonical orthonormal basis functions for the Hilbert space L 2 (S 1 ) are given by e k (z) = (2π) −1/2 z k , for k ∈ Z, while the Hardy space H 2 (S 1 ) is the subspace of L 2 (S 1 ) having orthonormal basis {e k } k 0 . The projection operator on L 2 (S 1 ) with range H 2 (S 1 ) is denoted by P. The linear map π : C(S 1 ) → B L 2 (S 1 ) given by π(f) = M f , the operator of multiplication by f on the Hilbert space L 2 (S 1 ), is an isometric * -representation of C(S 1 ) on L 2 (S 1 ), implying that the linear map ϕ : C(S 1 ) → B H 2 (S 1 ) defined by ϕ(f) = PM f|H 2 (S 1 ) is unital and completely positive. The operator ϕ(f) is denoted by T f , the Toeplitz operator with symbol f. In expressing a Toeplitz operator T f as a matrix with respect to the canonical orthonormal basis of H 2 (S 1 ), the result is an infinite Toeplitz matrix whose (ℓ, j)entry of the matrix is given byf(ℓ − j), for ℓ, j ∈ {0, 1, 2, . . . }.
For each n ∈ N, let which is an operator system of Toeplitz operators on H 2 (S 1 ).
, and if ϕ n = ϕ |C(S 1 ) (n) , for every n ∈ N, then ϕ n is a unital complete order isomorphism of C(S 1 ) (n) and T (n) .
Proof. The function ϕ n : C(S 1 ) (n) → B H 2 (S 1 ) is a unital completely positive linear map with range T (n) . In considering Fourier coefficients, the linear map ϕ n is clearly a linear isomorphism, and so we aim to prove that ϕ −1 n is completely positive.
First note that because ϕ −1 n : T (n) → C(S 1 ) (n) and C(S 1 ) (n) is an operator subsystem of the abelian C * -algebra C(S 1 ), the complete positivity of ϕ n is automatic, by [25, Theorem 3.9], once it has been shown that ϕ n is positive. To this end, let T f ∈ T (n) be a positive Toeplitz operator with symbol f ∈ C(S 1 ) (n) ; thus, the essential spectrum Sp e (T f ) of T f is a subset of [0, ∞). Because Sp e (T f ) = f(S 1 ) (see, for example, [6, §4.6]), the symbol f is a positive element of C(S 1 ) and, hence, of C(S 1 ) (n) , thereby proving that ϕ n is a positive map.
In passing to duals and applying Theorem 3.1, we obtain another result that seems curious upon first encountering it: the n × n Toeplitz matrices are dual to the infinite Toeplitz matrices arising from symbols in C(S 1 ) (n) .
There is another natural ucp map of interest: the one that maps a Toeplitz oper- (Equivalently, by Proposition 4.1, the map that sends the infinite Toeplitz matrix T f to its n × n leading principal submatrix t f .) If we denote this map by ψ n , then ψ n : C(S 1 ) (n) → C(S 1 ) (n) is a ucp bijection; however, ψ n is not a complete order isomorphism. For example, in the case n = 2, the function f(z) Proof. The stated assertion is a direct consequence of Lemma 2.5, Theorem 3.1, and the fact that the C * -envelope of C(S 1 ) (n) d ≃ C(S 1 ) (n) is abelian.
To illustrate this result above, consider the indecomposable positive linear map ψ : M 3 (C) → M 3 (C) (known as the Choi map; see [26]) given by By the Geršgorin circle theorem, the eigenvalues of the symmetric matrix g are nonnegative, and so the mapping ψ coincides on C(S 1 ) (3) with the completely positive map on M 3 (C) given by the Schur-Hadamard multiplication of y ∈ M 3 (C) by the positive (semidefinite) matrix g. Theorem 5.1 also admits a version for Toeplitz matrices over nuclear C * -algebras: see Corollary 7.4. Theorem 3.1 also leads to an alternative proof of the following result first established in [13] concerning the structure of unital linear isometries on the operator system of Toeplitz matrices.
Proof. Let W(y) denote the numerical range of a matrix y ∈ M n (C): Thus, W (φ(x)) = W(x) for every x ∈ C(S 1 ) (n) , since φ : C(S 1 ) (n) → M n (C) is a unital linear isometry. As this is also the case for the lower-triangular nilpotent shift matrix s ∈ C(S 1 ) (n) , the numerical ranges of the contractions s and φ(s) coincide; hence, by a theorem of Wu [27], there is a unitary v such that then φ(s) = v * sv.
Because an element x ∈ C(S 1 ) (n) is positive if and only if the numerical range of x is contained in [0, ∞), the unital linear isometry φ preserves positivity. Hence, by Theorem 5.1, φ is a completely positive linear map. Now consider the unital completely positive map ψ : M n (C) → M n (C) defined by ψ(y) = vφ(y)v * . The fixed point set F ψ = {y ∈ M n (C) | ψ(y) = y} contains s and s * , and so F ψ is an irreducible operator subsystem of M n (C) such that the restriction of ψ to F ψ is the identity map. This implies, by Arveson's Boundary Theorem [5,7,11], ψ(y) = y for all y ∈ M n (C); in particular, φ(x) = v * xv, for every x ∈ C(S 1 ) (n) .
Let A(S 1 ) (n) denote the set of analytic Toeplitz matrices, by which is meant those Toeplitz matrices that are lower-triangular. Note that A(S 1 ) (n) is a unital abelian algebra generated by the shift s. Proof. Because the operator system C( is well-defined and determines a unital completely positive linear map [25, Proposition 2.12, 3.5]. Further, because αφ(s) + β1 = αs + β1 for all α, β ∈ C (by hypothesis),φ(s) is a matrix of unit norm with numerical range equal to that of the shift s. Applying the proof of Proposition 5.2 toφ, we obtain the stated structure forφ and, hence, for φ.

POSITIVITY OF BLOCK TOEPLITZ MATRICES VIA TENSOR PRODUCTS
If S is any operator system, then with respect to the linear basis {r −n+1 , . . . , r n−1 } of C(S 1 ) (n) identified earlier, an arbitrary element x of the algebraic tensor product C(S 1 ) (n) ⊗ S can be written as for some s −n+1 , . . . , s n−1 ∈ S. Hence, the algebraic tensor product C(S 1 ) (n) ⊗ S is naturally identified with the * -closed complex vector space of n × n Toeplitz matrices with entries from S, whereby the element x above is represented as Such elements x ∈ C(S 1 ) (n) ⊗ S are, therefore, called block Toeplitz matrices. The purpose of this section is to consider various ways in which block Toeplitz matrices x can be said to be "positive," particularly in the cases where S = M m (C) or S = C(S 1 ) (m) , as these cases are of special interest in applied mathematics (for example, [19]).
With the case S = M m (C), the positivity of block Toeplitz matrices is determined by way of the following theoretical criterion, which is a direct consequence of (the proof of) Theorem 3.1.
Proposition 6.1. The following statements are equivalent for a matrix T ∈ M n (M m (C)) of the form where each τ k ∈ M m (C): (1) T is positive; (2) for every positive function F : for some matrices a k ∈ M m (C), the matrix is positive.
6.1. Operator system tensor products. One way to approach the positivity question for block Toeplitz matrices for operator systems S different from M m (C) is via tensor products of operator systems.
An operator system tensor product [22] R ⊗ σ S of operator systems S and T is an operator system structure ⊗ σ on the algebraic tensor product R ⊗ S such that: (1) (R ⊗ S, {C n } n∈N , e R ⊗ e S ) is an operator system, where C n ⊆ M n (R ⊗ S), for each n, and e R and e S denote the Archimedean order units for R and S; (2) a ⊗ b ∈ C nm , for all a ∈ M n (R) + , b ∈ M m (S) + , and n, m ∈ N; (3) for all n, m ∈ N and all ucp maps φ : R → M n (C) and ψ : S → M m (C), the linear map φ ⊗ ψ : R ⊗ σ S → M nm (C) is completely positive. Suppose that R 1 ⊆ S 1 and R 2 ⊆ S 2 are inclusions of operator systems. Let ι j : R j → S j denote the inclusion maps ι j (x j ) = x j for x j ∈ S j , j = 1, 2, so that the map ι 1 ⊗ ι 2 : R 1 ⊗ R 2 → S 1 ⊗ S 2 is a linear inclusion of vector spaces. If γ and σ are operator system structures on R 1 ⊗ R 2 and S 1 ⊗ S 2 respectively, then we use the notation is a (unital) completely positive map. This notation is motivated by the fact that ι 1 ⊗ ι 2 is a completely positive map if and only if, for every p ∈ N, the cone M p (R 1 ⊗ γ R 2 ) + is contained in the cone M p (S 1 ⊗ σ S 2 ) + . If, in addition, ι 1 ⊗ ι 2 is a complete order isomorphism onto its range, then we write In particular, if γ and σ are two operator system tensor-product structures on R⊗S, then R ⊗ γ S = R ⊗ σ S means that the identity map is a unital complete order isomorphism (equivalently, that the matrix positivity cones for R ⊗ γ S and R ⊗ σ S coincide).
6.2. Minimal and maximal tensor products of Toeplitz matrices. An operator system tensor product is described by indicating what the matrix positivity cones are.
The minimal tensor product ⊗ min is the familiar spatial tensor product in matrix and operator theory: if R ⊆ B(H) and S ⊆ B(K), where H and K are Hilbert spaces, then R ⊗ min S is the operator system arising from the natural inclusion of S ⊗ T into B(H ⊗ K). In this regard, Proposition 6.1 is a characterisation of the cone When considering the finite-dimensional operator system T n of Toeplitz operators T f acting on the Hardy space H 2 (S 1 ) with symbols f ∈ C(S 1 ) (n) , the operator system T n ⊗ min T m is the operator system of two-level Toeplitz operators T h acting on the Hardy space H 2 (S 1 × S 1 ) of the torus S 1 × S 1 using symbols from the set C(S 1 × S 1 ) (n,m) of all continuous functions h : The maximal tensor product ⊗ max is the operator system structure on R ⊗ S obtained through declaring a matrix x ∈ M p (R ⊗ S) to be positive if for each ε > 0 there are n, m ∈ N, a ∈ M n (R) + , b ∈ M m (S) + , and a linear map δ : C p → C n ⊗C m such that ε(e R ⊗ e S ) + x = δ * (a ⊗ b)δ. It is more difficult for a matrix to be "max positive" than "min positive." For example, if x is a strictly positive element of S ⊗ max R, then there exist N ∈ N, Therefore, for R ⊗ min S = R ⊗ max S to hold, at least one of S or R ought to be rich in positive elements and matrices. In general, for every operator system tensor product structure ⊗ σ on R ⊗ S.
The following theorem gives two fundamental results relating the min and max tensor products. Theorem 6.2. ( [14,22]) If R and S are finite-dimensional operator systems and T is an arbitrary operator system, then Concerning the maximal tensor product of Toeplitz matrices and operators, we have: Proposition 6.3. For all n, m ∈ N, Proof. This is an immediate consequence of Proposition 4.1 and Theorems 3.1 and 6.2.
Theorem 6.2 demonstrates that for all n, m ∈ N. Corollary 6.6 below shows that if we change the entries s j of the Toeplitz matrix x in (8) from arbitrary m × m complex matrices to arbitrary m × m Toeplitz matrices, then the two forms of matrix positivity (min and max) are distinct. To prepare for the proof, we require the following lemma.
Because p is prime, ζ ℓ is a primitive p-th root of unity, and so {(ζ ℓ ) k | k = 1, . . . , p} is the set of all p-th roots of unity. Thus, which completes the proof.
The following theorem was established in the case n = m = 2 in [12,Theorem 4.7]; the proof below draws from the proof of that result. Theorem 6.5. C(S 1 ) (n) ⊗ min C(S 1 ) (m) = C(S 1 ) (n) ⊗ max C(S 1 ) (m) , for all n, m 2.
For each j, n ∈ N such that j n, let ι j,n : C(S 1 ) (j) → C(S 1 ) (n) be the canonical inclusion map, and note that this map is completely positive. Because the tensor products ⊗ min and ⊗ max are functorial, the linear map ι j,n ⊗ ι k,m , for j n and k m, is a unital completely positive embedding of C( and let x ∈ C(S 1 ) (2) ⊗ min M 2 (C(S 1 ) (m) ) be given by , and b −1 = b * 1 . As shown in [12,Theorem 4.7], this matrix-valued function is a positive element of C(S 1 ) (2) ⊗ min M 2 (C(S 1 ) (m) ) and the eigenvalues of x(z, w), for (z, w) ∈ S 1 × S 1 , are uniformly bounded below by some δ > 0. Therefore, the matrix x is strictly positive in C(S 1 ) (n) ⊗ min M 2 (C(S 1 ) (m) ), which by hypothesis coincides with C(S 1 ) (n) ⊗ max M 2 (C(S 1 ) (m) ).
, for all n, m 2.
Proof. The dual of the equality C( , which is false by Theorem 6.5. 6.3. The commuting tensor product. The commuting tensor product ⊗ c is the operator system structure on R ⊗ S obtained by declaring a matrix X ∈ M p (R ⊗ S) to be positive if (φ · ψ) (p) (X) is a positive operator for all pairs of completely positive maps φ : R → B(H) and ψ : S → B(H) with commuting ranges, where φ · ψ denotes the linear map The relationship of ⊗ c to ⊗ min can be vexing to determine, even for operator systems of low dimension. (The problem of whether S 2 ⊗ min S 2 = S 2 ⊗ c S 2 , for the operator system S 2 generated by the unitary generators of the free group C * -algebra C * (F 2 ), is equivalent to the Connes Embedding Problem [20,Theorem 5.11].) In contrast, the relationship of ⊗ c to ⊗ max can often be discerned, as is the case with the Toeplitz operator system and its dual.
Proof. Via the complete order isomorphism (C(S 1 ) (n) ) d ≃ C(S 1 ) (n) , the equality of C(S 1 ) (n) ⊗ c C(S 1 ) (n) and C(S 1 ) (n) ⊗ max C(S 1 ) (n) is possible only if C(S 1 ) (n) is completely order isomorphic to a C * -algebra A [20,Proposition 4.3]. As the C *envelope is invariant under complete order isomorphism, this would imply that A = C * e (A) ∼ = C * e (C(S 1 ) (n) ) = C(S 1 ), yielding a linear isomorphism between the finite-dimensional vector space C(S 1 ) (n) and the infinite-dimensional vector space C(S 1 ), which is impossible.
The matrix ordering of S ⊗ c R is achieved through the canonical embedding of the algebraic tensor product S ⊗ R into the C * -algebra C * u (S) ⊗ max C * u (R) [22]; that is, S ⊗ c R ⊆ coi C * u (S) ⊗ max C * u (R). Even so, the universal C * -algebras of C(S 1 ) (n) and C(S 1 ) (n) are not sufficiently tractable (e.g., see Theorem 2.1) to draw additional information from.

THE MAXIMALLY ENTANGLED TOEPLITZ MATRIX
Definition 7.1. If ⊗ σ is an operator system tensor product structure on S⊗R, for operator systems S and R, then an element x ∈ (S ⊗ σ R) + is: (1) σ-separable, if there exist k ∈ N, s 1 , . . . , s k ∈ S + , and r 1 , . . . , r k ∈ R + such that (2) σ-entangled, if x is not σ-separable. In the case where ⊗ σ is the minimal operator system tensor product ⊗ min , then we simply refer to x ∈ (S ⊗ min R) + as being separable or entangled.
A beautiful result of Gurvits shows that positive block Toeplitz matrices with blocks coming from M m (C) are separable. Theorem 7.2 (Gurvits). Every positive element of C(S 1 ) (n) ⊗ min M m (C) is separable.
Proof. The proof of Gurvits' theorem given in [15,§III] yields the result for the operator system C(S 1 ) (n) ⊗ min M m (C), although the result is predominantly cited in the literature as pertaining to the matrix algebra M n (C) ⊗ min M m (C).
The second consequence of Gurvits' separation theorem extends Theorem 5.1, showing that every positive linear map ψ of a unital nuclear C * -algebra A is "Toeplitz completely positive."  Proof. Assume that x = [a k−ℓ ] n−1 ℓ,k=0 is a positive Toeplitz matrix over A and let ψ (n) = ι n ⊗ ψ, where ι n is the identity map on C(S 1 ) (n) . By Corollary 7.3, for each k ∈ N there is a positive separable Toeplitz matrix x k ∈ C(S 1 ) (n) ⊗ min A such that x − x k < 1/k. In writing x k as which is a positive element of C(S 1 ) (n) ⊗ min B(H). As ψ (n) is norm-continuous, Because the spectrum of each x k is nonnegative, the upper semicontinuity of the spectrum as a set-valued function implies that the selfadjoint operator ψ (n) (x) also has nonnegative spectrum. Hence, To apply this notion in the case where S = C(S 1 ) (n) , we denote the dual basis of the linear basis {r −n+1 , . . . , r n−1 } of C(S 1 ) (n) by {δ −n+1 , . . . , δ n−1 }. We have the identification C(S 1 ) (n) ≃ C(S 1 ) (n) d via the unital complete order isomorphism φ : C(S 1 ) (n) → C(S 1 ) (n) d given in Theorem 3.1. The effect of φ on the canonical linear basis of C(S 1 ) (n) is φ(r k ) = e −k , for every k = −n + 1, . . . , n − 1. Note that the linear basis {e −n+1 , . . . , e n−1 } of C(S 1 ) (n) d is dual to the linear basis {χ −n+1 , . . . , χ n−1 } of C(S 1 ) (n) . Therefore, if ψ is the unital complete order isomorphism that implements C(S 1 ) (n) d ≃ C(S 1 ) (n) , then ψ(δ k ) = χ −k , for each k = −n + 1, . . . , n − 1. Thus, in equation (9) above, we replace each δ k with χ −k and arrive at the following definition. Definition 7.5. The element ξ n ∈ C(S 1 ) (n) ⊗ C(S 1 ) (n) defined by is called the maximally entangled Toeplitz matrix.
Observe that ξ n can be view as the following function S 1 → C(S 1 ) (n) : its diagonal entry satisfies τ (j) 0 > 0; and because t j is not diagonal, there is a ℓ > 0 such that τ (j) ℓ = 0. Thus, equation (12) yields for every z ∈ S 1 . This equation above shows that α j (z) is continuous in z and for every z ∈ S 1 . Hence, equation (13) implies that the Fourier coefficients of f j and f j χ ℓ agree at every k ∈ Z, which can happen only if f j is identically zero. Because f j = 0 contradicts the hypothesis that the functions f 1 , . . . , f m ∈ C(S 1 ) be nonzero, it must be that ξ n is not separable. Proof.
, then it would be separable in C(S 1 ) (n) ⊗ min C(S 1 ) as well, in contradiction to Proposition 7.6.
Besides the notion of entanglement, another property of relevance to elements of convex cones is that of purity. Definition 7.8. An element x of a convex cone C is pure if the equation x = y + z, for y, z ∈ C, implies that z = λx and y = (1 − λ)x for some real number λ ∈ [0, 1].
Proof. If W is a finite-dimensional vector space, then the tensor product W ⊗ W d is linearly isomorphic to L(W), the vector space of linear transformations on W. If we apply this linear isomorphism to a finite-dimensional operator system C(S 1 ) (n) and its operator system dual C(S 1 ) (n) , then the cone CP(C(S 1 ) (n) ) in L(C(S 1 ) (n) ) of completely positive linear maps on C(S 1 ) (n) determines a cone in C(S 1 ) (n) ⊗ C(S 1 ) (n) : namely, C(S 1 ) (n) ⊗ min C(S 1 ) (n) + [22, §4].
The canonical linear isomorphism between R ⊗ R d and L(R) is the one that maps elementary tensors x ⊗ ψ ∈ R ⊗ R d to rank-1 linear transformations r → ψ(r)x, for r ∈ R. Let Γ be the inverse of this linear isomorphism and take R = C(S 1 ) (n) , thereby obtaining a linear isomorphism in which Γ CP(C(S 1 ) (n) ) = C(S 1 ) (n) ⊗ min C(S 1 ) (n) + .
Observe that φ ∈ CP(R) is pure in the cone CP(C(S 1 ) (n) ) if and only if Γ (φ) is pure in the cone C(S 1 ) (n) ⊗ min C(S 1 ) (n) + .
One final point of interest regarding the maximally entangled Toeplitz matrix: it is universal for all (spatially) positive Toeplitz matrices over C * -algebras. Theorem 7.10 (Ando). If A is a unital C * -algebra and x ∈ (C(S 1 ) (n) ⊗ min A) + , then there exists a completely positive linear map (possibly non-unital) φ : C(S 1 ) (n) ⊗ min C(S 1 ) (n) → C(S 1 ) (n) ⊗ min A such that φ(ξ n ) = x.

CONCLUSION
The identification in the operator system category of the operator system C(S 1 ) (n) of n × n Toeplitz matrices with the operator system dual of the space C(S 1 ) (n) of trigonometric polynomials of degree less than n has a number of striking consequences, including the implications that every positive linear map of the Toeplitz matrices is completely positive and every unital M n (C)-valued linear isometric map of C(S 1 ) (n) is completely isometric. This identification also allows for a clearer understanding of positivity for block Toeplitz matrices, distinguishing block Toeplitz matrices with blocks that are Toeplitz matrices from block Toeplitz matrices with blocks that are arbitrary complex matrices.
An operator system S has the double commutant expectation property if, for every unital complete order embedding map κ : S → B(H), there exists a unital completely positive linear map φ : B(H) → (κ(S)) ′′ such that φ • κ = κ. Using the work developed in [23], one can show that, for the Toeplitz operator system C(S 1 ) (n) , the double commutant expectation property is equivalent to the assertion that C(S 1 ) (n) ⊗ min B = C(S 1 ) (n) ⊗ max B for every unital C * -algebra B. (The equality C(S 1 ) (n) ⊗ min B = C(S 1 ) (n) ⊗ max B is known to hold for all nuclear C *algebras B [22].) The case n = 2 provides some insight into the general situation.