Pure UCP Maps on Finite Toeplitz Systems and
Quantum Gromov–Hausdorff Convergence
June 01, 2026
We study pure unital completely positive (UCP) maps on the finite Toeplitz operator system \(\mathcal{T}_{{\tt{d}}}\) of \({\tt{d}}\times {\tt{d}}\) Toeplitz matrices. This note makes three main contributions.
We give an explicit characterization of pure UCP maps on \(\mathcal{T}_{{\tt{d}}}\) taking values in \(n \times n\) matrices \(M_n\) in terms of positive \(n\times n\) matrix-valued trigonometric polynomials of degree at most \({\tt{d}}-1\). The characterization yields a checkable criterion for deciding whether a given UCP map is pure.
As a first application of this characterization, we prove that every pure UCP map on \(\mathcal{T}_{{\tt{d}}}\) taking values in \(M_n\) has 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 on \(\mathcal{T}_{{\tt{d}}}\) taking values in \(M_{n}\), 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.
Pure UCP maps and pure matrix states have been studied from several related viewpoints. The foundational background goes back to Arveson’s seminal work on operator systems and boundary representations [1], [2]. From the perspective of matrix convexity, pure UCP maps coincide with matrix extreme points; see [3], [4]. Pure states determine the norm of every element in an operator system [5]. Pure UCP maps also play an important role in the study of \(C^{*}\)-extreme points and \(C^*\)-convexity, where they appear as the basic irreducible building blocks in decompositions of \(C^*\)-extreme maps; see, for example, [6]–[9].
Definition 1. A nonzero completely positive (CP) map \(\varphi\) is said to be pure if every CP map dominated by \(\varphi\) is a scalar multiple of \(\varphi\); that is, whenever \(\psi\leq_{\mathrm{cp}}\varphi\), one has \[\psi = t\varphi\] for some \(t\in[0,1]\). Here \(\psi\leq_{\mathrm{cp}}\varphi\) means that \(\varphi-\psi\) is CP. Equivalently, a CP map is pure if it spans an extreme ray in the cone of CP maps.
Pure CP maps have an intrinsic connection with Stinespring dilation theory. A CP map defined on a \(C^{*}\) algebra is pure if and only if its minimal Stinespring dilation is irreducible. For operator systems, however, this representation-theoretic criterion does not by itself give a complete description. The restriction of a pure CP map to an operator subsystem need not remain pure. Thus, unlike the \(C^*\)-algebraic case, the characterization of pure CP maps is genuinely sensitive to the particular operator system under consideration; see, for instance, [4], [10].
There is another point of view on pure UCP maps. They play the role of noncommutative pure states for an operator system. At matrix level \(n\), a matrix state on an operator system \(\mathcal{S}\) is a UCP map \(\varphi:\mathcal{S}\to M_n\), and the family \(\coprod_{n=1}^\infty\operatorname{UCP}(\mathcal{S},M_n)_{n\geq 1}\) forms a matrix convex set [11], [12]. Farenick [3] showed that pure UCP maps are precisely the matrix extreme points of this matricial state space. In other words, they cannot be written as nontrivial matrix convex combinations of other matrix states. Hence, pure UCP maps are not merely extreme points in an ordinary convex set; they encode genuinely noncommutative boundary data. Thus, it is natural to characterize them.
For arbitrary operator systems, even finite-dimensional ones, a concrete characterization of pure UCP maps is generally delicate. We study the finite Toeplitz operator systems, where positive matrix-valued trigonometric polynomials encode UCP maps; see 3 and Proposition 2. The finite Toeplitz setting therefore provides a tractable but nontrivial class in which pure matrix states can be described explicitly through polynomial densities and Fejér–Riesz factorization.
Finite Toeplitz operator systems arise classically from compressions of multiplication operators on \(L^2(\mathbb{T})\), where \(\mathbb{T}\) denotes the unit circle. They encode finite sections of Toeplitz symbols, which makes their connection with trigonometric polynomials natural. Recently, Connes and van Suijlekom [10] studied these systems as spectral truncations of the unit circle in the framework of noncommutative geometry [13]. Farenick [14] subsequently established a complete order isomorphism between finite Toeplitz systems and the dual of the space of scalar trigonometric polynomials. In a related metric direction, Hekkelman [15] proved that the pure state spaces of finite Toeplitz systems converge, in the Gromov–Hausdorff sense, to the state space of \(C(\mathbb{T})\), equivalently to the space of Borel probability measures on \(\mathbb{T}\).
Equation 3 associates to a CP map \(\varphi\) a matrix-valued trigonometric polynomial \(P_\varphi\), while Proposition 2 shows that these polynomials serve as densities for the measures defining the corresponding CP maps. We call \(P_\varphi\) the polynomial density of \(\varphi\). Thus, the characterization of pure UCP maps becomes the problem of identifying the indecomposable polynomial densities among positive matrix-valued trigonometric polynomials. We carry this out for finite Toeplitz systems in Theorem 1.
Theorem 1. Let \(\varphi: \mathcal{T}_{{\tt{d}}} \to M_{n}\) be a UCP map with polynomial density \(P_{\varphi}\). Then \(\varphi\) is pure if and only if the following conditions hold:
There exists a row polynomial \[Q(z) \,=\,\begin{pmatrix} q_1(z)&q_2(z)&\cdots&q_n(z) \end{pmatrix}, \qquad q_j\in \mathbb{C}[z],\qquad \max_{j}\deg q_j = {\tt{d}}-1,\] such that \(P_\varphi(z)=Q(z)^*Q(z)\) for all \(z\in\mathbb{T}\).
If \(g_\varphi=\gcd(q_1,\ldots,q_n)\) denotes the greatest common divisor of the polynomials appearing in condition \((1)\), then all zeros of \(g_\varphi\) lie on \(\mathbb{T}\).
Remark 1. Several remarks related to Theorem 1 are in order.
In Subsection 4.3, we explain how the characterization in Theorem 1 leads to a practical criterion for deciding whether a given UCP map is pure. The discussion in that subsection also shows that the polynomial density of a pure UCP map admits a unique Fejér–Riesz factorization, up to left multiplication by a constant unit column vector.
For UCP maps of the form \[T\mapsto V^*TV, \qquad {T \in \mathcal{T}_{{\tt{d}}}},\] where \(V\) is an isometry, purity can be decided directly from \(V\); see Subsection 4.4.
Pure states (\(n=1\) case) on \(\mathcal{T}_{{\tt{d}}}\) were characterized in [10], [15]. Our treatment is self-contained and approaches the result through polynomial densities and Fejér–Riesz factorization.
Theorem 1 has two applications. The first concerns the unique extension phenomenon for pure UCP maps which we discuss below.
Let \(\mathcal{S}\) be an operator system and \(\mathcal{H}\) be a Hilbert space. A CP map \(\varphi: \mathcal{S} \to \mathcal{B}(\mathcal{H})\) is said to have a unique CP extension if there exists a unique CP map \(\tilde{\varphi}: \mathcal{C^*}(\mathcal{S}) \to \mathcal{B}(\mathcal{H})\) such that \(\tilde{\varphi}|_{\mathcal{S}} = \varphi\). The characterization of pure UCP maps leads to a unique extension phenomenon for pure UCP maps on finite Toeplitz systems. We first obtain a general criterion for a CP map on \(\mathcal{T}_{{\tt{d}}}\) to have a unique CP extension to \(C^*(\mathcal{T}_{{\tt{d}}})=M_{{\tt{d}}}\), expressed in terms of the Fejér–Riesz factorizations of its polynomial density. Combining this criterion with the characterization of pure UCP maps gives us the following result.
Theorem 2. If \(\varphi:\mathcal{T}_{{\tt{d}}}\to M_n\) is a pure UCP map, then \(\varphi\) has a unique CP extension to \(M_{{\tt{d}}}\).
Remark 2. Several remarks related to Theorem 2 are in order.
More generally, if the polynomial density of a CP map admits a unique Fejér–Riesz factorization up to left multiplication by a constant isometry, then the CP map admits a unique CP extension; see Proposition 5.
As a consequence of Theorem 2, we obtain the following uniqueness result. Let \(\varphi\) be a pure UCP map and suppose that \[\varphi(T) \,=\, V^*TV \,=\, W^*TW,\qquad T\in\mathcal{T}_{{\tt{d}}},\] for two isometries \(V,W:\mathbb{C}^n\to\mathbb{C}^{{\tt{d}}}\). Then \(V=\lambda W\) for some \(\lambda\in\mathbb{T}\).
Theorem 2 reflects a special rigidity of finite Toeplitz systems. We show by example that purity, even in a finite-dimensional hyperrigid operator system, need not imply a unique CP extension; see example 5.3.
Unique extension phenomena for pure states on subspaces of \(C^*\)-algebras have been studied in [16]. A comparison between Theorem 2 and Arveson’s unique extension property [1], [2] is given in Remark 6.
Gromov–Hausdorff convergence provides a natural way to compare metric spaces without requiring them to be embedded in a common ambient space. Noncommutative analogues of this convergence were developed in Rieffel’s theory of compact quantum metric spaces [17], [18], with matricial versions introduced by Kerr [19] and further studied by Kerr–Li [20]. This theme has since appeared in several directions, including propinquity-type distances [21], [22], convergence phenomena for spectral truncations [10], [23], [24], and examples arising from other noncommutative spaces [25]–[28]. Our next result, Theorem 3, belongs to this circle of ideas.
For fixed \(n\), once a description of pure UCP maps is available, it is natural to ask whether the finite-dimensional boundary objects \[\operatorname{PureUCP}(\mathcal{T}_{{\tt{d}}},M_n)\] are sufficiently rich to approximate the matrix-state space \[\operatorname{UCP}(C(\mathbb{T}),M_n)\] as \({\tt{d}}\uparrow \infty.\) To make this question precise, we equip \(\operatorname{PureUCP}(\mathcal{T}_{{\tt{d}}},M_n)\) with the matricial Connes distance, denoted by \(\rho_{{\tt{d}},n}\), and \(\operatorname{UCP}(C(\mathbb{T}),M_n)\) with the matricial Monge–Kantorovich distance, denoted by \(\rho_n\), both induced by Lipschitz seminorms on the corresponding operator systems; see Subsection 2.4 for more details. The following theorem shows that the metric spaces of pure UCP maps on finite Toeplitz systems approximate the metric space of UCP maps on \(C (\mathbb{T})\) in the Gromov–Hausdorff sense.
Theorem 3. Fix \(n\geq 2\). Then \[d_{GH} \left ( \bigl( \operatorname{PureUCP}(\mathcal{T}_{{\tt{d}}},M_n),\rho_{{\tt{d}},n} \bigr),\, \bigl( \operatorname{UCP}(C(\mathbb{T}),M_n), \rho_{n} \bigr) \right) \longrightarrow 0 \qquad \text{as} \qquad {\tt{d}}\uparrow \infty.\]
Remark 3. Two remarks related to Theorem 3 are in order.
The statement of Theorem 3 also holds in the scalar case \(n=1\), where it was proved by Hekkelman [15]. We point out, however, that the scalar and matrix-valued cases require rather different arguments. Our proof uses features specific to the genuinely matrix-valued setting \(n\geq 2\) and should therefore be viewed as complementary to Hekkelman’s scalar approach.
The convergence of the full matrix-state spaces \(\operatorname{UCP}(\mathcal{T}_{\tt{d}},M_n)\) to \(\operatorname{UCP}(C(\mathbb{T}),M_n)\) was proved in [29], although with respect to a different metric.
The paper is organized as follows. In Section 2, we recall the necessary background on finite Toeplitz operator systems, completely positive maps, matrix-valued trigonometric polynomials, matricial Connes–Kantorovich metrics, and Gromov–Hausdorff distance. In Section 3, we establish the connection between UCP maps on \(\mathcal{T}_{{\tt{d}}}\) and matrix-valued trigonometric polynomials. In Section 4, we prove the characterization of pure UCP maps on \(\mathcal{T}_{{\tt{d}}}\) stated in Theorem 1. We also explain how this characterization leads to a practical criterion for deciding whether a given UCP map is pure, and we discuss the case of UCP maps of the form \(T\mapsto V^*TV\), where \(V:\mathbb{C}^n\to\mathbb{C}^{{\tt{d}}}\) is an isometry. In Section 5, we prove the unique CP extension theorem for pure UCP maps and give an example showing that this phenomenon is special to the Toeplitz setting. In Section 6, we prove Hausdorff convergence after embedding the approximating spaces into the limiting space. Finally, in Section 7, we prove the Gromov–Hausdorff convergence result stated in Theorem 3.
For \({\tt{d}}\geq 1\), let \(\mathcal{T}_{{\tt{d}}}\subseteq M_{{\tt{d}}}\) denote the operator system consisting of all \({\tt{d}}\times {\tt{d}}\) Toeplitz matrices: \[\mathcal{T}_{\tt{d}} \, = \, \left\{ [a_{i-j}]_{i,j=0}^{{\tt{d}}-1}:a_{-({\tt{d}}-1)},\ldots,a_{{\tt{d}}-1}\in \mathbb{C} \right\} \,\subseteq\, M_{\tt{d}}.\] The \(C^*\)-algebra generated by \(\mathcal{T}_{\tt{d}}\) is \(M_{{\tt{d}}}.\) Note that \(\mathcal{T}_d \,=\, {\rm span} \{I,J, \ldots,J^{{\tt{d}}-1},J^*,\ldots,(J^*)^{{\tt{d}}-1}\},\) where \[J \, =\, \begin{bmatrix} 0&1&0&\cdots&0\\ 0&0&1&\cdots&0\\ \vdots& &\ddots&\ddots&\vdots\\ 0&0&\cdots&0&1\\ 0&0&\cdots&0&0 \end{bmatrix} \,\in\, M_{\tt{d}}.\]
Let \(C(\mathbb{T})\) denote the space of all continuous functions on the unit circle. We regard \(C(\mathbb{T})\) as a subalgebra of \(\mathcal{B}(L^2(\mathbb{T}))\) by identifying \(f\in C(\mathbb{T})\) with the multiplication operator \(M_f\) on \(L^2(\mathbb{T})\). We shall simply write \(f\) for \(M_f\) whenever no confusion can arise.
For \(k\in\mathbb{Z}\), let \(e_k(z)=z^k,\) \(z \in \mathbb{T}.\) Then \(\{e_k:k\in\mathbb{Z}\}\) is an orthonormal basis for \(L^2(\mathbb{T})\). Let \(P_{{\tt{d}}}\) denote the orthogonal projection onto \(H_{{\tt{d}}} := \operatorname{span}\{e_1,e_1,\ldots,e_{{\tt{d}}}\}.\) For \(f\in C (\mathbb{T})\), the compression \(P_{{\tt{d}}}fP_{{\tt{d}}}\) acts on \(H_{{\tt{d}}}\). With respect to the basis \(e_1,e_2,\ldots,e_{{\tt{d}}}\), it is represented by the \({\tt{d}}\times {\tt{d}}\) Toeplitz matrix \[\begin{bmatrix} \widehat f(0) & \widehat f(-1) & \cdots & \widehat f(-{\tt{d}}+1) \\ \widehat f(1) & \widehat f(0) & \cdots & \widehat f(-{\tt{d}}+2) \\ \vdots & \vdots & \ddots & \vdots \\ \widehat f({\tt{d}}-1) & \widehat f({\tt{d}}-2) & \cdots & \widehat f(0) \end{bmatrix}.\] Equivalently, since \(J=P_{{\tt{d}}}M_{\overline{z}}P_{{\tt{d}}}\), we can also write \[P_{{\tt{d}}}fP_{{\tt{d}}} = \widehat f(0)I + \sum_{k=1}^{{\tt{d}}-1}\widehat f(-k)J^k + \sum_{k=1}^{{\tt{d}}-1}\widehat f(k)(J^*)^k .\] Thus the finite Toeplitz operator system \(\mathcal{T}_{{\tt{d}}}\subseteq M_{{\tt{d}}}\) may be realized as \(\mathcal{T}_{{\tt{d}}} = \{P_{{\tt{d}}}fP_{{\tt{d}}}:f\in C(\mathbb{T})\}.\) Indeed, every \({\tt{d}}\times{\tt{d}}\) Toeplitz matrix arises in this way by choosing a trigonometric polynomial \(f\) of degree at most \({\tt{d}}-1\) whose Fourier coefficients agree with the entries on the relevant diagonals. Such spectral truncations have been studied in detail in [10], [15], [24].
Let \(\mathcal{S}\) be an operator system. We write \(\operatorname{UCP}(\mathcal{S},M_n)\) for the set of all the UCP maps from \(\mathcal{S}\) to \(M_n\). For CP maps \(\psi,\varphi:\mathcal{S}\to M_n\), we write \(\psi\leq_{\mathrm{cp}}\varphi\) if \(\varphi-\psi\) is CP.
The following extension result of Arveson [1] (see also [3]) is one of the main technical inputs in our characterization of pure UCP maps. It allows us to pass from pure UCP maps on an operator system to pure UCP maps on the generated \(C^*\)-algebra, thereby providing a necessary condition for purity in the operator system setting.
Theorem 1 (Pure extention theorem). Let \(\mathcal{S} \subseteq \mathcal{A} = C^*(\mathcal{S})\) be an operator system. If \(\varphi:\mathcal{S} \to M_n\) is a pure UCP map, then \(\varphi\) admits a pure UCP extension \(\tilde{\varphi}:\mathcal{A}\to M_n\).
In our setting, the operator system under consideration is the finite Toeplitz system \(\mathcal{T}_{{\tt{d}}}\), and \(C^*(\mathcal{T}_{{\tt{d}}})=M_{{\tt{d}}}\). Pure UCP maps \(M_{{\tt{d}}}\to M_n\) have a particularly concrete form, which follows as an elementary consequence of Stinespring dilation theory.
Lemma 1. Let \(\varphi:M_{{\tt{d}}} \to M_n\) be a UCP map. Then \(\varphi\) is pure if and only if there exists an isometry \(V:\mathbb{C}^n\to \mathbb{C}^{{\tt{d}}}\) such that \[\varphi(T) \;=\;V^*TV,\qquad T\in M_{{\tt{d}}}.\] In particular, a pure UCP map \(M_{{\tt{d}}}\to M_n\) can exist only if \(n\leq {\tt{d}}\).
Proof. The \(C^{*}\)-algebra \(M_{\tt{d}}\) has, up to unitary equivalence, only one irreducible representation, namely the identity representation on \(\mathbb{C}^{\tt{d}}\). A pure CP map on a \(C^{*}\)-algebra has irreducible minimal Stinespring representation. Hence, if \(\varphi\) is pure, its minimal Stinespring representation is unitarily equivalent to the identity representation of \(M_{\tt{d}}\), and therefore \[\varphi(T) \,=\, V^* T V,\qquad (T \in M_{\tt{d}}),\] for some operator \(V:\mathbb{C}^n\to\mathbb{C}^{\tt{d}}.\) Since \(\varphi\) is unital, \(I_n \,=\, \varphi(I_{\tt{d}}) \,=\, V^*V.\) Thus \(V\) is an isometry.
Conversely, if \(\varphi(T) = V^* T V\) with \(V^*V = I_n\), then the Stinespring representation is the identity representation of \(M_{\tt{d}}\), which is irreducible and minimal. Hence \(\varphi\) is a pure map. ◻
The next result is another fundamental tool used throughout the paper. In our setting, CP maps on the finite Toeplitz system are often studied through their extensions to the full matrix algebra \(M_{{\tt{d}}}\). Choi’s theorem then converts complete positivity of such extensions into an explicit positivity condition of a finite block matrix.
Theorem 2 (Choi’s theorem [30]). Let \(\varphi:M_{\tt{d}}\to M_n\) be a linear map, and let \(\{E_{ij}\}_{i,j=1}^{\tt{d}}\) denote the standard matrix units in \(M_{\tt{d}}\). Then \(\varphi\) is CP if and only if its Choi matrix \[C_\varphi \,=\, \bigl[\varphi(E_{ij})\bigr]_{i,j=1}^{\tt{d}} \in M_{\tt{d}}(M_n)\cong M_{{\tt{d}}\,n}\] is positive semidefinite.
For background on operator systems, CP maps, \(C^*\)-algebras, and their representations, we refer the reader to [31], [32].
Let \(\mathbb{T}=\{z\in\mathbb{C}: |z|=1\}\). Let \(P:\mathbb{T}\to M_n\) be a matrix-valued trigonometric polynomial of the form \[\label{eq:trig32poly} P(z)\,=\, \sum_{k=-{\tt{d}}}^{{\tt{d}}} A_k z^k, \qquad A_k\in M_n.\tag{1}\] We say that \(P\) is positive if \(P(z)\succeq 0\) for all \(z\in\mathbb{T}\), where, for an operator \(T\) on a Hilbert space \(\mathcal{H}\), the notation \(T\succeq 0\) means that \(T\) is positive semidefinite. We say that \(P\) has degree at most \({\tt{d}}\) if it is of the form 1 . If \(P\) is not identically zero, its degree is the largest integer \(|k|\) for which \(A_k\neq 0\).
A positive matrix-valued trigonometric polynomial \(P\) is said to be normalized if its constant coefficient is \(I_n\). Equivalently, \[\int_{\mathbb{T}} P(z)\,dm(z)=I_n,\] where \(dm\) denotes the normalized arc length measure on \(\mathbb{T}\).
The following result is one of the central tools of the paper. The scalar-valued Fejér–Riesz factorization asserts that every non-negative scalar-valued trigonometric polynomial of degree at most \({\tt{d}}\) can be written as the modulus square of an analytic polynomial of degree at most \({\tt{d}}\). Its matrix-valued and operator-valued extensions, due in particular to Rosenblum [33], play a fundamental role in Toeplitz operator theory and factorization theory. For a modern account and further perspectives on the operator Fejér–Riesz factorization, see [34]. We shall use the matrix-valued version repeatedly to pass from positive matrix-valued trigonometric polynomials to analytic polynomial factorizations.
Theorem 3 (Matrix-valued Fejér–Riesz theorem). Let \(P\) be an \(n\times n\) matrix-valued positive trigonometric polynomial of degree atmost \({\tt{d}}\). Then \[P(z) \,=\, Q(z)^*Q(z),\] where \(Q\) is an analytic matrix polynomial of degree at most \({\tt{d}}\).
Set \[\mathcal{X}_{{\tt{d}},n} \,:=\, {\rm PureUCP} (\mathcal{T}_{{\tt{d}}},M_n), \qquad \mathcal{Y}_{{\tt{d}},n}\,:=\, {\rm UCP} (\mathcal{T}_{{\tt{d}}}, M_{n}), \qquad \mathcal{Y}_n \,:=\, \operatorname{UCP}(C(\mathbb{T}),M_n).\] The set \(\mathcal{Y}_{{\tt{d}},n}\) can be equipped with a metric induced by the matricial Connes distance formula. It is defined by \[\rho_{{\tt{d}},n}(\varphi,\psi) \,:=\, \sup\left\{ \|\varphi(T)-\psi(T)\| \;:\; T\in \mathcal{T}_{{\tt{d}}},\;\left\|[D_{{\tt{d}}},T] \right\|\leq 1 \right\},\] where \(D_{{\tt{d}}}:=\operatorname{diag}(1,\ldots,{\tt{d}})\) and \([D_{{\tt{d}}},T]=D_{{\tt{d}}}T-TD_{{\tt{d}}}\) denotes the commutator. We shall denote this seminorm by \(L_{{\tt{d}}}.\) We also define the matricial Monge–Kantorovich metric on \(\mathcal{Y}_{n}\) by \[\label{eq:M-Kmetric} \rho_{n}(\varphi,\psi) \,:=\, \sup\left\{ \|\varphi(f)-\psi(f)\| \;:\; f\in C^{1}(\mathbb{T}),\;\left\|[D,f] \right\|\leq 1 \right\},\tag{2}\] where \(D=-i\frac{d}{dt}\) is the Dirac operator on \(L^2(\mathbb{T})\), with domain consisting of continuously differentiable functions \(C^{1}(\mathbb{T})\). The Connes Lip-norm associated with \(D\) is defined as \[\|[D, f]\| \,:= \, \|[D,M_f]\| \,=\, D M_{f} - M_{f} D,\] whenever the commutator \([D,M_f]\) extends to a bounded operator on \(L^2(\mathbb{T})\). For \(f\in C^{1}(\mathbb{T})\), we have \([D,M_f] \,=\, M_{-i f'},\) and therefore \[\| [ D, f] \| \,=\, \|[D,M_f]\| \,=\, \|f'\|_\infty.\] We shall simply write \({\rm Lip}(f)\) for \(\|[D,f]\|\).
The standard Connes distance formula and the Monge–Kantorovich metric are originally formulated on state spaces. Since we work at the matrix level, we use their matricial analogues on spaces of UCP maps. Such matricial metrics were studied by Kerr in [19].
Both \(L_{{\tt{d}}}\) and \({\rm Lip}\) are Lip-norms in the sense of [18]. Indeed, the metric induced by \({\rm Lip}\) on the state space of \(C(\mathbb{T})\) induces the weak\(^*\) topology, and \(L_{{\tt{d}}}\) is the corresponding finite-dimensional truncation of the Lipschitz seminorm. Therefore, by [19], the associated matricial metrics induce the point-norm topology on the corresponding matrix state spaces. Thus the metric spaces \((\mathcal{Y}_{{\tt{d}},n},\rho_{{\tt{d}},n})\) and \((\mathcal{X}_n,\rho_n)\) fit into the framework of matricial quantum metric spaces.
Let \((X,d_X)\) be a metric space. For a subset \(S\) in a metric space \(X\), denote the \(r\)-neighborhood of \(S\) by \(U_r(S)\), i.e., \[U_r(S) = \bigcup_{x \in S} B_r(x)\] where \(B_r(x)\) is the open ball of radius \(r\) centered at \(x\).
Definition 2. The Hausdorff distance between two subsets \(A\) and \(B\) of a metric space \((X,d_X)\) is defined as \[d_H(A, B) \,=\, \inf \{\, r > 0 \;:\; A \subseteq U_r(B) \text{ and } B \subseteq U_r(A) \,\}.\]
Definition 3. Let \((X, d_X)\) and \((Y,d_Y)\) be two metric spaces. The Gromov–Hausdorff distance between \(X\) and \(Y,\) to be denoted as \(d_{GH} (X,Y),\) is defined as the infimum of all \(r > 0\) such that there exists a metric space \(Z\) with subsets \(X_1, Y_1 \subseteq Z\) isometric to \(X\) and \(Y\), respectively, with \(d_H(X_1, Y_1) < r\), where \(d_H(X_1, Y_1)\) is the Hausdorff distance between \(X_1\) and \(Y_1\).
A useful formula to calculate the Gromov–Hausdorff distance is given in [35]. To state that formula, we need the following definitions.
Definition 4. Let \(X\) and \(Y\) be two sets. A correspondence between \(X\) and \(Y\) is a set \(R \subseteq X \times Y\) such that for every \(x \in X\) there exists at least one \(y \in Y\) with \((x, y) \in R\) and similarly for every \(y \in Y\) there exists an \(x \in X\) with \((x, y) \in R\).
Definition 5. Let \(R\) be a correspondence between metric spaces \((X, d_X)\) and \((Y, d_Y)\). The distortion of \(R\) is defined by \[\operatorname{dis} R \,=\, \sup \left\{\, |d_X(x,x') - d_Y(y,y')| \;:\; (x,y), (x', y') \in R \,\right\}.\]
Theorem 1. For any two metric spaces \(X\) and \(Y\), \[d_{GH}(X,Y) \,=\, \frac{1}{2}\, \inf_R \, (\operatorname{dis} R),\] where the infimum is taken over all correspondences \(R\) between \(X\) and \(Y\).
Let \(\varphi:\mathcal{T}_{\tt{d}}\to M_n\) be a linear map. Define an \(n \times n\) matrix-valued trigonometric polynomial \[\label{eq:P95phi} P_\varphi(z) \,=\, \varphi(I) + \sum_{k=1}^{{\tt{d}}-1}\varphi(J^k)z^k + \sum_{k=1}^{{\tt{d}}-1}\varphi((J^*)^k)z^{-k}, \qquad (z\in\mathbb{T}).\tag{3}\] Let \(C(\mathbb{T})_{({\tt{d}}-1)}^{(n)}\) be the vector space of \(n \times n\) matrix-valued trigonometric polynomials of degree at most \({\tt{d}}-1\) and \(\mathcal{L}(\mathcal{T}_{{\tt{d}}}, M_{n})\) be the vector space of all linear maps \(\varphi:\mathcal{T}_{\tt{d}}\to M_n\).
The following result is closely related to the complete order isomorphism between \(\mathcal{T}_{{\tt{d}}}\) and the dual of \(C(\mathbb{T})_{{\tt{d}}-1}^{(1)}\) established in [14] (see also [10]). We include a self-contained proof in our notation, based on Arveson’s extension theorem, the matrix-valued Fejér–Riesz factorization and Choi’s theorem.
Lemma 2. The map \[\Phi_{n}: \mathcal{L}(\mathcal{T}_{{\tt{d}}}, M_{n}) \to C(\mathbb{T})_{({\tt{d}}-1)}^{(n)}; \qquad \varphi \mapsto P_{\varphi}\] is a linear bijective map such that \(\varphi \in \mathcal{L}(\mathcal{T}_{{\tt{d}}}, M_{n})\) is CP if and only if \(P_{\varphi} \in C(\mathbb{T})_{({\tt{d}}-1)}^{(n)}\) is positive on the unit circle. Moreover, \(\varphi\) is UCP if and only if \(P_{\varphi}\) is normalized.
Proof. The injectivity of \(\Phi_{n}\) is immediate from the definition. The surjectivity follows from the fact that \[{\rm dim}\, \mathcal{L}(\mathcal{T}_{{\tt{d}}}, M_{n}) \,=\, (2 {\tt{d}}-1) n^{2} \,=\, {\rm dim}\, C(\mathbb{T})_{({\tt{d}}-1)}^{(n)}.\]
For the remaining part of the assertion, let \(\varphi \in \mathcal{L}(\mathcal{T}_{{\tt{d}}}, M_{n})\) be a CP map. By the Arveson extension theorem, \(\varphi\) extends to a CP map \(\tilde{\varphi}:M_{\tt{d}}\to M_n.\) Since \(\tilde{\varphi}\) is CP, its Choi matrix \(C_{\tilde{\varphi}} = [\tilde{\varphi}(E_{ij})]_{i,j=0}^{{\tt{d}}-1} \in M_{\tt{d}}(M_n)\) is positive. Thus there exist matrices \(Q_0,Q_1,\ldots,Q_{{\tt{d}}-1}\) of suitable size such that \[\tilde{\varphi} (E_{ij}) \, =\, Q_i^*Q_j \qquad 0\leq i,j\leq {\tt{d}}-1.\] This is just a Gram factorization of the positive block matrix \([\tilde{\varphi}(E_{ij})]_{i,j=0}^{{\tt{d}}-1}.\) Define \[Q(z) \,=\, Q_0 +Q_1z+\cdots+Q_{{\tt{d}}-1}z^{{\tt{d}}-1}.\] Then \(Q(z)^*Q(z) = \sum_{i,j=0}^{{\tt{d}}-1}Q_i^*Q_jz^{j-i}.\) We now compute the Fourier coefficients of \(Q(z)^*Q(z).\) First, the constant coefficient is \[\sum_{i=0}^{{\tt{d}}-1}Q_i^*Q_i \,=\, \sum_{i=0}^{{\tt{d}}-1}\tilde{\varphi}(E_{ii}) \,=\, \tilde{\varphi}(I) \,=\, \varphi(I).\] Next, for \(1\leq k\leq {\tt{d}}-1\), the coefficient of \(z^k\) in \(Q(z)^*Q(z)\) is \(\sum_{i=0}^{{\tt{d}}-1-k} Q_i^*Q_{i+k}.\) Using \(J^k = \sum_{i=0}^{{\tt{d}}-1-k}E_{i,i+k},\) we get \[\sum_{i=0}^{{\tt{d}}-1-k}Q_i^*Q_{i+k} \,=\, \sum_{i=0}^{{\tt{d}}-1-k}\tilde{\varphi}(E_{i,i+k}) \,=\, \tilde{\varphi}(J^k) \,=\, \varphi(J^k).\] Similarly, the coefficient of \(z^{-k}\) is \[\sum_{i=0}^{{\tt{d}}-1-k}Q_{i+k}^*Q_i \,=\, (\tilde{\varphi}(J^k))^{*} \,=\, \varphi((J^*)^k).\] Therefore \[Q(z)^*Q(z) \,=\, \varphi(I) + \sum_{k=1}^{{\tt{d}}-1}\varphi(J^k)z^k + \sum_{k=1}^{{\tt{d}}-1}\varphi((J^*)^k)z^{-k} \,=\, P_\varphi(z).\] Thus \(P_\varphi(z) \, =\, Q(z)^*Q(z)\geq 0\) for all \(z\in\mathbb{T}\).
Conversely, assume that \(P_\varphi(z)\geq 0\) for all \(z\in\mathbb{T}\) for some \(\varphi \in \mathcal{L} (\mathcal{T}_{{\tt{d}}} , M_{n}).\) By the matrix-valued Fejér–Riesz factorization, there exist matrices \(R_0,R_1,\ldots,R_{{\tt{d}}-1}\) of a common suitable size such that \(P_\varphi(z)\,=\,R(z)^*R(z),\) where \(R(z) = R_0+R_1z+\cdots+R_{{\tt{d}}-1}z^{{\tt{d}}-1}.\) Thus \(P_\varphi(z) = \sum_{i,j=0}^{{\tt{d}}-1}R_i^*R_jz^{j-i}.\) Define a linear map \(\psi_{R}:M_{\tt{d}}\to M_n\) on matrix units by \(\psi_R(E_{ij}) \,=\, R_i^*R_j\) for \(0\leq i,j\leq {\tt{d}}-1.\) Its Choi matrix is \[C_{\psi_{R}} \,=\, [\psi_{R} (E_{ij})]_{i,j=0}^{{\tt{d}}-1} \,=\, [R_i^*R_j]_{i,j=0}^{{\tt{d}}-1}.\] This block matrix is positive. By Choi’s theorem, \(\psi_{R}\) is CP. It remains to show that \(\psi_{R}|_{\mathcal{T}_{\tt{d}}} = \varphi.\) The constant coefficient of \(P_\varphi(z)\) is \(\varphi(I).\) On the other hand, the constant coefficient of \(R(z)^*R(z)\) is \(\sum_{i=0}^{{\tt{d}}-1}R_i^*R_i.\) Therefore \[\varphi(I) \,=\, \sum_{i=0}^{{\tt{d}}-1} R_i^*R_i \,=\, \sum_{i=0}^{{\tt{d}}-1} \psi_R (E_{ii}) \,=\, \psi_R(I).\] For \(1\leq k\leq {\tt{d}}-1\), the coefficient of \(z^k\) in \(P_\varphi(z)\) is \(\varphi(J^k).\) The coefficient of \(z^k\) in \(R(z)^*R(z)\) is \(\sum_{i=0}^{{\tt{d}}-1-k} R_i^*R_{i+k}.\) Therefore \(\varphi(J^k) = \sum_{i=0}^{{\tt{d}}-1-k} R_i^*R_{i+k}.\) But \(J^k = \sum_{i=0}^{{\tt{d}}-1-k} E_{i,i+k}.\) Hence \[\varphi(J^k) \,=\, \sum_{i=0}^{{\tt{d}}-1-k}R_i^*R_{i+k} \,=\, \sum_{i=0}^{{\tt{d}}-1-k}\psi_R(E_{i,i+k}) \,=\, \psi_R(J^k).\] Similarly, comparing the coefficients of \(z^{-k}\), we get \(\psi_R((J^*)^k) \,=\, \varphi((J^*)^k).\) This completes the proof. ◻
The following proposition gives an integral representation for CP maps on \(\mathcal{T}_{{\tt{d}}}\). It shows that every CP map \(\varphi:\mathcal{T}_{{\tt{d}}}\to M_n\) is represented by the positive matrix-valued trigonometric polynomial \(P_{\varphi}\) defined in 3 . We call this polynomial the polynomial density of \(\varphi\).
Proposition 2. Let \(\varphi:\mathcal{T}_{{\tt{d}}}\to M_n\) be a CP map. Then \[\varphi(P_{{\tt{d}}} f P_{{\tt{d}}}) \,=\, \int_{\mathbb{T}} f(z) P_{\varphi} (z)\, dm(z)\] for every \(f\in C(\mathbb{T}),\) where \(P_{\varphi}\) is defined in 3 .
Proof. By Arveson’s extension theorem, \(\varphi\) extends to a CP map \(\tilde{\varphi}:M_{{\tt{d}}}\to M_n\). Hence, by the finite-dimensional Stinespring–Kraus representation (Choi’s theorem), there exist operators \(V_1,\ldots,V_r:\mathbb{C}^n\to\mathbb{C}^{{\tt{d}}}\) such that \[\tilde{\varphi} (T) \,=\, \sum_{\ell=1}^r V_\ell^* T V_\ell, \qquad T\in M_{{\tt{d}}}.\]
The \((i,j)\)-entry of \(P_{{\tt{d}}} f P_{{\tt{d}}}\), for \(0\leq i,j\leq {\tt{d}}-1\), is \(\widehat f(i-j) = \int_{\mathbb{T}} f(z)z^{j-i}\,dm(z).\) Now define \[\eta(z) \,=\, \begin{pmatrix} 1 & \overline{z} & \overline{z}^2 \cdots \overline{z}^{{\tt{d}}-1} \end{pmatrix}^{t} \in \mathbb{C}^{{\tt{d}}}.\] Then the \((i,j)\)-entry of \(\eta(z)\eta(z)^*\) is \(z^{j-i}\). Therefore the \((i,j)\)-entry of \(\int_{\mathbb{T}} f(z)\eta(z)\eta(z)^*\,dm(z)\) is also \(\int_{\mathbb{T}}f(z)z^{j-i}\,dm(z)\). Hence \(P_{{\tt{d}}} f P_{{\tt{d}}} \,=\, \int_{\mathbb{T}} f(z)\eta(z)\eta(z)^*\,dm(z).\) Therefore \[\begin{align} \varphi(P_{{\tt{d}}} f P_{{\tt{d}}}) & \,=\, \tilde{\varphi}(P_{{\tt{d}}} f P_{{\tt{d}}}) \\ & \,=\, \sum_{\ell=1}^r V_\ell^* \left(\int_{\mathbb{T}} f(z)\eta(z)\eta(z)^*\,dm(z)\right) V_\ell \\ &= \int_{\mathbb{T}} f(z) \left(\sum_{\ell=1}^r V_\ell^*\eta(z)\eta(z)^*V_\ell\right) dm(z). \end{align}\] Thus the required density is \[\widetilde{P}_{\varphi}(z) \,:=\, \sum_{\ell=1}^r V_\ell^*\eta(z)\eta(z)^*V_\ell .\] This is a positive \(M_n\)-valued trigonometric polynomial of degree at most \({\tt{d}}-1\). We need to show that \(\widetilde{P}_{\varphi} = P_{\varphi}.\) Write \(\widetilde{P}_\varphi(z) = \sum_{k=-({\tt{d}}-1)}^{{\tt{d}}-1} A_k z^k .\) Since \(J = P_{{\tt{d}}} \overline{z} P_{{\tt{d}}}\), we have \(J^k \,=\, P_{{\tt{d}}} \overline{z}^k P_{{\tt{d}}}\) for \(|k| \leq {\tt{d}}-1.\) Hence \[\varphi(J^k) \,=\, \int_{\mathbb{T}} z^{-k} \widetilde{P}_\varphi(z)\,dm(z) \qquad (|k| \leq {\tt{d}}-1).\] Therefore \[\varphi(J^k) \,=\, \sum_{\ell =-({\tt{d}}-1)}^{{\tt{d}}-1}A_\ell \int_{\mathbb{T}} z^{\ell-k}\,dm(z) \,=\, A_k.\] Thus \(\varphi(J^k)\) is the Fourier coefficient of \(\widetilde{P}_\varphi\) corresponding to \(z^k\). Therefore \(\widetilde{P}_\varphi = P_{\varphi}.\) ◻
Let \(C(\mathbb{T})_{({\tt{d}}-1)}^{+}\) denote the cone of scalar-valued positive trigonometric polynomials of degree at most \({\tt{d}}-1\). We begin with a lemma which characterizes those elements of this cone that span extreme rays. Under the polynomial density correspondence, Proposition 2, this is equivalent to characterizing pure states on \(\mathcal{T}_{{\tt{d}}}\). This result should be compared with [10] and [15], where pure states on \(\mathcal{T}_{{\tt{d}}}\) are characterized. The proofs in these works implicitly contain the degree condition (see Lemma 3), although it is not stated explicitly in the assertions. We give a different proof below, making the degree condition explicit. Our proof uses only the scalar Fejér–Riesz factorization.
Lemma 3. Let \(g\in\mathbb{C}[z]\) be a nonzero polynomial of degree at most \({\tt{d}}-1\). Then the positive trigonometric polynomial \(|g|^2\) spans an extreme ray of \(C(\mathbb{T})_{({\tt{d}}-1)}^{+}\)if and only if \(\deg g = {\tt{d}}-1\) and every zero of \(g\) lies on the unit circle.
Proof. First suppose \(\deg g< {\tt{d}}-1.\) Then the polynomial \(\tilde{g}(z) \;=\;zg(z)\) has degree at most \({\tt{d}}-1\), and \(|\tilde{g}(z)| \,=\, |g(z)|\) on \(\mathbb{T}.\) Hence \[|g|^2 \,=\, \left|\frac{g+\tilde{g}}{2}\right|^2+\left|\frac{g-\tilde{g}}{2}\right|^2.\] The two summands are positive trigonometric polynomials of degree at most \({\tt{d}}-1\). They are not scalar multiples of \(|g|^2\), because \(g+\tilde{g}\) and \(g-\tilde{g}\) are not scalar multiples of \(g\). Thus \(|g|^2\) does not span an extreme ray.
Next suppose \(g\) has a zero \(\alpha\notin\mathbb{T}\). Write \(g(z)=(z-\alpha)h(z).\) Define \(\tilde{g}(z) = (1-\overline{\alpha}z)h(z).\) For \(z \in \mathbb{T}\), one has \(|z-\alpha|=|1-\overline{\alpha}z|.\) Therefore \(|\tilde{g}(z)| = |g(z)|,\) for all \(z \in \mathbb{T}.\) Again, \[|g|^2 \,=\, \left|\frac{g+\tilde{g}}{2}\right|^2+\left|\frac{g-\tilde{g}}{2}\right|^2.\] is a nontrivial decomposition inside \(C(\mathbb{T})_{({\tt{d}}-1)}^{+}\). Hence \(|g|^2\) is not extreme.
Conversely, assume \(\deg g= {\tt{d}}-1\) and every zero of \(g\) lies on \(\mathbb{T}\). Let \(0 \leq p \leq |g|^2\) with \(p\in C(\mathbb{T})_{({\tt{d}}-1)}^{+}\). By the Fejér–Riesz factorization, there is a polynomial \(h \in \mathbb{C}[z]\) of degree at most \({\tt{d}}-1\) such that \(p (z)=|h(z)|^2.\) The inequality \(|h(z)|^2 \leq |g(z)|^2\) implies \(|h(z)|\leq |g(z)|\) for all \(z \in \mathbb{T}\). If \(\lambda\in\mathbb{T}\) is a zero of \(g\) of multiplicity \(m\), then for \(z\neq \lambda\) we get \[\frac{|h(z)|}{|z-\lambda|^m} \,\leq\, \frac{|g(z)|}{|z-\lambda|^m}\] Since the right-hand side has a finite limit as \(z \in \mathbb{T}\) tends to \(\lambda,\) the left-hand side remains bounded in a neighbourhood of \(\lambda.\) Hence \(h\) must vanish at \(\lambda\) with multiplicity at least \(m\). Since all zeros of \(g\) lie on \(\mathbb{T}\), and \(\deg h\leq {\tt{d}}-1 = \deg g,\) it follows that \(h=cg\) for some scalar \(c\in\mathbb{C}\). Hence \(p=|c|^2|g|^2.\) Therefore the only positive trigonometric polynomials dominated by \(|g|^2\) are scalar multiples of \(|g|^2\), and so \(|g|^2\) spans an extreme ray. ◻
In the proposition below, we characterize pure UCP maps \(\tilde{\varphi}:M_{{\tt{d}}}\to M_n\) by looking at the polynomial density of their restriction to \(\mathcal{T}_{{\tt{d}}}\). Thus the purity of a UCP map on the full matrix algebra is translated into a condition on the polynomial density associated with \(\tilde{\varphi}|_{\mathcal{T}_{{\tt{d}}}}\).
Proposition 3. Let \(\varphi:\mathcal{T}_{{\tt{d}}}\to M_n\) be a UCP map and let \(P_\varphi\) be its associated polynomial density. Then the following are equivalent.
There exists an isometry \(V:\mathbb{C}^n\to \mathbb{C}^{{\tt{d}}}\) such that \[\varphi(T) \,=\, V^*TV, \qquad T\in \mathcal{T}_{{\tt{d}}}.\]
There exists a row polynomial \[\label{eq:Qdef} Q(z) \,=\,\begin{pmatrix} q_1(z)&q_2(z)&\cdots&q_n(z) \end{pmatrix}, \qquad q_j\in \mathbb{C}[z],\quad \deg q_j\leq {\tt{d}}-1,\qquad{(1)}\] such that \(P_\varphi(z)=Q(z)^*Q(z)\) for all \(z\in\mathbb{T}\).
Proof. Let \(\eta(z)= \begin{pmatrix} 1 & \overline{z} & \cdots & \overline{z}^{{\tt{d}}-1} \end{pmatrix}^{t} \in \mathbb{C}^{{\tt{d}}}.\) If \(\varphi(T) = V^*TV\), then, using \[P_{{\tt{d}}} f P_{{\tt{d}}} \,=\, \int_{\mathbb{T}} f(z)\eta(z)\eta(z)^*\,dm(z),\] we obtain \[\varphi(P_{{\tt{d}}} f P_{{\tt{d}}}) \,=\, V^*\left( \int_{\mathbb{T}} f(z)\eta(z)\eta(z)^*\,dm(z) \right) V \,=\, \int_{\mathbb{T}} f(z)V^*\eta(z)\eta(z)^*V\,dm(z).\] Hence \(P_\varphi(z) = V^*\eta(z)\eta(z)^*V.\) If we put \(Q(z) = \eta(z)^*V,\) then \(Q\) is a row polynomial of degree at most \({\tt{d}}-1\), and \(P_\varphi(z)=Q(z)^*Q(z).\)
Conversely, suppose \[P_\varphi(z) \,=\, Q(z)^*Q(z), \qquad Q(z) \,=\, Q_0+Q_1z+\cdots+Q_{{\tt{d}}-1}z^{{\tt{d}}-1}, \qquad Q_{j} \,\in\, M_{1,n}.\] Define \[V \,=\, \begin{pmatrix} Q_0\\ Q_1\\ \vdots\\ Q_{{\tt{d}}-1} \end{pmatrix} : \mathbb{C}^n\to \mathbb{C}^{{\tt{d}}}.\] Then \[V^*V \,=\, \sum_{j=0}^{{\tt{d}}-1}Q_j^*Q_j \,=\, \int_{\mathbb{T}}Q(z)^*Q(z)\,dm(z) \,=\, \int_{\mathbb{T}} P_{\varphi} (z) \,dm(z) \, =\, I_n,\] so \(V\) is an isometry. Also \(Q(z)=\eta(z)^*V\). \(P_\varphi(z) = V^*\eta(z)\eta(z)^*V.\) Therefore, for every \(f\in C(\mathbb{T})\), \[\varphi(P_{{\tt{d}}} f P_{{\tt{d}}}) \,=\, \int_{\mathbb{T}}f(z)P_\varphi(z)\,dm(z) \,=\, V^* \left( \int_{\mathbb{T}}f(z)\eta(z)\eta(z)^*\,dm(z) \right) V \,=\, V^*P_{{\tt{d}}} f P_{{\tt{d}}}V.\] Thus \(\varphi(T) = V^*TV\) for every \(T\in\mathcal{T}_{{\tt{d}}}\). ◻
Lemma 3 and Proposition 3 suggest a tempting but false criterion for purity of UCP maps from \(\mathcal{T}_{{\tt{d}}}\) to \(M_n\). One might expect that a UCP map \(\varphi:\mathcal{T}_{{\tt{d}}}\to M_n\) is pure whenever its polynomial density admits a factorization \(P_\varphi=Q^*Q\), with \(Q\) as in ?? , such that \(\max_j\deg q_j={\tt{d}}-1\) and each scalar polynomial \(q_j\) has all its zeros on \(\mathbb{T}\). The following example shows that this criterion is not necessary in the matrix-valued setting.
Example 1. Let \(Q(z)=(1,z)\) and \(P_\varphi(z) = Q(z)^*Q(z) = \begin{pmatrix} 1 & z\\ z^{-1} & 1 \end{pmatrix}.\) Let \(\varphi:\mathcal{T}_2\to M_2\) be the UCP map with polynomial density \(P_\varphi\). Then \(\varphi\) is pure.
Proof. Let \(\psi:\mathcal{T}_2\to M_2\) be CP with \(\psi\leq_{\mathrm{cp}}\varphi\). Let \(P_\psi\) be the polynomial density of \(\psi\). Since \(P_\varphi = Q^*Q\) with \(Q(z)=(1,z)\), we have \[0 \,\preceq\, P_\psi(z) \,\preceq\, Q(z)^*Q(z), \qquad z\in\mathbb{T}.\] Set \(u(z)= \begin{pmatrix} -z\\ 1 \end{pmatrix}.\) Then \(Q(z)u(z)=0\). Hence \[0 \,\leq\, \left\langle P_\psi(z)u(z),u(z) \right\rangle \,\leq\, \left\langle Q(z)^*Q(z)u(z),u(z) \right\rangle \,=\, \|Q(z)u(z)\|^2 \,=\, 0.\] Since \(P_\psi(z)\succeq 0\), it follows that \(P_\psi(z)u(z)=0\) for every \(z\in\mathbb{T}\). Write \[P_\psi(z)= \begin{pmatrix} a(z) & b(z)\\ b(z)^* & c(z) \end{pmatrix}.\] The equation \(P_\psi(z)u(z) = 0\) gives \(b(z)=za(z).\) Since \(P_\psi\) has degree at most \(1\), write \[a(z) \,=\, \alpha+\beta z+\overline{\beta}z^{-1}.\] Then \[b(z) \,=\, za(z) \,=\, \alpha z + \beta z^2+\overline{\beta}.\] But \(b\) also has degree at most \(1\), so \(\beta=0\). Hence \(a(z)=\alpha\) is constant. Consequently, \[b(z)=\alpha z, \qquad c(z)=\alpha,\] and therefore \[P_\psi(z) \,=\, \alpha \begin{pmatrix} 1 & z\\ z^{-1} & 1 \end{pmatrix} \,=\, \alpha\, Q(z)^*Q(z).\] Since \(0\preceq P_\psi\preceq P_\phi\), we have \(\alpha\in[0,1]\). Thus \(P_\psi=\alpha P_\varphi\), and hence \(\psi=\alpha\varphi\). Therefore \(\varphi\) is pure. ◻
The following proposition is the main ingredient in the proof of Theorem 1.
Proposition 4. Let \(\varphi:\mathcal{T}_{{\tt{d}}}\to M_n\) be a UCP map, and suppose that its polynomial density has the form \(P_\varphi(z)=Q(z)^*Q(z),\) where \(Q\) is as in \(\eqref{eq:Qdef}.\) Write \[Q(z)\,=\,g_\varphi(z)R(z), \qquad R(z) \,=\, \begin{pmatrix} r_1(z)&r_2(z)&\cdots&r_n(z)\end{pmatrix},\] where \(g_\varphi=\gcd(q_1,\ldots,q_n)\) is chosen so that \(\gcd(r_1,\ldots,r_n)=1\). Let \({\tt{d}}_r=\max_{1\leq j\leq n}\deg r_j.\) Then a CP map \(\psi:\mathcal{T}_{{\tt{d}}}\to M_n\) satisfies \(\psi\leq_{\mathrm{cp}}\varphi\) if and only if there exists a scalar polynomial \(h\in\mathbb{C}[z]\) with \(\deg h\leq {\tt{d}}-1-{\tt{d}}_r\) such that \[P_\psi(z) \,=\, |h(z)|^2 R(z)^*R(z), \qquad z\in\mathbb{T},\] and \[0 \,\leq\, |h(z)|^2 \,\leq\, |g_\varphi(z)|^2, \qquad z\in\mathbb{T}.\]
Proof. The reverse implication is straightforward. It remains to prove the forward implication. Since \(Q(z) = g_\varphi(z)R(z)\), we have \[P_{\varphi} (z) \,=\, Q(z)^*Q(z) \,=\, |g_\varphi(z)|^2 R(z)^*R(z).\] Since \(\psi\) is CP, \(P_{\psi}\) is positive. By the matrix-valued Fejér–Riesz factorization \(P_{\psi}(z) = F(z)^{*}F(z),\) where \(F\) is an \(m \times n\) matrix-valued polynomial of degree at most \({\tt{d}}-1\). Since \(0 \leq_{\mathrm{cp}} \psi \leq_{\mathrm{cp}} \varphi,\) we have \(0 \preceq P_{\psi} \preceq P_{\varphi}\) and thus \[0 \,\preceq\, F(z)^*F(z) \,\preceq \, Q(z)^*Q(z) \qquad (z \in \mathbb{T}).\] This implies that for every \(z\in\mathbb{T}\), the range of \(F(z)^*F(z)\) is contained in the range of \(Q(z)^*Q(z)\). But \[{\rm Range} (Q(z)^{*}Q(z)) \,=\, {\rm Range} (Q(z)^{*}) ,\] and \[{\rm Range} (F(z)^{*}F(z)) \,=\, {\rm Range} (F(z)^{*}) .\] Since \(Q(z)^*Q(z)\) has rank at most one, every row of \(F(z)\) must be pointwise proportional to \(Q(z)\). Write a row of \(F\) as \(f(z)= \begin{pmatrix} f_1(z)&\cdots&f_n(z) \end{pmatrix}.\) The fact that \(f(z)\) is pointwise proportional to \(Q(z)=g_\varphi(z)R(z)\) is equivalent to \[f_i(z)r_j(z) \,=\, f_j(z)r_i(z) \qquad \text{for all }i,j.\] Since \(\gcd(r_1,\ldots,r_n)=1\), there exist polynomials \(a_1,\ldots,a_n\in \mathbb{C}[z]\) such that \[\sum_{j=1}^n a_j(z)r_j(z) \,=\, 1.\] Using the identities \[f_i(z)r_j(z)=f_j(z)r_i(z), \qquad 1\leq i,j\leq n,\] we obtain \[f_i(z) = f_i(z)\sum_{j=1}^n a_j(z)r_j(z) = \sum_{j=1}^n a_j(z)f_i(z)r_j(z) = \sum_{j=1}^n a_j(z)f_j(z)r_i(z).\] If we set \(\alpha(z) = \sum_{j=1}^n a_j(z)f_j(z),\) then \(f_i(z) \,=\, \alpha(z) r_i(z)\) for \(1\leq i\leq n.\) Thus \(f(z) \,=\, \alpha(z)R(z).\) Applying this argument to each row of \(F\), we obtain \[F(z) \,=\, A(z)R(z),\] where \(A\) is a column vector whose entries are scalar polynomials. Consequently, \[F(z)^*F(z) \,=\, R(z)^*A(z)^*A(z)R(z).\] Writing \[A(z) \,=\, \begin{pmatrix} \alpha_1(z)\\ \vdots\\ \alpha_m(z) \end{pmatrix},\] we have \(A(z)^*A(z) = \sum_{\ell=1}^m |\alpha_\ell(z)|^2.\) By the scalar-valued Fejér–Riesz factorization, there exists a scalar polynomial \(h \in \mathbb{C}[z]\) such that \[\sum_{\ell=1}^m |\alpha_\ell(z)|^2 \,=\, |h(z)|^2, \qquad z\in\mathbb{T}.\] Therefore \[P_{\psi} \,=\, F(z)^*F(z) \,=\, |h(z)|^2 R(z)^*R(z).\] Since \(Q(z)^*Q(z) = |g_\varphi(z)|^2 R(z)^*R(z),\) the inequality \(F(z)^*F(z) \preceq Q(z)^*Q(z)\) is equivalent to \[0 \,\leq\, |h(z)|^2 \,\leq\, |g_\varphi(z)|^2, \qquad z\in\mathbb{T}.\] Here we use that \(R(z)\neq 0\) for every \(z\in\mathbb{T}\), which follows from \(\gcd(r_1,\ldots,r_n)=1\).
Finally, since \(\deg R={\tt{d}}_r\) and the entries of \(F\) have degree at most \({\tt{d}}-1\), each polynomial \(\alpha_\ell\) has degree at most \({\tt{d}}-1-{\tt{d}}_r\). Hence the scalar Fejér–Riesz factor \(h\) may also be chosen with \[\deg h \,\leq\, {\tt{d}}-1-{\tt{d}}_r.\] ◻
Proof. Let \(\varphi: \mathcal{T}_{{\tt{d}}} \to M_{n}\) be a pure UCP map with polynomial density \(P_{\varphi}.\) By pure extension theorem, there exists a pure UCP map \(\tilde{\varphi} : M_{{\tt{d}}} \to M_{n}\) such that \(\tilde{\varphi}|_{\mathcal{T}_{{\tt{d}}}} = \varphi.\) By Lemma 1, there exists an isometry \(V: \mathbb{C}^{n} \to \mathbb{C}^{{\tt{d}}}\) such that \[\varphi (T) \,=\, V^{*}TV, \qquad T \in \mathcal{T}_{{\tt{d}}}.\] This forces \(1 \leq n \leq {\tt{d}}.\) By Proposition 3, there exists a row polynomial \[Q(z) \,=\,\begin{pmatrix} q_1(z)&q_2(z)&\cdots&q_n(z) \end{pmatrix}, \qquad q_j\in \mathbb{C}[z],\quad \deg q_j \leq {\tt{d}}-1,\] such that \(P_{\varphi} = Q(z)^{*}Q(z)\) for all \(z \in \mathbb{T}.\) Write \[Q(z)\,=\,g_\varphi(z)R(z), \qquad R(z) \,=\, \begin{pmatrix} r_1(z)&r_2(z)&\cdots&r_n(z)\end{pmatrix},\] where \(g_\varphi=\gcd(q_1,\ldots,q_n)\) is chosen so that \(\gcd(r_1,\ldots,r_n)=1\). Let \({\tt{d}}_{r} = \max_{j} \deg r_{j}.\) Then \(\deg g_{\varphi} \leq {\tt{d}}-1- {\tt{d}}_{r}.\) We claim that \(|g_{\varphi}|^{2}\) spans an extreme ray of \(C(\mathbb{T})^{+}_{({\tt{d}}-1 - {\tt{d}}_{r})}.\) Let \(h\in \mathbb{C}[z]\) be polynomial of degree at most \({\tt{d}}-1-{\tt{d}}_{r}\) such that \(|h(z)| \leq |g_{\varphi} (z)|\) for all \(z \in \mathbb{T}.\) By Proposition 4, the CP map \(\psi : \mathcal{T}_{{\tt{d}}} \to M_{n}\) with polynomial density \[P_{\psi}(z) \,=\, |h(z)|^{2} R(z)^{*}R(z), \qquad z \in \mathbb{T},\] satisfies \(\psi \leq_{\rm cp} \varphi.\) Since \(\varphi\) is pure, \(\psi = t\, \varphi\) for some \(t \in [0,1].\) Thus \(|h(z)|^{2} = t\, |g_{\varphi} (z)|^{2}\) for all \(z \in \mathbb{T}.\) This proves our claim. Lemma 3 now implies that \(\deg g_{\varphi}={\tt{d}}-1-{\tt{d}}_r,\) and every zero of \(g_{\varphi}\) lies on \(\mathbb{T}\). Since \(\deg g_{\varphi}={\tt{d}}-1-{\tt{d}}_r,\) we get \(\max_{j} \deg q_{j} = {\tt{d}}-1.\)
Conversely, assume that \(\varphi: \mathcal{T}_{{\tt{d}}} \to M_{n}\) is a UCP map with polynomial density \(P_{\varphi}\) which satisfies conditions (1) and (2). Let \(\psi : \mathcal{T}_{{\tt{d}}} \to M_{n}\) be a CP map such that \(\psi \leq_{\rm cp} \varphi.\) By Proposition 4, there exists a scalar polynomial \(h \in \mathbb{C}[z]\) with \(\deg h \leq {\tt{d}}-1-{\tt{d}}_{r}\) such that \[P_{\psi}(z) \,=\, |h(z)|^{2} R(z)^{*} R(z), \qquad z \in \mathbb{T},\] and \(|h(z)| \leq |g_{\varphi}(z)|\) for all \(z \in \mathbb{T}.\) Since \(\deg g_\varphi = {\tt{d}}-1-{\tt{d}}_{r}\) and all zeros of \(g_{\varphi}\) lie on the circle, by Lemma 3, \(|g_{\varphi}|^{2}\) spans an extreme ray of \(C(\mathbb{T})^{+}_{{\tt{d}}-1-{\tt{d}}_{r}}.\) Therefore, there exists \(t \in [0,1]\) such that \(|h(z)|^{2} = t\, |g_{\varphi}(z)|^{2}\) for all \(z \in \mathbb{T}.\) This shows that \(\psi = t\, \varphi.\) Hence \(\varphi\) is pure. ◻
In the following lemma, we show that the row polynomial \(Q\) appearing in Theorem 1 is unique up to multiplication by a unimodular constant.
Lemma 4. Let \(\varphi: \mathcal{T}_{{\tt{d}}} \to M_{n}\) be a UCP map. Let \(Q(z)=\begin{pmatrix}q_1(z)&\cdots&q_n(z)\end{pmatrix}\) be a row polynomial satisfying conditions (1) and (2) of Theorem 1. If \(R(z)=\begin{pmatrix}r_1(z)&\cdots&r_n(z)\end{pmatrix}\) is a row polynomial of degree at most \({\tt{d}}-1\) such that \(P_{\varphi}(z) = R(z)^*R(z)\) for all \(z\in\mathbb{T}\), then there exists \(\lambda\in\mathbb{C}\) with \(|\lambda|=1\) such that \[R(z) \,=\, \lambda Q(z), \qquad z \in \mathbb{T}.\] In particular, if \(R\) is a row polynomial with \(P_{\varphi} = R^{*}R,\) then \(R\) satisfies conditions (1) and (2) of Theorem 1.
Proof. Write \[Q(z) \,=\, g(z) Q_0(z), \qquad Q_0(z) \,=\, \begin{pmatrix}q_1^0(z)&\cdots&q_n^0(z)\end{pmatrix},\] where \(\gcd(q_1^0,\ldots,q_n^0)=1\). Let \({\tt{d}}_{0}=\max_j \deg q_j^0.\) Since \(Q\) satisfies the full-degree condition, we have \(\deg g+{\tt{d}}_0={\tt{d}}-1.\) Since \(P_{\varphi}=R(z)^*R(z)\), the proof of Proposition 4 shows that there exists an analytic polynomial \(h\) of degree at most \({\tt{d}}-1-{\tt{d}}_0\) such that \[R(z) \,=\, h(z)Q_0(z),\qquad z\in\mathbb{T}.\] Now \[P_{\varphi}(z) \,=\, Q(z)^*Q(z) \,=\, R(z)^*R(z)\] implies \[|g(z)|^2 Q_0(z)^*Q_0(z) \,=\, |h(z)|^2 Q_0(z)^*Q_0(z),\qquad z\in\mathbb{T}.\] Since \(\gcd(q_1^0,\ldots,q_n^0)=1\), the row \(Q_0(z)\) is not identically zero at any point of \(\mathbb{T}\). Hence \[|h(z)| \,=\, |g(z)|,\qquad z\in\mathbb{T}.\] Since all zeros of \(g\) lie on \(\mathbb{T}\), by an argument similar to the proof of Lemma 3, it follows that \(h=\lambda g\) for some unimodular constant \(\lambda\). This completes the proof. ◻
Theorem 1 gives a characterization of pure UCP maps. In particular, it immediately provides abundant examples of such maps. However, so far it is less clear whether this characterization gives a checkable criterion: given a UCP map \(\varphi:\mathcal{T}_{{\tt{d}}}\to M_n\), can one decide whether \(\varphi\) is pure? At first glance, this seems nontrivial. Indeed, to show that \(\varphi\) is not pure, one would have to rule out every factorization of \(P_\varphi\) satisfying conditions (1) and (2).
We shall show that this difficulty is only apparent: any Fejér–Riesz factorization of \(P_\varphi\) contains enough information to decide whether \(\varphi\) is pure.
Let \[P_{\varphi}(z) = F(z)^{*} F(z), \qquad z \in \mathbb{T},\] be any Fejér–Riesz factorization of \(P_{\varphi}.\) So \(F\) is an \(m \times n\) matrix-valued analytic polynomial. Write \[F(z)= \begin{pmatrix} f_1(z)\\ \vdots\\ f_m(z) \end{pmatrix},\] where each \(f_\ell\) is a \(1\times n\) row polynomial. If \(\varphi\) is pure, then it follows from the proof of Proposition 4 that all rows of \(F\) are scalar multiples of a single nonzero row polynomial. Thus we obtain a first checkable obstruction: if the rows of \(F\) are not all scalar multiples of one nonzero row polynomial, then \(\varphi\) is not pure.
Now choose a nonzero row, say \(R(z)=f_p(z).\) Then for each \(\ell\) there is a scalar \(c_\ell\in\mathbb{C}\) such that \(f_\ell(z)=c_\ell R(z).\) Thus \[F(z) \,=\, cR(z), \qquad c= \begin{pmatrix} c_1\\ \vdots\\ c_m \end{pmatrix}.\] Let \[\alpha=\|c\|, \qquad h=\frac{c}{\alpha}, \qquad Q(z)=\alpha R(z).\] Then \(\|h\|=1\) and \(F(z)=hQ(z).\) Consequently \[P_\varphi(z)=F(z)^*F(z)=Q(z)^*Q(z).\] Now \(Q\) is a row polynomial with \(P_{\varphi}=Q^*Q.\) If \(\varphi\) is pure, then Lemma 4 shows that \(Q\) must satisfy conditions (1) and (2) of Theorem 1. Thus the purity of \(\varphi\) can be checked directly from \(Q\): the map \(\varphi\) is pure if and only if \(\max_j \deg q_j={\tt{d}}-1\) and \(\gcd(q_1,\ldots,q_n)\) has all its zeros on \(\mathbb{T}\).
A pure UCP map \(\varphi:\mathcal{T}_{{\tt{d}}}\to M_n\) must be of the form \[\varphi(T)=V^*TV,\qquad T\in\mathcal{T}_{{\tt{d}}},\] for some isometry \(V:\mathbb{C}^n\to\mathbb{C}^{\tt{d}}\). However, not every UCP map of this form is pure. To decide when such a map is pure, one needs a Fejér–Riesz factorization of \(P_\varphi\). In the following lemma, we associate to the isometry \(V\) a row polynomial \(Q_V\) such that \[P_\varphi(z) \,=\, Q_V^* (z) Q_V(z), \qquad z \in \mathbb{T} .\] Thus, the purity of \(\varphi\) can then be decided directly from this row polynomial.
Lemma 5. Let \(\varphi:\mathcal{T}_{{\tt{d}}}\to M_n\) be given by \(\varphi(T) = V^*TV,\) where \(V:\mathbb{C}^n\to\mathbb{C}^{{\tt{d}}}\) is an isometry. Write \[V \,=\, \begin{pmatrix} V_0\\ V_1\\ \vdots\\ V_{{\tt{d}}-1} \end{pmatrix}, \qquad V_i\in M_{1,n}.\] Define the row polynomial \(Q(z) \,=\, \sum_{i=0}^{{\tt{d}}-1}V_i z^i.\) Then the polynomial density of \(\varphi\) is \(P_\varphi(z)=Q(z)^*Q(z).\)
Proof. For \(0\leq k\leq {\tt{d}}-1\), we have \(J^k=\sum_{i=0}^{{\tt{d}}-1-k}E_{i,i+k}.\) Therefore \[\varphi(J^k) \,=\, V^*J^kV \,=\, \sum_{i=0}^{{\tt{d}}-1-k}V_i^*V_{i+k}.\] On the other hand, \[Q(z)^*Q(z) \,=\, \left(\sum_{i=0}^{{\tt{d}}-1}V_i z^i\right)^* \left(\sum_{j=0}^{{\tt{d}}-1}V_j z^j\right) \,=\, \sum_{i,j=0}^{{\tt{d}}-1}V_i^*V_j z^{j-i}.\] Hence the coefficient of \(z^k\) in \(Q(z)^*Q(z)\) is \[\sum_{i=0}^{{\tt{d}}-1-k}V_i^*V_{i+k} = \varphi(J^k),\] and the coefficient of \(z^{-k}\) is \(\varphi(J^k)^*\). This completes the proof. ◻
Remark 4. An interested reader may compare our scalar case \((n=1)\) with [15] and [15]. For a unit vector \(\xi=(\xi_0,\ldots,\xi_{{\tt{d}}-1})^{t}\), the vector state \(T \mapsto \langle T \xi , \xi \rangle\) leads to two different polynomial conventions. Hekkelman’s polynomial is \[Q_\xi(z) \,=\, \sum_{k=0}^{{\tt{d}}-1}\xi_k z^{{\tt{d}}-k-1},\] whereas Lemma 5 gives \[\widetilde{Q}_\xi(z) \,=\, \sum_{k=0}^{{\tt{d}}-1}\xi_k z^k.\] The condition that \(Q_\xi\) has full degree and all its zeros on \(\mathbb{T}\) is equivalent to the same condition for \(\widetilde{Q}_{\xi}\), and hence [15] agrees with our scalar characterization. However, the density of the vector state is \(|\widetilde{Q}_\xi|^2\), not in general \(|Q_\xi|^2\); for instance, when \({\tt{d}}=2\), these are \(|\xi_0+\xi_1z|^2\) and \(|\xi_0z+\xi_1|^2\), which need not be equal.
For a CP map \(\varphi:\mathcal{T}_{\tt{d}}\to M_n,\) we prove that \(\varphi\) has a unique CP extension if and only if all Fejér–Riesz factors of its polynomial density \(P_\varphi\) has the same coefficient Gram matrix.
Fejér–Riesz factors and coefficient Gram matrices.
Definition 6. A polynomial matrix \[F(z) \,=\, F_0 + F_1 z + \cdots + F_{{\tt{d}}-1} z^{{\tt{d}}-1}, \qquad F_j\in M_{m,n}(\mathbb{C}),\] is called a Fejér–Riesz factor of \(P_\varphi\) if \[P_\varphi(z) \;=\;F(z)^*F(z) \qquad (z\in\mathbb{T}).\] Here the number of rows \(m\) is allowed to depend on \(F\).
Given such a factor \(F\), define its coefficient Gram matrix by \[G_F \, =\, [F_i^*F_j]_{i,j=0}^{{\tt{d}}-1} \,\in\, M_{\tt{d}}(M_n).\] Thus \(G_F\) is a \({\tt{d}}\times {\tt{d}}\) block matrix whose \((i,j)\)-block is \((G_F)_{i,j} = F_i^*F_j.\) The coefficient of \(z^k\) in \(F(z)^*F(z)\) is \(\sum_{r=0}^{d-1-k}F_r^*F_{r+k},\) for \(0\leq k\leq d-1.\) Therefore \(P_\varphi=F^*F\) implies that \[\sum_{j=0}^{{\tt{d}}-1-k}F_j^*F_{j+k} \;=\;\widehat P_{\varphi}(k) \qquad(0\leq k\leq d-1).\] In particular, if \(\varphi\) is UCP, then \(\sum_{j=0}^{{\tt{d}}-1}F_j^*F_j \,=\, I_n.\)
Extensions and Choi matrices. A CP extension of \(\varphi\) to \(M_{\tt{d}}\) is a CP map \(\tilde{\varphi}:M_{\tt{d}}\to M_n\) such that \(\tilde{\varphi}|_{\mathcal{T}_{{\tt{d}}}} = \varphi.\) The Choi matrix of \(\tilde{\varphi}\) is \[C_{\tilde{\varphi}} \,=\, [\tilde{\varphi} (E_{i,j})]_{i,j=0}^{{\tt{d}}-1}\in M_{\tt{d}}(M_n).\] By Choi’s theorem, \(\tilde{\varphi}\) is CP if and only if \(C_{\tilde{\varphi}} \succeq 0.\)
Since \(J^k = \sum_{i=0}^{{\tt{d}}-1-k} E_{i, i+k},\) the condition that \(\tilde{\varphi}\) extends \(\varphi\) is exactly \[\sum_{i=0}^{{\tt{d}}-1-k}\tilde{\varphi} (E_{i,i+k}) \,=\, \varphi(J^k) \,=\, \widehat P_{\varphi} (k), \qquad 0\leq k\leq {\tt{d}}-1.\] In terms of the Choi matrix \(C_{\tilde{\varphi}} = [C_{i,j}]\), this becomes \[\sum_{i=0}^{{\tt{d}}-1-k} C_{i,i+k} \,=\, \widehat P_{\varphi}(k), \qquad 0\leq k\leq {\tt{d}}-1.\] Thus CP extensions of \(\varphi\) are the same thing as positive block matrices \(C=[C_{r,s}]_{r,s=0}^{{\tt{d}}-1}\in M_{\tt{d}}(M_n)\) satisfying \[\sum_{i=0}^{{\tt{d}}-1-k} C_{i,i+k} \,=\, \widehat P_{\varphi} (k), \qquad 0\leq k\leq {\tt{d}}-1.\]
Factorizations give extensions.
Lemma 6. Let \[P_\varphi(z) \,=\, F(z)^*F(z), \qquad F(z) \,=\;\sum_{j=0}^{{\tt{d}}-1}F_jz^j,\] be a Fejér–Riesz factor of \(P_\varphi\). Define \(\Psi_F:M_d\to M_n\) on matrix units by \[\Psi_F(E_{i,j}) \,=\, F_i^*F_j, \qquad 0\leq i,j \leq {\tt{d}}-1.\] Then \(\Psi_F\) is a CP extension of \(\varphi\), and its Choi matrix is \(C_{\Psi_F} = G_F = [F_i^*F_j]_{i,j=0}^{{\tt{d}}-1}.\)
Proof. The Choi matrix of \(\Psi_F\) is \(C_{\Psi_F} = [F_i^*F_j]_{i,j=0}^{{\tt{d}}-1}.\) This is a positive block matrix, because it is a Gram matrix. Thus \(\Psi_F\) is CP by Choi’s theorem.
Since \(P_\varphi(z)=F(z)^*F(z),\) the coefficient of \(z^k\) in \(P_\varphi\) is \(\sum_{i=0}^{{\tt{d}}-1-k}F_i^*F_{i+k}.\) But this coefficient is \(\widehat P_{\varphi} (k) = \varphi(J^k)\). Hence \(\sum_{i=0}^{{\tt{d}}-1-k} F_i^*F_{i+k} \,=\, \widehat P_{\varphi} (k).\) Therefore \[\Psi_F(J^k) \,=\, \Psi_F\left(\sum_{i=0}^{{\tt{d}}-1-k} E_{i,i+k}\right) \,=\, \sum_{i=0}^{{\tt{d}}-1-k}\Psi_F(E_{i,i+k}) \,=\, \sum_{i=0}^{{\tt{d}}-1-k}F_i^*F_{i+k} \,=\, \widehat P_{\varphi} (k) \,=\, \varphi(J^k).\] Similarly, \[\Psi_F((J^*)^k) \,=\, \varphi((J^*)^k).\]
Since \(\mathcal{T}_{{\tt{d}}}\) is spanned by \(I,J,\ldots,J^{{\tt{d}}-1},J^*,\ldots,(J^*)^{{\tt{d}}-1},\) we conclude that \(\Psi_F|_{\mathcal{T}_{{\tt{d}}}} = \varphi.\) Thus \(\Psi_F\) is a CP extension of \(\varphi\). ◻
Extensions give factorizations.
Lemma 7. Let \(\tilde{\varphi} : M_{\tt{d}}\to M_n\) be a CP extension of \(\varphi\). Then there exists a Fejér–Riesz factor \(F(z) = \sum_{j=0}^{{\tt{d}}-1}F_j z^j\) of \(P_\varphi\) such that \[\tilde{\varphi}(E_{i,j}) \,=\, F_i^* F_j \qquad 0\leq i,j\leq {\tt{d}}-1.\] Equivalently, \(C_{\tilde{\varphi}} = G_F.\)
Proof. Let \(C_{\tilde{\varphi}} = [\tilde{\varphi} (E_{i,j})]_{i,j=0}^{{\tt{d}}-1}\) be the Choi matrix of \(\tilde{\varphi}\). Since \(\tilde{\varphi}\) is CP, \(C_{\tilde{\varphi}} \succeq 0.\) Therefore \(C_{\tilde{\varphi}}\) has a Gram factorization. Hence there exist matrices \(F_0,\ldots,F_{{\tt{d}}-1}\) of a common size such that \[\tilde{\varphi} (E_{i,j}) \,=\, F_i^*F_j \qquad 0\leq i,j\leq {\tt{d}}-1.\]
Define \(F(z)=\sum_{j=0}^{{\tt{d}}-1}F_jz^j.\) Then \(F(z)^*F(z) = \sum_{i,j=0}^{{\tt{d}}-1}F_i^*F_j z^{j-i}.\) The coefficient of \(z^k\), for \(0\leq k\leq {\tt{d}}-1\), is \[\sum_{j=0}^{{\tt{d}}-1-k}F_j^*F_{j+k} \,=\, \sum_{j=0}^{{\tt{d}}-1-k}\tilde{\varphi} (E_{j,j+k}) \,=\, \tilde{\varphi} (J^k).\] Since \(\tilde{\varphi}\) extends \(\varphi\), \(\tilde{\varphi}(J^k)=\varphi(J^k)= \widehat P_{\varphi}(k).\) Thus the coefficient of \(z^k\) in \(F^*F\) agrees with the coefficient of \(z^k\) in \(P_\varphi\). The same argument for \((J^*)^k\) gives agreement of the negative Fourier coefficients. Therefore \[F(z)^*F(z) \,=\, P_\varphi(z) \qquad(z\in\mathbb{T}),\] so \(F\) is a Fejér–Riesz factor of \(P_\varphi\). Moreover, \[C_{\tilde{\varphi}} \,=\, [\tilde{\varphi} (E_{i,j})]_{i,j=1}^{{\tt{d}}-1} \,=\, [F_i^*F_j]_{i,j=1}^{{\tt{d}}-1} \,=\, G_F.\] ◻
Proposition 5. Let \(\varphi:\mathcal{T}_{{\tt{d}}} \to M_n\) be CP, and let \(P_\varphi\) be its polynomial density. Then \(\varphi\) has a unique CP extension if and only if for every pair of Fejér–Riesz factors \[P_\varphi(z) \,=\, F(z)^*F(z) \,=\, H(z)^*H(z),\] where \(F(z) = \sum_{j=0}^{{\tt{d}}-1} F_jz^j\) and \(H(z) \,=\, \sum_{j=0}^{{\tt{d}}-1} H_r z^r,\) one has \[[F_i^*F_j]_{i,j=0}^{{\tt{d}}-1} \,=\, [H_i^*H_j]_{i,j=0}^{{\tt{d}}-1}.\] Equivalently, \(G_F \,=\,G_H\) for all Fejér–Riesz factors \(F\) and \(H\) of \(P_\varphi\).
Proof. First, suppose that \(\varphi\) has a unique CP extension. Let \(P_\varphi \,=\, F^*F \,=\, H^*H\) be two Fejér–Riesz factorizations. By Lemma 6, \(F\) defines a CP extension \(\Psi_F:M_{\tt{d}}\to M_n\) of \(\varphi\), with Choi matrix \(C_{\Psi_F} = G_F = [F_i^*F_j]_{i,j=0}^{{\tt{d}}-1}.\) Similarly, \(H\) defines a UCP extension \(\Psi_H : M_{\tt{d}}\to M_n\) of \(\varphi\), with Choi matrix \(C_{\Psi_H} = G_H = [H_i^*H_j]_{i,j=0}^{{\tt{d}}-1}.\) Since \(\varphi\) has a unique CP extension, \(\Psi_F=\Psi_H.\) Therefore, their Choi matrices are equal \(G_F=G_G.\) Hence all Fejér–Riesz factors of \(P_\varphi\) have the same coefficient Gram matrix.
Conversely, assume that all Fejér–Riesz factors of \(P_\varphi\) have the same coefficient Gram matrix. Let \(\tilde{\varphi}_{1},\tilde{\varphi}_2 : M_{\tt{d}}\to M_n\) be two CP extensions of \(\varphi\). By Lemma 7, there exist Fejér–Riesz factors \(F(z)=\sum_{j=0}^{{\tt{d}}-1}F_jz^j\) and \(H(z)=\sum_{j=0}^{{\tt{d}}-1} H_j z^j,\) of \(P_\varphi\) such that \(C_{\tilde{\varphi}_1} = G_F = [F_i^*F_j]_{i,j=0}^{{\tt{d}}-1}\) and \(C_{\tilde{\varphi}_2} = G_H = [H_i^*H_j]_{i,j=0}^{{\tt{d}}-1}.\) By the assumed Gram rigidity, \(G_F=G_G.\) Hence \(C_{\tilde{\varphi}_1} = C_{\tilde{\varphi}_2}.\) Since a CP map \(M_{\tt{d}}\to M_n\) is determined by its Choi matrix, we get \(\tilde{\varphi}_1 = \tilde{\varphi}_2.\) Therefore \(\varphi\) has a unique CP extension to \(M_{\tt{d}}.\) This proves the equivalence. ◻
Remark 5. This criterion is purely polynomial. The Toeplitz data of \(\varphi\) fixes only the diagonal sums \[\sum_{i=0}^{{\tt{d}}-1-k}F_i^*F_{i+k} = \varphi(J^k), \qquad 0\leq k\leq {\tt{d}}-1.\] A CP extension to \(M_{{\tt{d}}}\), however, is determined by the full coefficient Gram matrix \([F_i^*F_j]_{i,j=0}^{{\tt{d}}-1}.\) Thus the unique CP extension requires precisely that, among positive Choi matrices with the prescribed Toeplitz diagonal sums, there is only one possible full Gram matrix. For pure UCP maps this uniqueness follows from the Fejér–Riesz rigidity of the rank-one density \(P_\varphi=Q^*Q\) as explained in the following theorem.
Proof. Let \(F\) be a Fejér–Riesz factor of \(P_{\varphi},\) that is, \(F\) is a matrix valued analytic polynomial of degree at most \({\tt{d}}-1\) satisfying \(F(z)^*F(z) = P_\varphi(z).\)
Since \(\varphi\) is pure, by Theorem 1, there exists a row polynomial \(Q\) with \(P_\varphi(z) = Q(z)^*Q(z),\) satisfying conditions (1) and (2) of Theorem 1. It follows from the discussion in Subsection 4.3 that there exists a constant unit column vector \(h_{F}\) such that \(F = h_{F}Q.\)
Write \(Q(z)=\sum_{i=0}^{{\tt{d}}-1}Q_i z^i.\) Then \(F_i = h_{F} Q_i,\) for \(0\leq i\leq {\tt{d}}-1.\) Since \(h_{F}^{*}h_{F} = 1\), we get \[F_i^*F_j \,=\, Q_i^*h_{F}^* h_{F} Q_j \,=\, Q_i^*Q_j \qquad 0\leq i,j\leq {\tt{d}}-1.\] It follows from Proposition 5 that \(\varphi\) has a unique CP extension to \(M_{{\tt{d}}}.\) ◻
Corollary 1. Let \(V,W:\mathbb{C}^n\to\mathbb{C}^{{\tt{d}}}\) be isometries, and suppose that \[V^*TV=W^*TW,\qquad T\in\mathcal{T}_{{\tt{d}}}.\] Assume that the UCP map \(\varphi(T)=V^*TV\) is pure. Then there exists \(\lambda\in\mathbb{T}\) such that \(W=\lambda V.\)
Proof. Since \(\varphi\) is pure, it has a unique UCP extension to \(M_{{\tt{d}}}=C^*(\mathcal{T}_{{\tt{d}}})\). The maps \[A\mapsto V^*AV \qquad\text{and}\qquad A\mapsto W^*AW\] are both UCP extensions of \(\varphi\) to \(M_{{\tt{d}}}\). By uniqueness, they are equal: \[V^*AV=W^*AW,\qquad A\in M_{{\tt{d}}}.\]
Now we use the uniqueness of minimal Stinespring dilations. Both maps are obtained by compressing the identity representation of \(M_{{\tt{d}}}\) on \(\mathbb{C}^{{\tt{d}}}\). This Stinespring representation is minimal. Hence there is a unitary \(U:\mathbb{C}^{{\tt{d}}}\to\mathbb{C}^{{\tt{d}}}\) such that \(UV=W\) and \(UA=AU\) for all \(A\in M_{{\tt{d}}}.\) But the commutant of \(M_{{\tt{d}}}\) is just the scalars. Therefore \(U=\lambda I_{{\tt{d}}}\) for some \(\lambda\in\mathbb{T}\). Hence \(W=UV=\lambda V.\) ◻
Remark 6. Theorem 2 should be compared with the usual unique extension property in noncommutative Choquet theory. In Arveson’s framework [1], [2], [36], and in the subsequent work of Dritschel and McCullough [37], maximal UCP maps are characterized by the unique extension property, where the unique extension to the generated \(C^*\)-algebra is a \(*\)-representation. This point of view plays a central role in the theory of boundary representations and the \(C^*\)-envelope, see also [5], [38], [39]. Our result is of a different nature. The pure UCP maps considered here need not be restrictions of representations; rather, they are typically compressions \(T\mapsto V^*TV\). Nevertheless, in the Toeplitz setting, purity forces a unique UCP extension to \(M_{\tt{d}}\), although this extension is generally not multiplicative unless the compression is trivial. Thus, the finite Toeplitz operator system exhibits a form of extension rigidity for pure matrix states which is weaker than maximality in Arveson’s sense, but stronger than what holds for general hyperrigid operator systems.
First, we give an example of a UCP map \(\varphi: \mathcal{T}_{2} \to M_{2}\) which is not pure but has a unique UCP extension.
Example 2. Let \[\xi \,=\, \frac{1}{\sqrt2}\begin{pmatrix}1\\1\end{pmatrix}, \qquad \rho(T)=\langle T\xi,\xi\rangle,\quad T\in\mathcal{T}_2.\] Define \(\varphi:\mathcal{T}_2\to M_2,\) by \(\varphi(T)=\rho(T)I_2.\) Then \(\varphi\) is a UCP map with a unique UCP extension, but it is not pure.
Proof. First, \(\rho\) is pure by [15]. Now let \(\tilde{\varphi}:M_2\to M_2\) be a UCP extension of \(\varphi\). Its Choi matrix has the form \[C_{\tilde{\varphi}} \,=\, \begin{bmatrix} D&\frac{1}{2} I_2\\ \frac{1}{2} I_2&I_2-D \end{bmatrix} \succeq 0,\] where \(D=\tilde{\varphi} (E_{11})\). For every \(x\in\mathbb{C}^2\), \[\Big\langle C_{\tilde{\varphi}} \begin{pmatrix}x\\-x\end{pmatrix}, \begin{pmatrix}x\\-x\end{pmatrix} \Big\rangle \,=\, \left\langle Dx,x \right\rangle+ \left\langle (I_2-D)x,x \right\rangle - \left\langle I_2x,x \right\rangle \,=\,0.\] Since \(C_{\tilde{\varphi}} \succeq 0\), this implies \(C_{\tilde{\varphi}} \begin{pmatrix}x\\-x\end{pmatrix}=0.\) Looking at the first component gives \(Dx-\frac{1}{2}x=0\) for every \(x\). Hence \(D=\frac{1}{2} I_2.\) Therefore \(C_{\tilde{\varphi}} = \frac{1}{2} \begin{bmatrix} I_2&I_2\\ I_2&I_2 \end{bmatrix},\) so the Choi matrix of any UCP extension is uniquely determined. Hence \(\varphi\) has a unique UCP extension.
Finally, \(\varphi\) is not pure. Let \(P= \begin{bmatrix} 1&0\\ 0&0 \end{bmatrix}.\) Define \(\psi:\mathcal{T}_2\to M_2\) by \(\psi(T)=\rho(T)P.\) Then \(\psi\) is CP and \(\varphi-\psi=\rho(\cdot)(I_2-P)\) is also CP. Thus \(\psi\leq_{\rm cp} \varphi.\) But \(\psi\) is not a scalar multiple of \(\varphi\), since \(\psi(I)=P\) is not a scalar multiple of \(\varphi(I)=I_2.\) Therefore, \(\varphi\) is not pure. ◻
Now we give an example of a state on \(\mathcal{T}_{2}\) which does not have a unique UCP extension.
Example 3. Define \(\varphi: \mathcal{T}_{2} \to \mathbb{C}\) by \(\varphi \begin{bmatrix} a&b\\ c&a \end{bmatrix} =a.\) Then \(\varphi\) is a state but it does not have a unique UCP extension.
Proof. For each \(t\in[0,1]\), define \(\Psi_t:M_2\to\mathbb{C}\) by \(\Psi_t \begin{pmatrix} a&b\\ c&d \end{pmatrix} = ta+(1-t)d.\) Each \(\Psi_t\) is a state on \(M_2\). Moreover, \(\Psi_t|_{\mathcal{T}_2} = \varphi\) for every \(t\in[0,1]\).
However, if \(s\neq t\), then \[\Psi_t(E_{11}) \,=\, t \,\neq\, s\,=\,\Psi_s(E_{11}).\] Thus \(\Psi_t\neq \Psi_s\). Therefore, \(\varphi\) has infinitely many UCP extensions to \(M_2=C^*(\mathcal{T}_2)\), and hence \(\varphi\) does not have a unique UCP extension. ◻
Next, we give an example of a finite-dimensional hyperrigid operator system \(\mathcal{S} \subseteq M_4\) and a pure state \(\varphi:\mathcal{S} \to \mathbb{C}\) which does not have a unique UCP extension. Thus, the hyperrigidity of \(\mathcal{S}\) does not imply that every pure UCP map on \(\mathcal{S}\) has a unique UCP extension.
Example 4. Let \(C=\frac{1}{\sqrt2} \begin{bmatrix} 1&1\\ 1&-1 \end{bmatrix}.\) Define two unitaries \(U,W\in M_4\) by \(U = \begin{bmatrix} 0&I_2\\ C&0 \end{bmatrix}\) and \(W = \operatorname{diag}(1,1,-1,i).\) Let \[\mathcal{S} \,=\, \text{span}\{I,U,U^*,W,W^*\} \,\subseteq\, M_4.\] Then \(\mathcal{S}\) is a hyperrigid operator system, but the vector state \[\varphi:\mathcal{S} \to \mathbb{C}, \qquad \varphi(a) \,=\, \langle a e_1,e_1\rangle,\] is pure and does not have a unique UCP extension.
Proof. The operator system is hyperrigid: First, we show that \(C^*(\mathcal{S}) = C^{*}(U,W) = M_4.\) Let \(X\in M_4(\mathbb{C})\) commute with both \(U\) and \(W\). Since \(W=\operatorname{diag}(1,1,-1,i),\) the eigenspaces of \(W\) are \({\rm span}\{e_1,e_2\},\, \mathbb{C} e_3,\, \mathbb{C} e_4.\) Therefore \(XW = WX\) implies that \(X\) has the block form \(X= \begin{bmatrix} A&0\\ 0&B \end{bmatrix},\) where \(A\in M_2(\mathbb{C})\) and \(B= \begin{bmatrix} b&0\\ 0&c \end{bmatrix}.\) Now imposing \(XU = UX\) we get \(A=B\) and \(BC = CA.\) Since \(A=B\), this becomes \(BC = CB.\) But \(B= \begin{bmatrix} b&0\\ 0&c \end{bmatrix}\) commutes with \(C=\frac{1}{\sqrt2} \begin{bmatrix} 1&1\\ 1&-1 \end{bmatrix}\) if and only if \(b=c\). Therefore \(B=bI_2\) and hence \(A=B=bI_2\). Thus \(X=bI_4\).
So the commutant of \(C^*(U, W)\) is only the scalars. Since \(C^*(U,W)\) is a finite-dimensional unital \(^*\)-subalgebra of \(M_4\), it follows that \(C^*(U,W)=M_4(\mathbb{C}).\)
Now we prove the hyperrigidity. Let \(\pi:M_4(\mathbb{C})\to \mathcal{B}(\mathcal{H})\) be a representation, and let \(\Psi:M_4 \to \mathcal{B}(\mathcal{H})\) be UCP such that \(\Psi|_\mathcal{S}=\pi|_\mathcal{S}.\) Since \(U,W\in \mathcal{S}\), we have \[\Psi(U) \,=\,\pi(U), \qquad \Psi(W) \,=\, \pi(W).\] The operators \(\pi(U)\) and \(\pi(W)\) are unitaries. Hence \(\Psi(U)^*\Psi(U)=I=\Psi(U^*U),\) and \(\Psi(U)\Psi(U)^*=I=\Psi(UU^*).\) Therefore \(U\) lies in the multiplicative domain of \(\Psi\). Similarly, \(W\) lies in the multiplicative domain of \(\Psi\).
Since \(U\) and \(W\) generate \(M_4(\mathbb{C})\), the multiplicative domain of \(\Psi\) contains all of \(M_4(\mathbb{C})\). Thus \(\Psi\) is a \(^*\)-homomorphism on \(M_4\). Because it agrees with \(\pi\) on the generators \(U\) and \(W\), it follows that \(\Psi=\pi.\) Hence, every representation of \(M_4\) has the unique extension property relative to \(\mathcal{S}\). Thus \(\mathcal{S}\) is hyperrigid.
\(\varphi\) is a pure state on \(\mathcal{S}\): Let \[h \,=\, {\rm Re}\, W=\frac{W+W^*}{2} =\operatorname{diag}(1,1,-1,0)\,\, \preceq I.\] Moreover \(\varphi(h)=1.\) Consider the exposed face of the state space of \(\mathcal{S}\) \[F \,=\, \{\omega\in {\rm UCP}(\mathcal{S}, \mathbb{C}) \;:\; \omega(h)=1\}.\] We claim that \(F=\{\varphi\}.\) Let \(\omega\in F\). Extend \(\omega\) to a state \(\tilde{\omega}:M_4(\mathbb{C})\to\mathbb{C}.\) Since \(I-h = \operatorname{diag}(0,0,2,1)\succeq 0\) and \(\tilde{\omega}(I-h)=1-\omega(h)=0,\) the state \(\tilde{\omega}\) is supported on the kernel of \(I-h\), namely on the subspace \({\rm span}\{e_1,e_2\}.\) Equivalently, if \(P\) denotes the projection onto \({\rm span}\{e_1,e_2\}\), then \(\tilde{\omega}(a)=\tilde{\omega}(PaP),\) \(a\in M_4.\) Now \(PWP=P\) and \(PUP=0.\) Therefore \(\omega(W)=1, \omega(U)=0.\) Also \(\omega(W^*)=1\) and \(\omega(U^*)=0.\) Since \(S\) is spanned by \(I,U,U^*,W,W^*\), these values determine \(\omega\) uniquely. They are exactly the values of \(\varphi\). Hence \(\omega=\varphi.\) Thus \(F=\{\varphi\}.\) Since \(\{\varphi\}\) is an exposed face of the state space of \(S\), \(\varphi\) is an extreme point of the state space. Therefore \(\varphi:\mathcal{S} \to \mathbb{C}\) is a pure state.
The state \(\varphi\) does not have unique CP extension: Define two states on \(M_4(\mathbb{C})\) by \[\tilde{\varphi}_1(a) \,=\, \langle ae_1,e_1\rangle, \qquad \tilde{\varphi}_2(a) \,=\, \langle ae_2,e_2\rangle.\] These are distinct states on \(M_4\). However, they agree on \(\mathcal{S}\). ◻
First we recall some notations \[\mathcal{X}_{{\tt{d}},n} \,:=\, {\rm PureUCP} (\mathcal{T}_{{\tt{d}}},M_n), \qquad \mathcal{Y}_{{\tt{d}},n}\,:=\, {\rm UCP} (\mathcal{T}_{{\tt{d}}}, M_{n}), \qquad \mathcal{Y}_n \,:=\, {\rm UCP}(C(\mathbb{T}),M_n).\] We embed \(\mathcal{Y}_{{\tt{d}},n}\) into \(\mathcal{Y}_n\) by sending each \(\varphi\in\mathcal{Y}_{{\tt{d}},n}\) to the UCP map \(C(\mathbb{T})\to M_n\) defined by \[f \,\mapsto\, \int_{\mathbb{T}} f(z)P_\varphi(z) \, dm(z), \qquad f\in C(\mathbb{T}),\] where \(P_{\varphi}\) is the polynomial density of \(\varphi.\)
Definition 7. Let \(\mathcal{A}_{{\tt{d}},n}\) be the set of UCP maps \(\varphi:C(\mathbb{T})\to M_n\) of the form \[\varphi (f) \,=\, \int_{\mathbb{T}} f(z) P(z) \,dm(z),\] where \(P\) is a normalized \(n \times n\) matrix-valued positive trigonometric polynomial of degree at most \({\tt{d}}-1\).
Definition 8. Let \(\mathcal{P}_{{\tt{d}},n}\subset \mathcal{A}_{{\tt{d}},n}\) be the subset consisting of those maps \[\varphi_Q(f) \,=\, \int_{\mathbb{T}}f(z)Q(z)^*Q(z)\,dm(z),\] where \(Q=(q_1,\ldots,q_n)\) is a row polynomial of degree at most \({\tt{d}}-1\) with normalized \(Q^{*}Q\) which satisfies \[\max_j\deg q_j={\tt{d}}-1, \quad \text{and} \quad \gcd(q_1,\ldots,q_n)=1.\]
Note that \(\mathcal{A}_{{\tt{d}},n}\) is the image of \(\mathcal{Y}_{{\tt{d}},n}\) under the embedding, while \(\mathcal{P}_{{\tt{d}},n}\) is a subset of the image of \(\mathcal{X}_{{\tt{d}},n}\) under the same embedding.
Proposition 6 is the main result of this section, where we prove that \(\mathcal{P}_{{\tt{d}},n}\) converges to \(\mathcal{Y}_n\) in the Hausdorff sense with respect to the matricial Monge–Kantorovich metric \(\rho_n\) defined in 2 . We start with an elementary density lemma for coprime row polynomials.
Lemma 8. Let \(n\geq 2\), and let \(Q(z)=\begin{pmatrix}q_1(z)&\cdots&q_n(z)\end{pmatrix}\) be a row of polynomials of degree at most \({\tt{d}}-1 .\) Fix \(N \geq {\tt{d}}\) and \(\varepsilon>0.\) Then there exists \(P(z)=\begin{pmatrix}p_1(z)&\cdots&p_n(z)\end{pmatrix}\) satisfying
\(\max_j \deg p_j=N,\)
\(\gcd(p_1,\ldots,p_n)=1,\) and
\(\|P-Q\|_{\infty} <\varepsilon.\)
Proof. Since \(n\geq 2\), it is enough to perturb the first two components. Choose a small nonzero complex number \(\delta_1\) and set \[p_1(z) \,=\, q_1(z)+\delta_1 z^N.\] Since \(N \geq {\tt{d}}\), we have \(\deg p_1 = N.\) Moreover, by choosing \(|\delta_1|\) sufficiently small, \(p_1\) is as close to \(q_1\) as desired in sup norm.
Let \(\alpha_1,\ldots,\alpha_m\) be the distinct roots of \(p_1\). We now choose \(p_2\). Put \[p_2(z) \,=\, q_2(z)+\delta_2,\] where \(\delta_2\in\mathbb{C}\) will be chosen small. We need \(p_1\) and \(p_2\) to have no common zero. Since the zeros of \(p_1\) are \(\alpha_1,\ldots,\alpha_m\), this is equivalent to requiring \(p_2(\alpha_\ell)\neq 0\) for all \(\ell=1,\ldots,m.\) But \(p_2(\alpha_\ell)=q_2(\alpha_\ell)+\delta_2.\) Hence the forbidden values of \(\delta_2\) are precisely \(-q_2(\alpha_1),\ldots,-q_2(\alpha_m).\) This is a finite set. Therefore we can choose \(\delta_2\) arbitrarily small such that \(\delta_2\notin \{-q_2(\alpha_1),\ldots,-q_2(\alpha_m)\}.\) For this choice, \(p_2(\alpha_\ell)\neq 0\) for all \(\ell=1,\ldots,m.\) Thus \(p_1\) and \(p_2\) have no common zero, and therefore \(\gcd(p_1,p_2) = 1.\)
For \(j \geq 3\), set \(p_j = q_j.\) Then \(\gcd(p_1,\ldots,p_n)=1,\) because already \(\gcd(p_1,p_2)=1.\) Also \(\max_j\deg p_j=N,\) because \(\deg p_1=N.\) Finally, by choosing \(|\delta_1|\) and \(|\delta_2|\) sufficiently small, we ensure \(\|Q-Q_0\|_{\infty}<\varepsilon.\) This proves the lemma. ◻
In the next step, we approximate a UCP map \(C(\mathbb{T})\to M_n\) by UCP maps with positive polynomial densities, using a standard Fejér kernel approximation argument. By the operator-valued Riesz–Markov representation theorem [32], every UCP map \(\varphi:C(\mathbb{T})\to M_n\) is of the form \[\varphi(f) \,=\, \int_{\mathbb{T}} f\,d\mu,\] where \(\mu\) is a positive \(n\times n\) matrix-valued regular Borel measure satisfying \(\mu(\mathbb{T})=I_n\). We approximate \(\mu\) by a positive trigonometric polynomial density.
Lemma 9. Let \(\varphi \in \mathcal{Y}_{n}\) and let \(\varepsilon>0\). Then there exists a normalized positive \(n \times n\) matrix-valued trigonometric polynomial \(P\) such that the UCP map \[\psi_P(f) \,:=\, \int_{\mathbb{T}}f(z)P(z)\,dm(z)\] satisfies \(\rho_n(\varphi, \psi_P)<\varepsilon.\)
Proof. Let \(\mu\) be the positive \(n \times n\) matrix-valued measure representing \(\varphi\). Let \(F_N\) be the scalar Fejér kernel \[F_{N} (z) \,=\, \sum\limits_{k= -N}^{N} \left( 1 - \frac{|k|}{N+1} \right) z^{k} \,=\, \frac{1}{N+1} \left| \sum_{k=0}^{N} z^{k} \right|^{2}, \qquad (z \in \mathbb{T}).\] Therefore, the scalar Fejér kernel satisfies \(F_{N}(z) \geq 0\) for all \(z \in \mathbb{T}\) and \(\int_{\mathbb{T}}F_{N}(z)\,dm(z) \,=\, 1 .\)
Now define the convolution of \(F_{N}\) with the \(n \times n\) matrix-valued measure \(\mu\) by \[P_N (z) \,=\, F_N*\mu (z) \,:=\, \int_{\mathbb{T}}F_{N}(z \overline{w})\,d\mu(w), \qquad (z \in \mathbb{T}).\] Then \(P_N\) is an \(n \times n\) matrix-valued trigonometric polynomial. Indeed, \[P_{N}(z) \,=\, \sum\limits_{k= -N}^{N} \left( 1 - \frac{|k|}{N+1} \right) z^{k} \int_{\mathbb{T}} \overline{w}^{k} \,d\mu(w).\] Moreover, \(P_{N}\) is positive. Fix \(z \in \mathbb{T}\) and \(\xi \in \mathbb{C}^{n}.\) Define the scalar positive measure \(\mu_{\xi} (E) = \langle \mu(E) \xi , \xi \rangle.\) Then \(\langle P_{N}(z) \xi ,\xi \rangle = \int_{\mathbb{T}} F_{N}(z \overline{w}) d\mu_{\xi}(w) .\) Since \(F_{N}(z \overline{w}) \geq 0\) and \(\mu_{\xi}\) is a positive scalar measure, we get \(\langle P_{N}(z) \xi ,\xi \rangle \geq 0.\) Thus \(P_{N}(z) \geq 0\) for all \(z \in \mathbb{T}.\) Also \[\int_{\mathbb{T}}P_N(z)\,dm(z) \,=\, \mu(\mathbb{T}) \,=\, I_n.\]
Let \(\psi_N(f):=\int_{\mathbb{T}}f(z)P_N(z)\,dm(z).\) Substituting the definition of \(P_{N}(z)\) we get \[\psi_{N}(f) \,=\, \int\limits_{\mathbb{T}} f(z) \Big( \int\limits_{\mathbb{T}} F_{N}(z \overline{w}) \, {\tt{d}}\mu(w) \Big) dm(z).\] Since \(f\) and \(F_{N}\) are bounded and \(\mu\) is finite, we can interchange the order of integration \[\psi_{N}(f) \,=\, \int\limits_{\mathbb{T}} \Big( \int\limits_{\mathbb{T}} f(z) F_{N}(z \overline{w}) \, dm(z) \Big) d\mu(w) \,=\, \int_{\mathbb{T}}(F_N*f)(w)\,d\mu(w).\] Therefore \[\psi_N(f) - \varphi(f) \,=\, \int_{\mathbb{T}}\bigl((F_N*f)(w) - f(w) \bigr)\, d\mu(w).\] Since \(\mu(\mathbb{T})=I_n\), we have \(\left\| \int_{\mathbb{T}}h(z)\,d\mu(z) \right\| \leq \|h\|_\infty\) for scalar continuous \(h\). Hence \[\| \psi_N(f) - \varphi(f) \| \,\leq\, \|F_N*f-f\|_\infty.\] The Fejér kernels form an approximate identity, and the convergence \(\|F_N*f - f\|_\infty\to 0\) is uniform over the class \[\{f\in C^1(\mathbb{T})\;:\; {\rm Lip}(f)\leq 1,\;f(1)=0 \}.\] Since the metric \(\rho_n\) is insensitive to constant functions, it is enough to consider this normalized class. Thus, for \(N\) sufficiently large, we obtain \(\rho_n(\varphi,\psi_N)<\varepsilon.\) Finally, setting \(P=P_N\) completes the proof. ◻
In the next step, we approximate a positive matrix-valued polynomial density by another such density which admits a Fejér–Riesz factorization with a row polynomial factor.
Let \(P\) be a normalized \(n\times n\) matrix-valued positive trigonometric polynomial. By the matrix-valued Fejér–Riesz factorization, there is an \(m\times n\) polynomial matrix \[H(z) \,=\, \begin{pmatrix} h_1(z)\\ \vdots\\ h_m(z) \end{pmatrix},\] where each \(h_\ell\) is an analytic row polynomial, such that \[P(z)=H(z)^*H(z)=\sum_{\ell=1}^m h_\ell(z)^*h_\ell(z).\]
We now encode the rows of \(H\) into one row polynomial by separating their frequencies.
Lemma 10. Let \(P(z)=H(z)^*H(z)\) be as above, and let \(\varepsilon>0\). Then there exists a row polynomial \(Q(z)=\begin{pmatrix}q_1(z)&\cdots&q_n(z)\end{pmatrix}\) such that \(Q^*Q\) is normalized and \(\rho_n(\varphi_Q,\varphi_H)<\varepsilon,\) where \(\varphi_{Q}\) and \(\varphi_H\) are elements of \(\mathcal{Y}_{n}\) with polynomial densities \(Q^{*}Q\) and \(H^*H\), respectively.
Proof. Write the rows of \(H\) as \(h_1,\ldots,h_m.\) Choose integers \(N_1 < N_2 < \cdots < N_m\) with gaps so large that \(|N_\ell - N_r| > \deg(h_\ell^*h_r)\) for \(\ell \neq r .\) Define \[Q(z) \,=\, \sum_{\ell=1}^{m} z^{N_\ell}h_\ell(z).\] Then \[Q(z)^* Q(z) \,=\, \sum_{\ell=1}^m h_\ell(z)^*h_\ell(z) + \sum_{\ell\neq r}z^{N_r - N_\ell}h_\ell(z)^*h_r(z).\] The first sum is \(P(z)\). The second sum consists of cross terms with large nonzero frequencies. Because the gaps are larger than the degrees of the polynomials \(h_\ell^*h_r\), the cross terms have no constant Fourier coefficient. Therefore \[\int_{\mathbb{T}}Q(z)^*Q(z)\,dm(z) \,=\, \int_{\mathbb{T}}P(z)\,dm(z) \,=\, I_n.\]
It remains to make the cross terms small against \(C^{1}(\mathbb{T})\) functions. For \(f \in C^{1}(\mathbb{T})\), its Fourier coefficients satisfy \[|\hat{f}(k)| \,\leq\, \frac{1}{|k|}, \qquad(k\neq 0),\] whenever \(\operatorname{Lip}(f)\leq 1\); see [40]. Each cross term is a finite sum of matrix coefficients multiplied by frequencies of the form \[s + N_r - N_\ell ,\] where \(s\) ranges over a fixed finite set depending only on \(H\). By choosing all gaps \(|N_r - N_\ell|\) sufficiently large, all these frequencies become large. Hence the integral of every cross term against \(f\) is uniformly small over \(\{ f \in C^{1}(\mathbb{T}) : {\rm Lip} (f) \leq 1, \, f(1) = 0\}\). Therefore the total cross-term contribution is less than \(\varepsilon\). ◻
The row polynomial \(Q\) constructed in Lemma 10 need not define a pure map on a finite Toeplitz system. For \(n\geq 2\), we can perturb it slightly so that its scalar polynomial entries are coprime.
Lemma 11. Let \(n\geq 2\). Let \(Q(z)=\begin{pmatrix}q_1(z)&\cdots&q_n(z)\end{pmatrix}\) be a row polynomial with normalized \(Q^{*}Q.\) Let \({\tt{d}}:= \max_{j} \deg q_{j}\) and \(\varepsilon>0\). Then for all \(N > {\tt{d}}+1\), there exists a row polynomial \(P(z)=\begin{pmatrix}p_1(z)&\cdots&p_n(z)\end{pmatrix}\) with normalized \(P^{*}P\) satisfying
\(\rho_{n} (\varphi_{P}, \varphi_{Q}) < \varepsilon,\) where \(\varphi_{P}\) and \(\varphi_Q\) are elements of \(\mathcal{Y}_{n}\) with polynomial densities \(P^{*}P\) and \(Q^*Q\), respectively.
\(\max_j\deg p_j = N - 1\),
\(\operatorname{gcd}(p_1,\ldots,p_n)=1\).
Consequently, \(P^{*}P\) is the polynomial density of a pure UCP map \(\mathcal{T}_{N}\to M_n.\)
Proof. Fix \(N > {\tt{d}}+1\). By Lemma 8, for any \(\delta>0\) we can find a row polynomial \(\widetilde{Q}_{\delta}(z)\) of degree at most \(N-1\) such that \[\max_j\deg \tilde{q}_{\delta,j} = N-1, \qquad {\rm gcd}(\tilde{q}_{\delta,1},\ldots,\tilde{q}_{\delta,n}) = 1, \quad \text{and} \quad \| \widetilde{Q}_{\delta} - Q \|_{\infty} < \delta.\]
Set \(G_{\delta} =\int_{\mathbb{T}}\widetilde{Q}_{\delta}(z)^*\widetilde{Q}_{\delta}(z)\,dm(z).\) Clearly \(G_{\delta}\geq 0\). Now we show that \(G_{\delta} \to I_n\) as \(\delta\to 0\). Since \(\|\widetilde{Q}_{\delta} - Q\|_{\infty}<\delta,\) we get \(\|\widetilde{Q}_{\delta} -Q\|_{\infty}\to 0\) as \(\delta\to 0.\) Therefore \(\widetilde{Q}_{\delta}(z) \to Q(z)\) uniformly on \(\mathbb{T}\). Hence \(\widetilde{Q}_{\delta}(z)^*\widetilde{Q}_{\delta}(z) \to Q(z)^*Q(z)\) uniformly on \(\mathbb{T}\). Integrating gives \[G_{\delta} \,=\, \int_{\mathbb{T}}\widetilde{Q}_{\delta}(z)^*\widetilde{Q}_{\delta}(z)\,dm(z) \longrightarrow \int_{\mathbb{T}}Q(z)^*Q(z)\,dm(z) \,=\, I_{n}.\] Therefore \(G_{\delta}\to I_n.\) In particular, for \(\delta>0\) sufficiently small, \[\|G_{\delta}-I_n\|<\frac{1}{2}.\] Since \(G_{\delta} \geq 0,\) the spectrum of \(G_{\delta}\) is contained in the interval \((1/2,3/2).\) In particular, \(G_{\delta}\) is invertible.
Now define \[P_{\delta}(z) \,=\, \widetilde{Q}_{\delta} (z)G_{\delta}^{-1/2}.\] Then \[\int_{\mathbb{T}}P_{\delta}(z)^*P_{\delta}(z)\,dm(z) \,=\, G_{\delta}^{-1/2}G_{\delta}G_{\delta}^{-1/2} \,=\, I_n.\] Right multiplication by the invertible matrix \(G_{\delta}^{-1/2}\) does not change the polynomial subspace spanned by the scalar polynomial entries of \(\widetilde{Q}_{\delta}\). Therefore the row polynomial \[P_{\delta}(z)\,=\, \begin{pmatrix}p_{\delta,1}(z)&\cdots&p_{\delta,n}(z)\end{pmatrix}\] satisfies \[\max_j\deg p_{\delta,j} = N-1, \quad \text{and} \quad {\rm gcd}(p_{\delta,1},\ldots,p_{\delta,n})=1.\] Finally, since \(P_{\delta}\to Q\) in sup norm as \(\delta\to 0\), we have \(\| P_{\delta}^*P_{\delta} - Q^{*} Q \|_{\infty} \to 0 .\) Hence \(\rho(\varphi_{P_{\delta}}, \varphi_{Q})\) can be made smaller than \(\varepsilon\) by choosing \(\delta\) sufficiently small.
It follows from Theorem 1, that \(P_{\delta}^{*}P_{\delta}\) is the polynomial density of a pure UCP map \(\mathcal{T}_N\to M_n .\) ◻
We can now prove the main result of this section.
Proposition 6. Fix \(n\geq 2\). Then \[\lim_{{\tt{d}}\to\infty} \operatorname{dist}_{H}^{\rho_n} \left( \mathcal{P}_{{\tt{d}},n}, \mathcal{Y}_{n} \right) = 0.\]
Proof. Let \(\varphi\in \mathcal{Y}_{n}\) and \(\varepsilon>0\). Let \(\mu\) be the positive \(n \times n\) matrix-valued measure representing \(\varphi\), so that \[\varphi(f) \,=\, \int_{\mathbb{T}}f\,d\mu, \qquad \mu(\mathbb{T})=I_n.\] By Lemma 9, there exists a normalized positive \(n \times n\) matrix-valued trigonometric polynomial \(P\) such that the UCP map \[\psi_{P}(f) \,:=\, \int_{\mathbb{T}} f(z) P(z) \, dm(z)\] satisfies \(\rho_{n}(\varphi, \psi_{P}) < \varepsilon /3.\)
By the matrix-valued Fejér–Riesz factorization, there exists an \(m\times n\) matrix-valued polynomial \(H\) such that \(P(z)=H(z)^*H(z).\) Hence, if \({\tt{d}}\) is larger than \(\deg H\), the map \[\varphi_H(f) \,:=\, \int_{\mathbb{T}}f(z)H(z)^*H(z)\,dm(z)\] belongs to \(\mathcal{A}_{d,n}\). Thus \[\lim_{{\tt{d}}\to\infty} \operatorname{dist}_{H}^{\rho_n} \left( \mathcal{A}_{{\tt{d}},n}, \mathcal{Y}_{n}) \right) \,=\, 0.\]
Now by Lemma 10, there exists a row polynomial \(Q(z) = \begin{pmatrix} q_{1}(z) & \cdots & q_{n}(z) \end{pmatrix}\) such that \(Q^{*}Q\) is normalized and the UCP map \[\varphi_{Q}(f) \,:=\, \int_{\mathbb{T}} f(z) Q(z)^{*}Q(z) \, dm(z)\] satisfies \(\rho_{n}(\psi_{P}, \varphi_{Q}) < \varepsilon /3.\)
Now by Lemma 11, for any \(N > \max_{j} \deg q_{j} +1,\) there exists a row polynomial \(\widetilde{Q}(z) = \begin{pmatrix} \tilde{q}_{1}(z) & \cdots & \tilde{q}_{n}(z) \end{pmatrix}\) with normalized \(\widetilde{Q}^{*} \widetilde{Q}\) satisfying \[\max_{j} \deg \tilde{q}_{j} = N-1, \quad \text{and} \quad \gcd (\tilde{q}_{1}, \ldots, \tilde{q}_{n}) = 1,\] and the UCP map \[\varphi_{Q}(f) \,:=\, \int_{\mathbb{T}} f(z) \widetilde{Q}(z)^{*} \widetilde{Q}(z) \, dm(z)\] satisfies \(\rho_{n}(\varphi_{Q}, \varphi_{\widetilde{Q}}) < \varepsilon /3.\) Note that \(\varphi_{\widetilde{Q}} \in \mathcal{P}_{N,n}.\)
Finally we get \(\rho_n(\varphi, \varphi_{\widetilde{Q}}) < \varepsilon\) for some \(\varphi_{\widetilde{Q}} \in \mathcal{P}_{N,n}.\) This proves the desired Hausdorff convergence. ◻
Let \(\mathcal{S}_{1}\) and \(\mathcal{S}_{2}\) be two unital operator systems. Suppose that \(L_i: \mathcal{S}_{i} \to [0,\infty] ,\) \(i=1,2\) are seminorms satisfying \(L_i(\lambda 1_{\mathcal{S}_{i}})=0\) for all \(\lambda\in\mathbb{C}\).
For \(n\geq 1\) and \(i=1,2,\) define \[\rho_{\mathcal{S}_{i},n} (\varphi,\psi) \,=\, \sup\{\|\varphi(a)-\psi(a)\| \;:\;a \in \mathcal{S}_{i}, \, L_{i}(a)\leq 1\},\] for \(\varphi,\psi\in {\rm UCP}(\mathcal{S}_{i},M_n).\)
Assume there are unital maps \[R_{1} : \mathcal{S}_{1} \to \mathcal{S}_{2}, \qquad R_{2} : \mathcal{S}_{2} \to \mathcal{S}_{1}\] such that \(R_{1}\) is UCP, \(R_{2}\) is unital and linear, and for some \(\varepsilon > 0\), \[\begin{align} {2} L_{2}(R_{1}a) &\leq L_{1}(a), \qquad & a &\in \mathcal{D}_{1}, \\[4pt] L_{1}(R_{2}b) &\leq L_{2}(b), \qquad & b &\in \mathcal{D}_{2}, \\[4pt] \|b-R_{1}R_{2}b\| &\leq \varepsilon L_{2}(b), \qquad & b &\in \mathcal{D}_{2}. \end{align}\] where \(\mathcal{D}_{1}\) and \(\mathcal{D}_{2}\) are dense subspaces of \(\mathcal{S}_{1}\) and \(\mathcal{S}_{2}\) respectively. The following lemma follows from a straightforward calculation as a consequence of the above estimates.
Lemma 12. Given \(\varphi\in {\rm UCP}(\mathcal{S}_{2},M_n),\) define \[\tilde{\varphi} \,=\, \varphi\circ R_{1} \in {\rm UCP}(\mathcal{S}_{1},M_n).\] Then, for every \(\varphi,\psi \in {\rm UCP}(\mathcal{S}_{2},M_n),\) one has \[\label{eq:first32comparison} \rho_{\mathcal{S}_{1}, n} (\tilde{\varphi}, \tilde{\psi}) \,\leq\, \rho_{ \mathcal{S}_{2}, n} (\varphi,\psi),\tag{4}\] and \[\label{eq:second32comparison} \rho_{\mathcal{S}_{2}, n}(\varphi,\psi) \,\leq\, (1+\varepsilon) \rho_{\mathcal{S}_{1}, n}(\tilde{\varphi}, \tilde{\psi}) + 2 \varepsilon.\tag{5}\]
We now apply these estimates in the setting of finite Toeplitz systems. Let \(\mathcal{S}_{1} = C(\mathbb{T})\) with its usual Lipschitz seminorm \(L(f) = {\rm Lip}(f),\) and let \(\mathcal{S}_{2} = \mathcal{T}_{\tt{d}}\) be the operator system of \({\tt{d}}\times {\tt{d}}\) Toeplitz matrices, equipped with Connes’ truncated seminorm \(L_{\tt{d}}\), see Subsection 2.4 for the definition.
Recall that \(P_{{\tt{d}}}\) is the orthogonal projection onto the subspace spanned by the orthonormal set \(\{ e_{1}, \ldots, e_{{\tt{d}}}\},\) where \(e_{k} (z) = z^{k}.\) Define \(R_{{\tt{d}}} : C(\mathbb{T}) \to \mathcal{T}_{\tt{d}}\) by \(f \mapsto P_{{\tt{d}}} f P_{{\tt{d}}} .\) Clearly \(R_{{\tt{d}}}\) is a UCP map. By [24], we have \[L_{{\tt{d}}}(R_{{\tt{d}}} f) \, \leq\, {\rm Lip}(f) \qquad \text{for all } f \in C^{\infty} (\mathbb{T}).\] Define \(S_{\tt{d}}:\mathcal{T}_d\to C(\mathbb{T})\) by \[S_{{\tt{d}}}(T) (z) \,=\, \frac{1}{{\tt{d}}} \sum\limits_{i,j =1}^{{\tt{d}}} T_{i,j} z^{i-j},\] where \(T = (T_{i,j})_{i,j =0}^{{\tt{d}}-1}.\) Clearly \(S_{{\tt{d}}}\) is linear and unital. By [24], we have \[{\rm Lip} (S_{{\tt{d}}} T) \,\leq\, L_{{\tt{d}}}(T) \qquad \text{for all } T \in \mathcal{T}_{{\tt{d}}}.\] Moreover, by [24], there exists a sequence of positive real numbers \(\{\varepsilon_{{\tt{d}}}\}\) converging to \(0\) such that \[\| T - R_{{\tt{d}}} S_{{\tt{d}}} (T) \| \, \leq\, \varepsilon_{{\tt{d}}} L_{{\tt{d}}} (T) \qquad \text{for all } T \in \mathcal{T}_{{\tt{d}}}.\]
Define \(E_{{\tt{d}},n} : \mathcal{X}_{{\tt{d}},n} \to \mathcal{Y}_{n}\) by \(\varphi \mapsto \varphi \circ R_{{\tt{d}}}.\)
Corollary 2. For every fixed \(n\geq1\), \[\rho_{n} (E_{{\tt{d}},n}(\varphi), E_{{\tt{d}},n}(\psi)) \, \leq\, \rho_{{\tt{d}},n} (\varphi,\psi),\] and \[\rho_{{\tt{d}},n}(\varphi,\psi) \leq (1+\varepsilon_{\tt{d}})\, \rho_{n}(E_{{\tt{d}},n}(\varphi), E_{{\tt{d}},n}(\psi)) + 2\varepsilon_{\tt{d}}.\]
Proof. We first compare the metric \(\rho_{d,n}\) with the pulled-back metric from \(\mathcal{Y}_{n}.\)
For \(\varphi,\psi\in \mathcal{X}_{{\tt{d}},n} ,\) we have \[\rho_n( E_{{\tt{d}},n}(\varphi), E_{{\tt{d}},n}(\psi)) \, \leq\, \rho_{{\tt{d}},n} (\varphi,\psi),\] and \[\rho_{{\tt{d}},n}(\varphi,\psi) \leq (1+\varepsilon_d) \rho_n (E_{{\tt{d}},n} (\varphi), E_{{\tt{d}},n}(\psi)) + 2\varepsilon_d.\]
Let \(D_n=\operatorname{diam}(\mathcal{Y}_{n},\rho_n).\) This is finite because \(\mathcal{Y}_{n}\) is compact in the metric \(\rho_n ,\) see [19]. Thus we have \[0 \,\leq\, \rho_{{\tt{d}},n} (\varphi,\psi) - \rho_n (E_{{\tt{d}},n} (\varphi), E_{{\tt{d}},n}(\psi)) \,\leq\, \varepsilon_{\tt{d}}D_n + 2\varepsilon_{\tt{d}}.\] Hence the distortion of \(E_{\tt{d}}: \mathcal{X}_{{\tt{d}},n} \to \mathcal{Y}_{n}\) satisfies \[\operatorname{dis}(E_{\tt{d}}) \,:=\, \sup_{\varphi, \psi\in \mathcal{X}_{{\tt{d}},n}} \left| \rho_{{\tt{d}},n}(\varphi,\psi) - \rho_n(E_{\tt{d}}(\varphi), E_{\tt{d}}(\psi)) \right| \, \leq \, \varepsilon_{\tt{d}}D_n + 2\varepsilon_{\tt{d}},\] Since \(\varepsilon_d\to0\), we have \(\operatorname{dis}(E_{\tt{d}})\to 0.\)
Now we use the standard Gromov–Hausdorff estimate. Suppose that \(F:X \to Y\) is a map between compact metric spaces with distortion at most \(\delta .\) Suppose moreover that for every \(y\in Y\), there exists \(x\in X\) such that \(d_Y(y,F(x))\leq \eta .\) Then \[d_{GH}(X,Y)\leq \eta+\frac{\delta}{2},\] see [35].
Apply this estimate with \[X \,=\, \mathcal{X}_{{\tt{d}},n}, \qquad Y \,=\, \mathcal{Y}_{n}, \qquad F \,=\, E_{\tt{d}}.\] Let \(\eta_{{\tt{d}}}\) be the Hausdorff distance from \(E_{{\tt{d}}} (\mathcal{X}_{{\tt{d}},n})\) to \(\mathcal{Y}_{n}.\) Then \[\sup_{\psi \in {\mathcal{Y}_{n}}} \inf_{\varphi \in \mathcal{X}_{{\tt{d}},n}} \rho_{n} (\psi, E_{{\tt{d}}} (\varphi)) \, \le \, \eta_{{\tt{d}}}.\] This means that for every \(\psi \in \mathcal{Y}_{n},\) there exists some \(\varphi \in \mathcal{X}_{{\tt{d}},n}\) such that \[\rho_{n} (\psi, E_{{\tt{d}}} (\varphi)) < \eta_{{\tt{d}}} + \epsilon,\] for arbitarily small \(\epsilon >0.\) In particular, we can take \(\epsilon = \eta_{{\tt{d}}}.\) Therefore \[d_{GH}\bigl( ( \mathcal{X}_{d,n},\rho_{d,n}), (\mathcal{Y}_n,\rho_n)\bigr) \,\leq\, 2 \eta_d +\frac{1}{2}\operatorname{dis}(E_d) \longrightarrow 0.\] This proves the theorem. ◻
Acknowledgments. The first author would like to thank her research supervisor, Prof. Tirthankar Bhattacharyya, for useful discussions. The second author thanks Dr. Poornendu Kumar for suggestions that helped improve the presentation of the paper.