Rational Cuntz states peak on the free disk algebra

We apply realization theory of noncommutative (NC) rational multipliers of the Fock space, or free Hardy space of square–summable power series in several noncommuting variables to the convex analysis of states on the Cuntz algebra. We show, in particular, that a large class of Cuntz states that arise as the “NC Clark measures” of isometric NC rational multipliers are peak states for Popescu's free disk algebra in the sense of Clouâtre and Thompson.


Introduction
This paper applies non-commutative (NC) analysis to a question in non-commutative convexity.Our main result constructs states on the Cuntz algebra, O d , which peak at non-commutative rational inner functions in Popescu's free disk algebra, A d .We also provide a novel characterization of unital quantum channels in terms of NC rational inner functions.
Non-commutative convexity first appeared in the seminal work of Arveson [5].In [5], Arveson extended classical Choquet theory to a non-commutative, operator-algebraic setting.Classical Choquet theory studies the extreme boundary of compact convex sets and representing measures.Namely, let K be a compact convex set; a probability measure µ on K represents a point x ∈ K if the restriction of the corresponding state from C(K), the continuous functions on K, to the space of continuous affine functions on K, is the functional of evaluation at x. Bauer characterized the extreme points of K as those that admit only one representing measure (necessarily δ x ).This characterization lends itself to an extension to a more general setting of unital subspaces 1 ∈ M ⊂ C(X), where X is a compact Hausdorff space.The Choquet boundary of such a subspace is the collection of all points x ∈ X, such that δ x | M admits a unique extension to C(X).One way to obtain points in the Choquet boundary is to find peak points.A point x ∈ X is an M −peak point, if there exists an f ∈ M , such that |f (x)| = f and |f (y)| < f for any y = x.In particular, if 1 ∈ A ⊂ C(X) is a uniform algebra, and X is metrizable, then an important result of Bishop states that the Choquet boundary of A is precisely the set of peak points of A.
The field of non-commutative convexity has expanded quickly with contributions from Wittstock, Effros and his collaborators, and many others.The reader is referred to the monograph of Davidson and Kennedy [16] and the references therein for further details on non-commutative convexity theory.Of particular interest is the non-commutative Choquet theory first introduced and developed by Arveson [2][3][4][5].Suppose B is a unital C * -algebra and 1 ∈ A ⊂ B is an operator algebra.In this case, we say that an irreducible representation π : B → B(H) is a boundary representation if π| A has a unique extension to B. The image of the direct sum of all boundary representations gives the C * −envelope, C * min (A) of A. This is a C * −algebra in which A embeds, completely isometrically, and it is universal and minimal in the sense that if A embeds completely isometrically into any C * −algebra, B, then there is a * −homomorphism from B onto C * min (A) which intertwines the embeddings.Existence and construction of the C * −envelope via boundary representations was a long-unsolved problem in operator algebra theory until it was resolved in full generality by Davidson and Kennedy [2,5,15,22,26].Arveson also introduced the concept of a peaking representation in [3].His ideas were further extended by Clouâtre [11], Clouâtre and Thompson [12,13], and Davidson and Passer [18].In particular, Clouâtre and Thompson propose to study 'peaking states'.Precise definitions and details on peaking states can be found in Subsection 2.3.Clouâtre and Thompson [13] suggest that studying peaking phenomena on the Cuntz algebra will be interesting.We will provide a family of examples of peaking states and corresponding representations in this setting.
In more detail, we will study states on the Cuntz algebra O d , which peak at elements of Popescu's free or non-commutative disk algebra, A d .Here, E d := C * {I, L 1 , • • • , L d }, is the Cuntz-Toeplitz algebra, the unital C * −algebra generated by the left creation operators, L k , on the full Fock space, H 2 d .Here, H 2 d can be defined as the Hilbert space of complex square-summable power series in several NC variables, equipped with the 2 −inner product of the complex power series coefficients.In this viewpoint the L k = M L z k act as isometric left multiplications by the d independent formal NC variables, z = (z 1 , • • • , z d ).This algebra contains the compact operators, K (H 2 d ), on H 2 d and the Cuntz algebra is then defined as the quotient C * −algebra, is the unital norm-closed algebra generated by the left creation operators.Moreover, A d can be identified, completely isometrically with Alg{I, S 1 , • • • , S d }, where the S j denote the generators of the Cuntz algebra.
Our primary tools are free analysis and non-commutative function theory.These are rapidly growing fields in modern analysis.Free or non-commutative analysis was initially motivated by the study of analytic functional calculus of several commuting and non-commuting operators, as pioneered by Taylor [46,47], Voiculescu's free probability theory [48,49], and Takesaki's extension of Gelfand duality to arbitrary, noncommutative C * −algebras in [45].Popescu [37,39,41,42] studied non-commutative functions to extend the classical Sz.Nagy-Foias theory of dilations and von Neumann's inequality to the multivariable setting.He introduced the full Fock space as an analog of the classical Hardy Hilbert space of analytic function in the complex unit disk, D. We provide the necessary definitions and background theory in Subsection 2.1.
A particularly well-studied class of NC functions is the set of all non-commutative rational functions.These functions arise naturally in many different branches of pure and applied mathematics, such as the theory of localizations and quasideterminants in non-commutative rings [1,14,25], the theory of formal languages [9,27], and systems theory [6,7].This paper will focus on non-commutative rational functions that are elements of the weak operator topology (WOT)-closed algebra generated by the left creation operators on the full Fock space.Such functions were extensively studied in [31,32].In this paper, there is a critical interplay between non-commutative rational inner functions, i.e.NC rational functions which define isometric (inner) multipliers on the Fock space, states on the Cuntz algebra which peak on the free disk algebra, and finite-dimensional row coisometries.A row contraction is a contractive linear map from several copies of a Hilbert space into one copy.The necessary background and details are in Subsection 2.2.
Given an irreducible and finite-dimensional row coisometry, T = (T 1 , • • • , T d ), and a unit vector x, we can define a linear functional on A d via µ(f ) = x, f (T )x .One can apply a Gelfand-Naimark-Segal construction to (µ, A d ), and this yields a GNS-Hilbert space, H 2 d (µ) and a * −representation of the Cuntz-Toeplitz C * −algebra, π µ .Since T is a row-coisometry, one can show that Π µ := π µ (L) is a Cuntz, i.e. surjective row isometry, so that µ admits a unique extension to a state on O d , μ by [33,Proposition 5.11].This state, μ, is a finitely-correlated state, as introduced by Bratteli and Jørgensen [10].Our main result is Theorem 3.5, which states that every such finitely-correlated state is an A d -peak state.In particular, by results of Clouâtre, this implies that every such state is an exposed extreme point of the state space of the non-commutative or free disk operator system.The proof of the theorem constructs a non-commutative rational inner, such that the state peaks at it.However, this inner is constructed from , the coordinate-wise transpose of T .This duality leads us to Theorem 3.8, which shows that a non-commutative rational inner b arises from a quantum channel if and only if the 'transpose' b t , obtained by reversing the order of all products in monomials in the power series of b, is an inner as well.En route to these results, we obtain some results on spectra of non-commutative rational functions regular at the origin, which refine and extend our previous results in [32].

Non-commutative Hardy space and its multipliers
Given d ∈ N, the full Fock space is the Hilbert space direct sum H 2 d = ⊕ ∞ n=0 C d ⊗n , with the usual convention of C d ⊗0 ∼ = C.We will call H 2 d the free or non-commutative (NC) Hardy space as it has many properties in common with the classical Hardy space, H 2 , of square-summable power series in the complex unit disk.In particular, the elements of H 2 d can be viewed as NC functions on the NC unit ball The interpretation of H 2 d as a space of non-commutative functions was first considered by Popescu in [41].A natural way to see H 2 d as a space of functions is by noting that H 2 d is the completion of the free algebra, C z 1 , . . ., z d = C z , of NC or free polynomials, with respect to the inner product that makes the monomials orthonormal.The NC monomials correspond to words in the alphabet {1, . . ., d}.Given a word If the word is empty, we define z ∅ := 1.We will also denote the length of α = i 1 • • • i n by |α| = n.For a d-tuple of operators T = (T 1 , . . ., T d ), we set T α = T i1 • • • T in and T ∅ = I.Note that for any free polynomial, p ∈ C z , we can evaluate p on any d-tuple of matrices n and obtain a function with the following properties: (iii) respects similarities: for every X ∈ C d n and S ∈ GL n , we have The elements of H 2 d , thus, can be viewed as power series in non-commuting variables.The power series converge uniformly and absolutely on all tuples of matrices of norm ≤ r, for every 0 < r < 1.The properties above hold, except that the third property needs to be modified to hold only if S −1 XS ∈ B d N .Therefore, we obtain a space of NC functions on B d N .In direct analogy with classical Hardy space theory, (left) multiplication by any of the d independent variables define isometries, L j = M L zj on H 2 d with pairwise orthogonal ranges.The 'row' operator, is then a row isometry, i.e. an isometry from several copies of a Hilbert space into one copy.The operators L j are called the left creation operators or the left free shifts.The WOT-closed algebra H ∞ d , the free Hardy algebra, generated by the L j was extensively studied by Popescu [39,41,42] and .The free Hardy algebra is completely isometrically isomorphic to the algebra of all uniformly bounded NC functions on B d N with the uniform norm.
The space H 2 d is an NC reproducing kernel Hilbert space (RKHS) in the sense of Ball, Marx, and Vinnikov [8].The free Hardy algebra can then be identified as the algebra of left multipliers of this NC-RKHS.d and an outer NC function is one that defines a left multiplier with a dense range.Recall that the free disk algebra, A d , is the unital norm-closed algebra generated by the left creation operators on the full Fock space.We assume throughout that d ≥ 2. This algebra is a very close NC analogue of the classical disk algebra, A 1 = A(D).The Cuntz algebra is the universal C * -algebra of a surjective row isometry.That is, if S = (S 1 , . . ., S d ) denotes the row isometry of generators of the Cuntz algebra, then • is an operator system that embeds, completely isometrically, into O d , but does not coincide with the Cuntz algebra.In particular, A d is not an algebra or a C * −algebra.Namely, by [40,Theorem 3 • are then completely isometrically isomorphic.For the remainder of the paper, we identify A d with A d (S) and A d with A d (S) so that the free disk algebra and the free disk system are viewed as subspaces of the Cuntz algebra, . This property enables one to perform a Gelfand-Naimark-Segal (GNS)-type construction directly from positive linear functionals on A d .Jury and the first author [28] have developed an NC extension of the classical Alexandrov-Clark measure theory.In this NC theory, the positive linear functionals on the free disk system, denoted by (A † d ) + , play the role of positive measures on the unit circle.There is, in particular, a one-to-one correspondence between states on A d and contractive functions in H ∞ d that vanish at 0. We will be primarily interested in the positive finitely-correlated states or 'NC measures' which arise as the 'NC Clark measures' of NC rational multipliers of the Fock space, and we will introduce these in more detail in the following section.

Non-commutative rational functions
The theory of non-commutative rational functions has been developed independently in pure and applied disciplines ranging from pure algebra to computational and systems theory.In particular, the algebra of all NC rational functions is the free skew field as constructed by Amitsur [1] and Cohn [14] and is denoted by C < ( z > ).(In NC algebra, Amitsur and Cohn proved that this 'free skew field' is the universal 'field of fractions' of the free algebra, C z , of free or non-commutative complex polynomials in the d NC variables, ) The domain of an NC rational function is roughly the largest collection of d-tuples of matrices to which our NC rational function can be continued.We will denote the domain of r by Dom r.This paper will focus on NC rational functions that are defined and bounded on B d N .This assumption simplifies much of the theory.The interested reader should consult [34,35,50] and the references therein for more detail on the theory of NC rational functions.
Since 0 ∈ B d N , we will assume that our NC rational functions always have 0 in their domains, and we will denote the algebra of all such NC rational functions by C 0 < ( z > ).This assumption simplifies the definition of an NC rational function somewhat.We will say that an NC rational function in One should note that this is not the original definition of an NC rational function but rather a result in the sense that an NC rational function is usually defined as a certain equivalence class of valid 'NC rational expressions' obtained by applying the arithmetic operations '+, •, and, −1 ', to the free algebra, C z .One can then prove that any such NC rational function in C 0 < ( z > ) obeys a 'realization formula' as in Equation (2.1) above.Such a triple, (A, b, c) ∈ C d n × C n × C n is called a descriptor realization of the NC rational function r.For every NC rational function, there exist many such descriptor realizations.However, there exists one with n minimal, the minimal realization.The article "the" is justified because two realizations of r with minimal n are jointly similar.Namely, given two minimal realizations (A, b, c) and where F d denotes the free monoid of all words in the d letters {1, • • • , d}.We will write L A (z) := I − d j=1 z j Ãj whose inverse appears in the above expression.This object, L A (Z), is called a linear pencil (it is affine linear).It is a result of Vinnikov-Kaliuzhnyi-Verbovetskyi [34] and Volčič [50] that the domain of our function r can be described as the collection of all Z, such that det L A (Z) = 0, where A comes from the minimal realization.
NC rational functions in H 2 d and H ∞ d were studied by Jury, and the authors in [31].In fact, r ∈ H 2 d if and only if the tuple A = (A 1 , . . ., A d ) appearing in its minimal realization has joint spectral radius strictly less than 1.Here, the joint spectral radius of A was defined by Popescu [43] as a natural multivariate analogue of Beurling's spectral radius formula, It further follows, by Popescu's multi-variable Rota-Strang theorem, that r ∈ H 2 d if and only if it has a row ball of radius strictly greater than 1 in its domain [43].In particular, if this is the case, then r ∈ A d ⊂ H ∞ d .See [31, Theorem A] for several characterizations equivalent to membership of an NC rational function in the full Fock space.
Every NC rational contractive function can be associated (essentially) uniquely to an NC Clark measure, i.e., a positive functional on A d .Jury and the authors in [32] characterized such linear functionals or 'NC rational Clark measures'.It turns out that the NC Clark measures that arise from inner NC rational functions in H ∞ d with r(0) = 0 are precisely the finitely-correlated states studied by Bratteli and Jørgensen [10] and later by Davidson, Kribs, and Shpigel [17].Here, note that an NC Clark measure, µ b , corresponding to any b ∈ [H ∞ d ] 1 can be any positive linear functional on the free disk system, and this functional will be a state, i.e. µ b (I) = 1, if and only if b(0) = 0.Moreover, [32,Theorem 4.1] provides a complete description of all NC rational inners in terms of finite-dimensional row coisometries.Since we will apply this description, we will recall it: Let b be a contractive NC rational function in the unit row-ball with b(0) = 0. (Again, b(0) = 0 ensures that its NC Clark measure, µ b , is a state.)Then, such an b is inner if and only if there exists a row coisometry and a unit vector x ∈ C n , such that x is T and T * −cyclic and b is given by the realization formula, Here we set H 0 = α =∅ T * α x, P 0 is the orthogonal projection on H 0 , and for 1 ≤ j ≤ d, T * 0,j = T * j (I − xx * )| H0 .In particular, if T is irreducible (the co-ordinate matrices of T generate C n×n as an algebra), then every unit vector will give rise to such a realization and H 0 = C n .This form of realization is called a Fornasini-Marchesini (FM) realization.More generally, an FM realization is one of the form Note that D = r(0), so that if we assume that r(0) = 0, we obtain the preceding form of Equation (2.2).We will denote the FM realization as (A, B, C, D).One can pass from a descriptor realization to an FM one quite easily.The following lemma is well-known to experts, but since we do not have a reference, we chose to include it.
obtained as the component-wise adjoint of Z.The row operator row(Z t ) is defined similarly.We will also consider the operation (•) : C n → C n , defined by x → x, where x denotes entry-wise complex conjugation with respect to the standard basis of , where Z j := (•)•Z j •(•), so that Z j is obtained by entry-wise complex conjugation of the matrix Z j and Z = (Z * ) t = (Z t ) * is a row d−tuple of matrices.
The following lemmas give us a useful condition on the spectra of NC rational functions with the origin in their domains.Proof.Since the claim is symmetric, we will prove only the forward implication.Let us assume that 0 = v ∈ ker(A − BD −1 C).Then, 0 = Av = BD −1 Cv.In particular, D −1 Cv = 0. Therefore, To prove the last part of the claim, we let Φ = D −1 C| ker(A−BD −1 C) .As we saw above, Φ is well-defined and injective.Similarly, let Ψ = A −1 B| ker(D−CA −1 B) .Then, for every v ∈ ker(A − BD −1 C), we have that Similarly, in the other direction.Hence, we obtain our isomorphism.
, where If v is an eigenvector of Z ⊗ A (λ) corresponding to the eigenvalue 1, then I ⊗ Cv is an eigenvector of r(Z) with corresponding eigenvalue 1.
Proof.Consider the following matrix Note that one Schur complement is The other is The claim now follows from the preceding lemma.
The previous proposition extends and refines [32,Proposition 5.5].In [32, Proposition 5.6], we applied these spectral results to prove that any NC rational inner, b, has eigenvalues of modulus 1 when evaluated at certain points on the boundary of the unit row-ball, B d N .Here, recall that NC rational inner multipliers of the Fock space, b, are in bijective correspondence with pairs (T, x), where T ∈ C d n is a row coisometry and x is any vector which is both T and T * cyclic.In particular, if b = b T,x is the unique NC rational inner corresponding to such a pair, then for any ζ ∈ ∂D, the NC rational inner ζb corresponds to a pair (T (ζ), x), where T (ζ) is a rank-one row coisometric perturbation of T , see Equation ( By vectorization, this happens if and only if there is a matrix, X, so that Here we have used the fact that T (ζ) is a row coisometry.

Peaking states and representations
In this section we follow the work of Clouâtre and Thompson in non-commuative convexity in operator algebra and operator system theory [13].Let A be a unital operator algebra and let B be a C * -cover of A.
Namely, we have a unital, completely isometric embedding ι : A → B, such that B = C * (ι(A)).Let us denote by K(B) the state space of B. Clouâtre and Thompson say that µ ∈ K(B) is A−peaking, or an A−peak state, if there exists a contraction a ∈ A, such that µ(a * a) = 1 and ν(a * a) < 1, for all ν ∈ K(B) \ {µ}.If µ is A-peaking, then it is quite immediate that µ is pure and µ has the unique extension property (UEP), i.e., the functional µ| A has a unique Hahn-Banach extension to B. In fact, any µ ∈ (A † d ) + with the property that its GNS row isometry is a Cuntz row isometry has the UEP by [33,Proposition 5.11].If π µ : B → B(H µ ) is a GNS representation of µ with cyclic vector ξ µ , then, by Cauchy-Schwarz, π µ (a)ξ µ = ξ µ .This implies that there is a finite-dimensional subspace F ⊂ H µ , such that P F π µ (a)| F = 1, where P F is the projection onto F .Similarly, in [11], if B is a C * −algebra and S ⊆ B is an operator system, a state, µ ∈ K(B) is said to be S −peaking if there is a self-adjoint element s ∈ S so that s = 1 and |λ(s)| < µ(s) = 1 for every λ ∈ K(B) \ {µ}.
An irreducible representation π : B → B(H π ) is called local A-peak by Clouâtre and Thompson, if there exists n ∈ N, T ∈ M n (A) := A ⊗ M n with T = 1, such that for every irreducible representation σ : B → B(H σ ) unitarily inequivalent to π and every finite-dimensional subspace G ⊂ H ⊕n σ , P G σ (n) (T )| G < 1.This implies, that there is a finite-dimensional subspace F ⊂ H ⊕n π , such that Example 2].One can say slightly more about the connection between the two notions.This is the goal of the following two lemmas.In particular, ϕ(a * a) = 1 and since it is a selfadjoint operator on a finite-dimensional space, there exists ξ ∈ G, such that ϕ(a * a)ξ = ξ.Since π(a * a) is a contraction, we have that π(a * a)ξ = ξ.Lemma 2.5.Let B be a unital C * -algebra and 1 ∈ A ⊂ B an operator algebra, such that B = C * (A).Let a ∈ A be a contraction and set Then, F a is a closed face of K(B).Moreover, if π : A → B(H) is local A-peak with witness a, then F a = ∅ and all the extreme points of F a are states arising from π. Lastly, ∂ e F a is in one to one correspondence with vectors η ∈ H, such that π(a * a)η = η.
Proof.Since a * a defines a weak*-continuous contractive real functional on (B sa ) * , then F a is the closed face where this functional attains its maximum on K(B).Let π be a local A-peak representation with witness a.
Then, by the previous lemma, there exists ξ ∈ G, such that π(a * a)ξ = ξ.Hence, the state µ(b) = ξ, π(b)ξ is in F a .Since F a is a closed face, its extreme points are pure states.Let ϕ ∈ F a be a pure state and (σ, K, η) be its GNS representation.Then, it is immediate that σ(a * a)η = η.Thus, if G ⊂ K is the space spanned by η and σ(a)η, then P G σ(a)| G = 1 and thus σ is unitarily equivalent to π In this paper, we are interested in the case when A = A d and B = O d .Since A d is semi-Dirichlet, we can say more about A d -peak states and connect them to peaking states for operator systems. 2 (1 + b), we may assume that b is invertible.Thus, b is factorizable in the sense of Popescu [38].Namely, there exists c ∈ H ∞ d , such that b = c * c.However, we do not know that c ∈ A d .This is true if we assume that b is the real part of an NC rational function.In this case, c is NC rational by the NC rational Fejér-Riesz theorem of [30,Theorem 6.5].However, we do not need this result for our purposes and only require the following observation to construct a large class of examples of Proof.Since b is inner, we have that: However, this contradicts the assumption that |ν(b)| < 1.

The above observation suggests the following strategy:
To construct examples of states on the Cuntz algebra which peak on the free disk algebra, we will show that if b ∈ H ∞ d is NC rational and inner, then there are finite points, A, on the boundary of the unit row-ball so that b(A) has 1 as an eigenvalue of multiplicity one.

Main result
A relationship between spectra and intertwiners is encoded in the following lemmas.Let Z ∈ C d n be an irreducible row contraction of row norm 1.
Then, there exists 0 = T ∈ B(C n , H), such that ψ Y,Z (T ) = T .We may assume that T = 1.Then, since n < ∞, there is a unit vector v ∈ C n , such that T v = 1 and hence Since T , Z, and Y are contractions, we conclude from the equality clause of Cauchy-Schwarz that Y * T v = (I ⊗ T )Z * v. Therefore, for every 1 ≤ n ≤ d.The same argument can be applied to the self-compositions This implies that v is cyclic.Therefore, for every w ∈ C n , there exists an NC polynomial p, such that p(Z * )v = w.Hence, for every We conclude that T is a homomorphism of C z -modules from C n to H.In particular, since the kernel of a homomorphism is a submodule.T must be injective.Thus, we have obtained the following lemma: We can do slightly better, assuming that both Z and Y are coisometries.Under the assumptions of the lemma, let T be the intertwinner obtained above.Then, However, since Z is irreducible, a result of Farenick [24,Theorem 2] implies that the map A → Z(I ⊗ A)Z * is irreducible and thus, by the Perron-Frobenius theorem for positive maps of Evans and Høegh-Krohn [23, Theorem 2.3], we know that there is a unique (up to scalar multiplication) eigenvector of this map that corresponds to eigenvalue 1.However, since Z is a coisometry, the corresponding map is a ucp.Hence, T * T is a scalar multiple of the identity of norm 1.Therefore, T * T = I and T is an isometry.We summarize Lemma 3.2.Let Z ∈ C d n be an irreducible row coisometry and let Y ∈ B(H) d be a row coisometry.If 1 ∈ σ p (ψ Y,Z ), then there exists a unique isometry V :

Note that we can canonically identify
Then, we can define an operator Now via identification, the right-hand map is Therefore, we get that the map ψ Y,X corresponds to the product n is an irreducible row co-isometry.By [32, Lemma 2.1], row(T t ) has joint spectral radius 1 and by [44,Lemma 4.10], row(T t ) is jointly similar to an irreducible row co-isometry, Z ∈ B d n .Fix a unit vector, x ∈ C n .Then there is a unique NC rational inner function, b = b T,x , corresponding to the pair (T, x) by [32,Theorem 4.1].Let S ∈ GL n be an invertible matrix, such that In particular, the dimension of the eigenspace corresponding to 1 of b(T t ) and b(Z) are the same.
Lemma 3.3.In the above setting, b(Z) has an eigenspace of dimension 1 corresponding to eigenvalue 1.
Proof.Consider the map ψ T (X) = d j=1 T j XT * j .This map corresponds to the tensor d j=1 T j ⊗ T j .Since the map is unital, we have that 1 ∈ σ d j=1 T j ⊗ T j .Moreover, 1 is the Perron-Frobenius eigenvalue of ψ T , and since T is irreducible, the dimension of the corresponding eigenspace is 1.Taking the adjoint, we get the tensor d j=1 T t j ⊗ T * j .By Lemma 2.3, we know that the dimension of the eigenspace corresponding to 1 of b(T t ) is 1, as well.The claim for Z now follows from the observation preceding the lemma.
Let y be a unit vector in this one-dimensional eigenspace of b(Z) to eigenvalue 1 and define the finitelycorrelated Cuntz state µ = µ Z,y ∈ (A † d ) + by µ(L ω ) := y * Z ω y.Let V denote the minimal row isometric dilation of Z on H ⊇ C n .By [17,Theorem 6.5], V is an irreducible Cuntz row isometry.Given any µ ∈ (A † d ) + , we can, as in [32], apply a Gelfand-Naimark-Segal (GNS) construction to (µ, A d ) to obtain a GNS-Hilbert space, H 2 d (µ), and a row isometry, Π µ , acting on H 2 d (µ), where H 2 d (µ) is defined as the completion of the free algebra, C z , modulo vectors of zero-length, with respect to the pre-inner product, p, q µ := µ(p(L) * q(L)).
Equivalence classes of free polynomials, p + N µ , where N µ denotes the left ideal of zero-length vectors with respect to • µ , are dense in H 2 d (µ) by construction.This construction also comes equipped with a row isometry, Π µ , defined by left multiplications by the independent variables on H then the pair (Z, V ) are jointly unitarily equivalent to (Z µ , Π µ ) by a unitary which sends y to 1 + N µ .In particular, H µ is finite-dimensional, Z µ is an irreducible row co-isometry, and Π µ is its minimal row isometric dilation, and this is irreducible and Cuntz.Hence, if we set ϕ(X) = d j=1 π(S j )XZ * j .Then, 1 ∈ σ p (ϕ).Therefore, by Lemma 3.2, we get that there exists a unique isometry V : C n → H, such that π(S) * V = (I ⊗ V )Z * .Theorem 3.5.Let T ∈ C d n be an irreducible row co-isometry and x ∈ C n a unit vector so that b = b T,x is the unique NC rational inner function corresponding to (T, x).Let Z be the irreducible row co-isometry which is jointly similar to row(T t ) via an invertible matrix, S, and let y ∈ C n be the unique unit eigenvector of b(Z) to eigenvalue 1.Then the finitely-correlated Cuntz state µ := µ Z,y ∈ (A † d ) + is an A d −peak state which peaks at b ∈ A d .
Proof.As described above, since µ(L ω ) := y * Z ω y and b(Z)y = y, where the unit vector, y, spans the eigenspace for b(Z) corresponding to eigenvalue 1, it follows that µ(b) = y * b(Z)y = 1.Note that µ is a pure Cuntz state since π µ is an irreducible representation of the Cuntz algebra.By the equality in the Cauchy-Schwarz inequality, it follows as before that π µ (b) and we conclude that b(Z µ )h 1 = h 1 .Since y is the unique (up to scalars) eigenvector of b(Z) to eigenvalue 1, 1 + N µ is the unique eigenvector of b(Z µ ) so that h 1 = α1 + N µ for some α ∈ C. It further follows that this is a pure isometry and we conclude that h 2 = 0.That is, 1 + N µ is the unique eigenvector (up to non-zero scalars) of b(Π µ ) to eigenvalue 1.Now let λ ∈ K(O d ) be another Cuntz state which peaks at b, λ(b) = 1.As before equality in the Cauchy-Schwarz inequality implies that b(Π λ )1 + N λ = 1 + N λ .By Lemma 3.4 and Lemma 3.2, there is a unique isometry, V : , and Π λ is a row-isometric dilation of Z, which must be minimal as Π λ is irreducible.Hence, by uniqueness of the minimal dilation, Π λ Π µ .Let U : H 2 d (λ) → H 2 d (µ) be the unitary implementing this equivalence, U Π α λ = Π α µ U .Then, to eigenvalue 1 so that by the previous arguments, U 1 + N λ = ζ(1 + N µ ) for some ζ ∈ ∂D.This proves that λ = µ so that µ = µ Z,y is an A d −peak state.
The following corollary follows immediately from the preceding theorem and Lemma 2.6.
Corollary 3.6.Every finitely-correlated state µ on A d that arises from an irreducible finite-dimensional row coisometry and a unit vector is an exposed extreme point of the state space of A d .
Corollary 3.7.If µ = µ T,x , for a finite irreducible row coisometry T and unit vector x, let Z be the unique irreducible finite row co-isometry which is jointly similar to row(T t ) and let y be the eigenvector of b T,x (Z) corresponding to the multiplicity one eigenvalue, 1.Then the finitely-correlated state µ Z,y peaks at the NC rational inner b T,x and µ T,x peaks at the NC rational inner b Z,y .
If we define the unital, completely positive map, then this map is a unital quantum channel, i.e.Ad T * ,T is also unital, if and only if row(T * ) is also a row coisometry, or, again equivalently, row(T t ) is a row coisometry.We require another definition to describe the class of NC rational inner functions associated with unital quantum channels.Let α Namely, α t is the reversal of α.We define a unitary on H 2 d by (z α ) t = z α t .This unitary is important in realization theory of NC rational functions [31].It is proved in [32,Lemma 2.2] that if r is an NC rational function, so is r t .However, it need not be the case that if b is an NC rational inner, that b t is inner.A simple example is b(z) = (1 + z 1 )z 2 .It is an immediate calculation, that b(L) * b(L) = I.However, b t (z) = z 2 (1+z 1 ).Here the inner part of b t (L) is L 2 , and the outer part is 1+L 1 .In particular, it is easily checked that b t (L) = √ 2 so that b t ∈ H ∞ d is not even contractive, see [29,Example 3.4].The following theorem identifies the class of all NC rational functions b, such that b t is also inner, as precisely those that arise from quantum channels.Theorem 3.8.Let b := b T,x be the NC rational inner generated by the pair (T, x), where T is a finitedimensional row co-isometry on H and x ∈ H is both T and T * −cyclic.Then, row(T t ) is also a row co-isometry if and only if b t is also NC rational inner and in this case b t = b row(T t ),x .If x is a unit vector and both T and T t are irreducible row coisometries then the NC rational Clark states µ T,x and µ T t ,x peak at the NC rational inners b t and b, respectively.This shows that G t has the minimal descriptor realization (T , x, x), where T t * = T .If row(T t ) is also finite row coisometry then by [32,Theorem 4.1], it follows that b t is also NC rational inner with minimal FM realization: A j := T j (I − xx * ), B j := T j x, C := x * , and D := b t (0) = 0.
Conversely, as above, given any NC rational inner b = b T,x , a minimal descriptor realization of (1 − b t ) −1 = G t is given by (T , x, x).Assuming that b t is also NC rational inner, [32, Theorem 3.2, Remark 3.4] implies that there is a finite row co-isometry, W , and a vector, y, which is both W and W * −cyclic so that (W, y, y) is also minimal descriptor realization of G t so that for any word, ω ∈ F d , y * W * ω y = x * T ω x.
Equivalently, if T t row is the row-norm of row(T t ), then for any word, ω ∈ F d , 1 If row(T t ) > 1, then 1 T t row row(T t ) and 1 T t row W are both row contractions.In either case, [32, Proposition 3.6, Lemma 3.9] implies that row(T t ) and W are jointly unitarily equivalent via a unitary U which sends y to x. Hence row(T t ) is a row coisometry.It is easily checked that row(T t ) and row(S t ) are both row coisometries so that, by the previous theorems, if x is any unit vector then b T,x , b t T,x = b row(T t ),x and b S,x , b t S,x are all NC rational inner, µ T,x , µ row(T t ),x and µ S,x , µ row(S t ),x are all Cuntz states which peak at b t T,x , b T,x , b t S,x and b S,x , respectively.The following example illustrates what happens if we drop the assumption that T is irreducible but require still that its minimal isometric dilation is irreducible.
Example 3.10.Consider the following coisometry: Let us write e 1 , e 2 , and e 3 for the vectors of the standard basis of C 3 .I is now easy to check that T 1 e 1 = − 1 2 (e 1 + e 2 + e 3 ), T 2 e 1 = 1 2 (e 1 − e 2 − e 3 ), T 2 T 1 e 1 = 1 2 e 2 .This implies that e 1 is T -cyclic.Similarly, T * 1 e 1 = − 1 2 (e 1 + e 3 ), T * 2 e 1 = 1 2 (e 1 − e 2 ), and T * 1 T * 2 e 1 = − 1 2 e 3 .This implies that e 1 is also T * -cyclic and, moreover, that α =∅ T * α e 1 = C 3 .Therefore, by [32], the following NC rational function is an inner ).Therefore, we have the following expression for the pencil Since we know T * 1 e 1 and T * 2 e 1 , we conclude that the expression for our function is In particular, we have that r(−1, 0) = 1.Now we observe that T * 1 and T * 2 have a common eigenvector.In fact, set Then, We note that r(T t ) has eigenvalue 1 of multiplicity 1.The corresponding eigenvector is e 1 + e 3 .The above calculation shows that only the semi-simple part of T t is in the closed ball.The similarity orbit of T t itself never intersects the closed ball.
d j=1 S j S * j = I.The quotient map from the Cuntz-Toeplitz algebra, E d onto O d restricts to a complete isometry on A d , and since the Cuntz algebra is simple, it is the C * -envelope of A d .Classically, C(T) is the C * -envelope of A(D).Moreover, C(T) = A(D) + A(D) * .In the NC setting, A d := (A d + A * d ) −

:
By [50, Theorem 3.5], the domain of r is the complement of the singularity locus of the linear pencil L Â(Z ), where ( Â, b, c) is a minimal descriptor realization of r.A minimal FM realization, (A , B , C , D ) of r can then be constructed by setting H := ω =∅ Âω c, A := Â| H , B := Âc, C = (P H b) * and D := r(0).By uniqueness of minimal realizations (uniqueness also holds for minimal FM realizations), we can assume that (A , B , C , D ) = (A, B, C, D).Since H has codimension at most 1, if H H, then Â decomposes asÂ = A * 0 a , where a ∈ C d .Again, by [50, Theorem 3.5], if Z ∈ Dom r then 0 = det L Â(Z ) = det(L A (Z))det L a (Z), so that det L A (Z) = 0. Conversely if det L A (Z)is not 0 for some Z ∈ C d n , then r(Z) is well-defined as the transfer function,DI n + I n ⊗ CL A (Z) −1 Z ⊗ B, so that Z ∈ Dom r.If r ∈ C 0 < ( z > ) with minimal descriptor realization (A, b, c), we will allow the domain of r to include d−tuples of operators in an infinite dimensional Hilbert space.We will denote such operator d−tuples byC d ∞ or B(H) d , where H is a separable Hilbert space.Namely, given Z = (Z 1 , • • • , Z d ) ∈ C d ∞ , we will say that Z ∈ Dom r if L A (Z) is invertible.It will be convenient to introduce some basic notations; we view any Z ∈ C d n , n ∈ N as a row d−tuple of operators, Z = (Z 1 , • • • , Z d ), Z j ∈ C n×n .Any such row defines a bounded linear map from C n ⊗ C d into C n .Hence, by Z * , we mean the 'column operator', Z * := C n → C n ⊗C d , obtained as the Hilbert space adjoint of the linear map Z.Similarly, Z t := row operator, Z, with respect to the standard bases of C n and C n ⊗ C d .We will, however, also have occasion to consider the row operator row(Z *

Lemma 2 . 2 .
Let H and K be Hilbert spaces.Let A ∈ B(H) and D ∈ B(K) be invertible and B ∈ B(K, H) and C ∈ B(H, K).Then, the Schur complement, A − BD −1 C, has a non-trivial kernel if and only if D − CA −1 B has a non-trivial kernel.Moreover, the map, D −1 C| ker(A−BD −1 C) : ker(A − BD −1 C) → ker(D − CA −1 B), is an isomorphism.
2.2) and[32, Theorem 4.1].Proposition 5.6 of[32] then states:Proposition.Given ζ ∈ ∂D, let A * ζ be a column-isometric restriction of T (ζ) * to an invariant subspace.Then ζ is an eigenvalue of b T,x (A t ζ ).There is a minor error in the proof of[32, Proposition 5.6].Fortunately, the proposition statement is still correct, and the proof can be readily fixed: Proof.( [32, Proposition 5.6]) We have that ζ ∈ ∂D ∩ σ(b(Z)) if and only if

Lemma 2 . 6 .
If µ ∈ K(O d ) is an A d -peak state, then there exists b ∈ A d positive, such that µ(b) = 1 and ν(b) < 1, for all states ν = µ.In particular, µ is an A d −peak state, µ| A d is a weak* exposed extreme point of the unit ball of (S * d ) sa and it has the unique extension property.Proof.Let µ ∈ K(O d ) be an A d -peak state.Then, there is a contraction a ∈ A d , such that µ(a * a) = 1 and for every state ν = µ, ν(a * a) < 1.However, since A is semi-Dirichlet, a * a ∈ A d .Hence, we are done.The second part follows from [11, Theorem 3.2].One could try and argue the converse claim.Let b ∈ A d be a positive element, such that µ(b) = 1 and for every ν ∈ K(O d ) \ {µ}, ν(b) < 1.By replacing b by 1