A UNIQUENESS RESULT CONCERNING SCHUR IDEALS

JAMES P. SOLAZZO

Abstract. A subset of the set of all positive semi-definite matrices of a given size which is invariant under Schur (componentwise) multiplication by an arbitrary positive semi-definite matrix is said to be a Schur ideal. A subset of k-dimensional complex space is said to be hyperconvex if it arises as the set of possible values (w_{1}, …, w_{k}) = (f(α_{1}), …, f(α_{k})) arising from restricting contractive elements f from some uniform algebra A to a finite set {α_{1}, …, α_{k}} in the domain. When the uniform algebra is the disk algebra, the hyperconvex set is said to be a Pick body. Motivated by the classical Pick interpolation theorem, Paulsen has introduced a natural notion of duality between Schur ideals and hyperconvex sets. By using some recently developed results in operator algebras (matricial Schur ideals), we show that each Pick body has a unique affiliated Schur ideal.

1. Introduction

The classical Nevanlinna-Pick problem is the following: given {α_{1}, ..., α_{k}} a k-tuple of distinct points with |α_{i}| < 1 and given {w_{1}, ..., w_{k}} a k-tuple of complex numbers what are the necessary and sufficient conditions for the existence of a function f, analytic in the open disk 𝔻, ∥f∥_{∞} ≤ 1 and such that f(α_{i}) = w_{i}? When such a function f exists, the function f is said to interpolate the given sequences {α_{i}} and {w_{i}}. This question was first answered by G. Pick in 1916 in his paper [11] where he states explicitly when one can interpolate the given sequences {α_{i}} and {w_{i}}. In 1919 (1929) R. Nevanlinna contributed to the study of interpolation by bounded analytic functions in his paper(s) [7] ([8]) by providing an inductive procedure which enables one to construct a bounded analytic function in 𝔻 which interpolates the given sequences {α_{i}} and {w_{i}}. We will now state Pick’s theorem.

Theorem 1. If α_{1}, ..., α_{k}, w_{1}, ..., w_{k} ∈ 𝔻, then there exists an analytic function f : 𝔻 → 𝔻 with f(α_{i}) = w_{i} if and only if the matrix

(\frac{1 − w̄_{i}w_{j}}{1 − ᾱ_{i}α_{j}} )_{i,j=1}^{k}

is positive semi-definite.

Fix α_{1}, ..., α_{k} in 𝔻. B. Cole, K. Lewis, and J. Wermer in [4] define a P ick body, denoted P(α_{1}, ..., α_{k}), as follows:

P(α_{1}, ..., α_{k}) = {w⃗ ∈ ℂ^{k} : ∃ f ∈ H^{∞}(𝔻), ∥f∥_{∞} ≤ 1, f(α_{i}) = w_{i}}.

This is precisely the set of all k-tuples (w_{1}, ..., w_{k}) in ℂ^{k} which satisfy the necessary and sufficient conditions of Pick’s theorem. It turns out that Pick bodies are a special class of sets contained a more general class of sets called “hyperconvex”. To this end, let X be a compact Hausdorff space and let C(X) denote the continuous complex-valued functions on X. We will call A ⊆ C(X) a uniform algebra provided that A is closed, contains the identity, and separates points in X. Fix x_{1}, ..., x_{k} in X. B. Cole, K. Lewis, and J. Wermer in [4], define an interpolation body, denoted D(A; x_{1}, ..., x_{k}), in the following way: a point w⃗ = (w_{1}, ..., w_{k}) in ℂ^{k} belongs to D(A; x_{1}, ..., x_{k}) if for each ϵ > 0, there exists f in A with ∥f∥_{∞} ≤ 1 + ϵ such that f(x_{i}) = w_{i}. The set D(A; x_{1}, ..., x_{k}) is the unit ball of the Banach algebra ℂ^{k} (under coordinate-wise multiplication) for some norm on ℂ^{k}. Sets in the closed k-polydisk arising as D(A; x_{1}, ..., x_{k}) for some A and x_{1}, ..., x_{k} in X are called hyperconvex in [4]. If we let A(𝔻) denote the analytic functions in 𝔻 which are continuous on 𝔻^{−} and let α_{1}, ..., α_{k} in 𝔻, then in [4] they show that the Pick body P(α_{1}, ..., α_{k}) = D(A(𝔻); α_{1}, ..., α_{k}).

Schur ideals are a relatively new concept which act as a natural “dual” object for hyperconvex sets. A precise definition of the Schur ideal will be given in Section 2. In general, there are often many Schur ideals that could serve as the “dual” of a hyperconvex set. In this paper we shall show that for this special class of hyperconvex sets (i.e, Pick bodies) there is a unique “dual” Schur ideal.

2. Preliminaries

Let A contained in C(X) be a uniform algebra and let I_{x} denote the ideal of functions vanishing at the points x_{1}, ..., x_{k} in X. Then the quotient algebra A = A/I_{x} is an operator algebra [3]. Since A separates points in X there exist functions f_{1}, ..., f_{k} in A with f_{i}(x_{j}) = δ_{i,j} where δ_{i,j} denotes the Kronecker delta. Thus in A we have that E_{i} = [f_{i}] are a family of k commuting idempotents satisfying E_{i}E_{j} = δ_{i,j}E_{j}, E_{1} + … E_{k} = 1, and which span A. Therefore, when studying the properties of an interpolation body, one can choose to work with the uniform algebra A or its corresponding quotient algebra (operator algebra) A since we have the following equalities:

D(A; x_{1}, ..., x_{k}) = {w⃗ ∈ ℂ^{k} : ∃ f ∈ A, ∥f∥ ≤ 1 and f(x_{i}) = w_{i}}^{−}
= {w⃗ ∈ ℂ^{k} : ∃ f ∈ A, ∥[f]∥ ≤ 1 and f(x_{i}) = w_{i}}
= {(w_{1}, ..., w_{k}) ∈ ℂ^{k} : ∥w_{1}E_{1} + … + w_{k}E_{k}∥ ≤ 1}.

For this reason we will abbreviate D(A; x_{1}, ..., x_{k}) as D(A) and often interchange the two without mention.

Recall if we let M_{n} denote the C^{∗}−algebra of n × n matrices and let M_{n}(A) denote the algebra of n × n matrices over A, then M_{n}(A) is endowed with the norm

∥F ∥ = sup{∥F (x)∥_{M_{m}} : x ∈ X},

where F = (f_{ij}) is in M_{n}(A). Moreover if x_{1}, ..., x_{k} in X and A = A/I_{x}, then M_{n}(A) is endowed with the quotient norm by identifying

M_{n}(A) = M_{n}(A)/M_{n}(I_{x}).

Now given points x_{1}, ..., x_{k} in X, the abstract interpolation problem is concerned with determining the set of k-tuples W⃗ = (W_{1}, ..., W_{k}) of n × n matrices, denoted

D_{n}(A; x_{1}, ..., x_{k}) = {W⃗ : ∃F ∈ M_{n}(A), ∥F ∥ ≤ 1, F (x_{i}) = w_{i}}^{−}
= {W⃗ : ∃F ∈ M_{n}(A), ∥[F]∥ ≤ 1, F (x_{i}) = w_{i}}
= {(W_{1}, ..., W_{k}) : ∥ ∑_{i=1}^{k} W_{i} ⊗ E_{i}∥ ≤ 1}.

Thus the sets {D_{n}(A)}, n = 1, 2, ..., are just a natural coordinatization of the closed unit balls of M_{n}(A) for an operator algebra A generated by k-idempotents satisfying the above relations.

Throughout the paper M_{k} will denote the k × k matrices with complex entries and M_{k}^{+} will denote the closed cone of positive semi-definite matrices in M_{k}. If P is in M_{k}, then it will be understood that P is the matrix (p_{ij})_{i,j=1}^{k}. If P, Q are in M_{k}, then the Schur product of the matrices P and Q, denoted as P ∗Q, is defined to be (p_{ij})∗(q_{ij}) = (p_{ij}q_{ij}).

The following theorem is due to Cole and Wermer [5], in which they give conditions on a point (w_{1}, ..., w_{k}) in ℂ^{k} in order that (w_{1}, ..., w_{k}) belong to the interpolation body D(A; x_{1}, ..., x_{k}) where A is an arbitrary uniform algebra.

Theorem 2. If A ⊆ C(X) is a uniform algebra and x_{1}, ..., x_{k} ∈ X, then there exists a set S ⊆ M_{k}^{+} such that

(w_{1}, ..., w_{k}) ∈ D if and only if ((1 − w̄_{i}w_{j})p_{ij}) ≥ 0 for all (p_{ij}) ∈ S.

In [9], V. Paulsen observes that Theorem 2 suggests the following “dualities” between closed subsets of the closed k-polydisk and subsets of M_{k}^{+}. Given a non-empty set S ⊆ M_{k}^{+} define

S^{⊥} = {(w_{1}, ..., w_{k}) ∈ ℂ^{k} : ((1 − w̄_{i}w_{j})p_{ij}) ≥ 0 ∀ (p_{ij}) ∈ S}.

Similarly, given a subset D of the closed k-polydisk with 0 ∈ D define

D^{⊥} = {(p_{ij}) : ((1 − w̄_{i}w_{j})p_{ij}) ≥ 0 ∀ (w_{1}, ..., w_{k}) ∈ D}.

Since we insist that 0 ∈ D then D^{⊥} is always a set of positive semidefinite matrices. Observing further that the set D^{⊥} has certain properties, V. Paulsen [9] introduces the concept of a Schur ideal. By studying this duality between hyperconvex sets and Schur ideals and using some results from the theory of abstract operator algebras, V. Paulsen is able to generalize J. Agler’s scalar-valued interpolation results [1] for the bidisk to more general product domains [9]. For another approach to such problems see the paper of A. Tomerlin [14].

Definition 1. Let I ⊆ M_{k}^{+} be a non-empty set. Then I will be called a Schur ideal provided that:

(1) A, B ∈ I ⇒ A + B ∈ I.

(2) A ∈ I, P ∈ M_{k}^{+} ⇒ A ∗ P ∈ I.

If D is a subset of the closed k-polydisk with 0 ∈ D, then D^{⊥} is a Schur ideal. Moreover if D is a hyperconvex set, then D = D^{⊥⊥}.

Definition 2. Let A ⊆ C(X), x_{1}, ..., x_{k} ∈ X, and S ⊆ M_{k}^{+}. Then the set S is said to be affiliated with the interpolation body D(A; x_{1}, ..., x_{k}) provided that S^{⊥} = D(A; x_{1}, ..., x_{k}).

Note that each interpolation body D(A; x_{1}, ..., x_{k}) has an affiliated set (Schur ideal), namely D(A; x_{1}, ..., x_{k})^{⊥}.

Example. Fix α_{1}, ..., α_{k} in 𝔻 and let P = (\frac{1}{1−ᾱ_{i}α_{j}} ). Note that in Theorem 2 if A = A(𝔻) and x_{i} = α_{i}, then S = {P }. It follows from Theorem 1 that P(α_{1}, ..., α_{k}) = {P }^{⊥}. However if v_{1}, ..., v_{k} in ℂ^{k} such that v_{i} ≠ 0 and (w_{1}, ..., w_{k}) in {P }^{⊥}, then we have that

(\frac{1 − w̄_{i}w_{j}}{1 − ᾱ_{i}α_{j}} ) ≥ 0 ⟹ D_{v}^{∗} (\frac{1 − w̄_{i}w_{j}}{1 − ᾱ_{i}α_{j}} ) D_{v} ≥ 0
⟹ (1 − w̄_{i}w_{j}) ∗ D_{v}^{∗} (\frac{1}{1 − ᾱ_{i}α_{j}} ) D_{v} ≥ 0
⟹ (w_{1}, ..., w_{k}) ∈ {D_{v}^{∗}P D_{v}}^{⊥}.

Similarly if (w_{1}, ..., w_{k}) in {D_{v}^{∗}P D_{v}}^{⊥}, then (w_{1}, ..., w_{k}) in {P }^{⊥}. Here D_{v} denotes the matrix with diagonal entries v_{1}, ..., v_{k} and zeros elsewhere.

Thus, given an interpolation body D, an affiliated set S need not be unique. This is why we study Schur ideals. In this example we are simply producing different generating sets for the same Schur ideal. This is why the “right” question is whether or not there is a unique Schur ideal affiliated with a given hyperconvex set. For Pick bodies we have uniqueness. We now state our main result.

Theorem A. If I ⊆ M_{k}^{+} is a closed Schur ideal such that I^{⊥} = P(α_{1}, ..., α_{k}), then I = ⟨(\frac{1}{1−ᾱ_{i}α_{j}} )⟩.

Here ⟨(\frac{1}{1−ᾱ_{i}α_{j}} )⟩ denotes the Schur ideal generated by the matrix

(\frac{1}{1−ᾱ_{i}α_{j}} ) (i.e, (\frac{1}{1−ᾱ_{i}α_{j}} ) ∗ M_{k}^{+}). So despite the fact that the Pick body P(α_{1}, ..., α_{k}) does not have a unique affiliated set S ⊆ M_{k}^{+}, it does have a unique affiliated Schur ideal. Theorem A can really be thought of as just a matrix theory result. However, it is surprising for a proof we need to invoke the abstract theory of operator algebras (matricial Schur ideals) to solve this matrix theory problem. The reader is encouraged to try and find an alternate proof. By solving this problem we are able to gain some insight into understanding the following problem:

Problem. Let I ⊆ M_{k}^{+} be a closed Schur ideal such that I^{⊥} = D(A(𝔻^{2}); (α_{1}, β_{1}), ..., (α_{k}, β_{k})). Is I = ⟨(\frac{1}{1−ᾱ_{i}α_{j}} )⟩∩⟨(\frac{1}{1−β̄_{i}β_{j}} )⟩?

In joint work with V. Paulsen and G. Weiss, we have shown that for three particular points say (α_{1}, β_{1}), (α_{2}, β_{2}), (α_{3}, β_{3}) in the bidisk, ⟨(\frac{1}{1−ᾱ_{i}α_{j}} )⟩ ∩ ⟨(\frac{1}{1−β̄_{i}β_{j}} )⟩ is not finitely generated. Note, that

D(A(𝔻^{2}); z_{1}, ..., z_{k})^{⊥} = ⟨(\frac{1}{1−ᾱ_{i}α_{j}} )⟩ ∩ ⟨(\frac{1}{1−β̄_{i}β_{j}} )⟩, see [9]. If the above Problem is answered in the affirmative, then this would show that J. Agler’s bidisk formula [1] is not computable in the following sense.

Definition 3. The set D(A; x_{1}, ..., x_{k}) is said to be weakly computable provided that there exists a finitely generated Schur ideal I with I^{⊥} = D(A; x_{1}, ..., x_{k}).

3. Matricial Schur Ideals

In this section we give a survey of those results in V. Paulsen’s paper, Operator Algebras of Idempotents [10](preprint), necessary to prove Theorem A. Since this is just a survey, the proofs will be omitted. The theorem (lemma, proposition, and definition) numbers in parenthesis coincide with the numbers in [10].

Definition 4. (1) Fix a natural number k. A sequence of sets S = {S_{n}}, S_{n} ⊆ M_{k}(M_{n})^{+} will be called a matricial Schur ideal provided that:

(i) (Q_{ij}), (P_{ij}) ∈ S_{n} ⇒ (Q_{ij} + P_{ij}) ∈ S_{n}.

(ii) (Q_{ij}) ∈ S_{n} and B_{1}, ..., B_{k} are n × m matrices ⇒ (B_{i}^{∗}Q_{ij}B_{j}) ∈ S_{m}.

Let E_{1}, ..., E_{k} be bounded operators on a Hilbert space H satisfying E_{i}E_{j} = δ_{i,j}E_{i} and E_{1} + … + E_{k} = I. We let A denote the (matrixnormed) operator algebra generated by this set. We shall call such an algebra a k − idempotent algebra. Given a concrete k-idempotent algebra on a Hilbert space, let A^{∗}A = span{E_{i}^{∗}E_{j} : i, j = 1, ..., k} denote the corresponding operator system.

Lemma 1. (2.1) Let A_{ij} ∈ M_{n}. Then ∑_{i,j=1}^{k} A_{ij}⊗E_{i}^{∗}E_{j} ≥ 0 on ℂ^{n}⊗H if and only if (A_{ij} ⊗ E_{i}^{∗}E_{j}) ≥ 0 on ℂ^{nk} ⊗ H.

Definition 5. (2) Let A be a k-idempotent algebra. Set

S_{n}(A^{∗}A) = {(Φ(E_{i}^{∗}E_{j})) : Φ : A^{∗}A → M_{n} is completely positive}.

Theorem 3. (2.2) Let A be a k-idempotent algebra. Then (1) S(A^{∗}A) = {S_{n}(A^{∗}A)} is a matricial Schur ideal.

(2) Let A_{ij} ∈ M_{n}, i, j = 1, ..., k. Then (A_{ij} ⊗ E_{i}^{∗}E_{j}) ≥ 0 if and only if (A_{ij} ⊗ Q_{ij}) ≥ 0 for all (Q_{ij}) ∈ S_{n}(A^{∗}A).

(3) Let B = span{F_{i} : i = 1, ..., k} be another k-idempotent algebra. Then the map E_{i}^{∗}E_{j} → F_{i}^{∗}F_{j} is n-positive if and only if S_{n}(B^{∗}B) ⊆ S_{n}(A^{∗}A).

Definition 6. (3) Let A be a k-idempotent algebra. Set S_{n}(A) = {(Q_{ij}) ∈ M_{k}(M_{n})^{+} : ((I − W_{i}^{∗}W_{j}) ⊗ Q_{ij}) ≥ 0 ∀(W_{1}, ..., W_{k}) ∈ D_{m}(A), m arbitrary}.

Theorem 4. (2.3) Let A be a k-idempotent algebra. Then (1) S(A) = {S_{n}(A)} is a matricial Schur ideal.

(2) Let W_{i} ∈ M_{n}. Then ∥ ∑ W_{i} ⊗ E_{i}∥ ≤ 1 if and only if ((I_{n} − W_{i}^{∗}W_{j}) ⊗ Q_{ij}) ≥ 0 for all (Q_{ij}) ∈ S_{n}(A).

(3) S_{n}(A^{∗}A) ⊆ S_{n}(A) for all n.

Let B = span{F_{i} : i = 1, ..., k} be another k-idempotent algebra.

(4) The following are equivalent (i) E_{i} → F_{i} is n contractive.

(ii) D_{n}(A) ⊆ D_{n}(B). (iii) S_{n}(B) ⊆ S_{n}(A).

Consequently, the map E_{i} → F_{i} is a completely isometric isomorphism if and only if S(A) = S(B).

Let P in M_{k}^{+} with p_{ij} ≠ 0 for all i and j and define A(P) = {E_{i} = P^{1/2}E_{ii}P^{−1/2} : i = 1, ..., k} where E_{ij} denote the canonical matrix units. In Example 2.4 [10], V. Paulsen shows that

S_{n}(A(P)^{∗}A(P)) = {(p_{ij}R_{ij}) : (R_{ij}) ∈ M_{k}(M_{n})^{+}}.

In particular, S_{1}(A(P)^{∗}A(P)) = ⟨P ⟩ . In Example 2.6 [10], it is shown when P is the special matrix (\frac{1}{1−ᾱ_{i}α_{j}} ) that S_{n}(A(P)) = S_{n}(A(P)^{∗}A(P))

for all n. Moreover when P = (\frac{1}{1−ᾱ_{i}α_{j}} ), every isometric representation of A(P) is completely isometric. One can see this by utilizing the matrix-valued Nevanlinna-Pick Theorem [12] from which one can verify that A(P) is completely isometrically isomorphic to A(𝔻)/I_{α}. Thus if B is isometrically isomorphic to A(P), then B is isometrically isomorphic to A(𝔻)/I_{α}. Hence by a theorem of W. Arveson [2], B is completely isometrically isomorphic to A(𝔻)/I_{α} and the result follows. We close this section with the following proposition.

Proposition 5. (2.8) Let S ⊆ M_{k}(M_{n})^{+} satisfy (Q_{ij}) ∈ S implies (B_{i}^{∗}Q_{ij}B_{j}) ∈ S whenever B_{1}, ..., B_{k} are in M_{n}, and let A_{ij} ∈ M_{n}. Then (A_{ij} ⊗ Q_{ij}) ≥ 0 for all (Q_{ij}) ∈ S if and only if ∑_{i,j=1}^{k} T r(A_{ij}^{t}Q_{ij}) ≥ 0 for all (Q_{ij}) ∈ S, where Tr denotes the trace of the matrix and A^{t} denotes the transpose of the matrix A.

4. Proof of Theorem A

Before proving Theorem A we will state and prove 3 lemmas from which the proof of Theorem A will follow. The following notation will remain fixed for the remainder of the paper.

Fix α_{1}, ..., α_{k} in 𝔻. Let I ⊆ M_{k}^{+} such that I^{⊥} = P(α_{1}, ..., α_{k}). Let P = (\frac{1}{1−ᾱ_{i}α_{j}} ) and define A(P) as in Section 3. If we let R = \{Q ∈ I : Q is invertible}, then ⟨R⟩^{−} = I since I^{⊥} = P(α_{1}, ..., α_{k}), where ⟨R⟩ consists of all finite sums P_{1} ∗ Q_{1} + … + P_{m} ∗ Q_{m} (m ∈ ℕ; Q_{1}, ...Q_{m} ∈ R; P_{1}, ..., P_{m} ∈ M_{k}^{+}). Define A(R) as follows:

A(R) = span{F_{i} = ⨁_{Q∈R} Q^{1/2}E_{ii}Q^{−1/2} : 1 ≤ i ≤ k}.

Since P(α_{1}, ..., α_{k}) is the unit ball of some norm on ℂ^{k} we have that there exits δ > 0 so that |w_{i}| < δ implies (w_{1}, ..., w_{k}) ∈ P(α_{1}, ..., α_{k}).

Thus, for all |w_{i}| < δ and for all (q_{ij}) ∈ R we have that ((1−w̄_{i}w_{j})q_{ij}) ≥ 0. This yields (q_{ij}) ≥ (w̄_{i}q_{ij}w_{j}) for all |w_{i}| < δ and for all (q_{ij}) ∈ R. Integrating this inequality around the torus of radius δ yields (q_{ij}) ≥ δ^{2}Diag(q_{ii}) for all (q_{ij}) ∈ R. Thus, δ^{2}q_{mm}E_{mm} ≤ (q_{ij}) for each 1 ≤ m ≤ k and for each (q_{ij}) ∈ R. Hence if F_{i} = ⨁_{Q∈R} Q^{1/2}E_{ii}Q^{−1/2}, then F_{i}^{∗}F_{i} = ⨁_{Q∈R} Q^{−1/2}q_{ii}E_{ii}Q^{−1/2} ≤ δ^{−2} ⨁_{Q∈R} I_{k×k} and we have that ∥F_{i}∥ ≤ δ^{−1}. Thus each F_{i} is bounded and A(R) is a k-idempotent operator algebra.

Lemma A. S_{n}(A(P)) = S_{n}(A(R)) for all n ≥ 1.

Proof: We have the following logical equivalences.

∥ ∑_{i=1}^{k} w_{i}F_{i}∥ ≤ 1 ⟺ ∥ ⨁_{Q∈R} Q^{1/2}D_{w}Q^{−1/2}∥ ≤ 1
⟺ ((1 − w̄_{i}w_{j})q_{ij}) ≥ 0 for all Q ∈ R
⟺ (w_{1}, ..., w_{k}) ∈ P(α_{1}, ..., α_{k})
⟺ ∥ ∑_{i=1}^{k} w_{i}E_{i}∥ ≤ 1.

Therefore A(R) is isometrically isomorphic to A(P). Hence A(R) is completely isometrically isomorphic to A(P) (see section 3). Thus by Theorem 4, S_{n}(A(P)) = S_{n}(A(R)) for all n ≥ 1. □

Lemma B. S_{n}(A(P)^{∗}A(P)) ⊆ S_{n}(A(R)^{∗}A(R)) for all n ≥ 1.

Proof: By Theorem 3, S_{n}(A(P)^{∗}A(P)) ⊆ S_{n}(A(R)^{∗}A(R)) for all n ≥ 1 if and only if the map F_{i}^{∗}F_{j} ⟶ E_{i}^{∗}E_{j} is completely positive. The map π : A(P) ⟶ ⨁_{Q∈R} M_{k}, where π(E_{i}) = F_{i}, is a complete isometry (Lemma A) and by a theorem of M. Hamana [6] there exists a ∗−epimorphism γ_{2} such that the following diagram commutes.

[figure: C^{∗}(A(P)); ι; A(P); π; C^{∗}(A(R)); π_{e}; C_{e}^{∗}(A(P)); γ_{1}; γ_{2}]

In the above diagram C^{∗}(A(P)) denotes the C^{∗}-algebra generated by the idempotents E_{1}, ..., E_{k} and C^{∗}(A(R)) is defined analogously. We also have that C_{e}^{∗}(A(P)) denotes the C^{∗}-envelope of the operator algebra A(P). The C^{∗}-envelope of an operator algebra B is a C^{∗}- quotient of any C^{∗}-algebra that contains B completely isometrically. Thus the C^{∗}-envelope is in a precise sense the ”minimal” C^{∗}-algebra generated by B. The concept of the C^{∗}-envelope is due to W. Arveson [2] and M. Hamana [6].

Note that C^{∗}(A(P)) is ∗ − isomorphic to M_{k}, but M_{k} is simple and hence C^{∗}(A(P)) must be C_{e}^{∗}(A(P)). Thus we have that γ_{1} is ∗−isomorphism. If we define γ : C^{∗}(A(R)) ⟶ C^{∗}(A(P)) to be γ = γ_{1}^{−1}∘γ_{2}, then γ is a ∗−epimorphism. Therefore γ is completely positive and S_{n}(A(P)^{∗}A(P)) ⊆ S_{n}(A(R)^{∗}A(R)) for all n ≥ 1 by Theorem 3. □

Lemma C. S_{1}(A(R)^{∗}A(R)) = ⟨R⟩^{−}.

Proof: We will first show that ⟨R⟩^{−} ⊆ S_{1}(A(R)^{∗}A(R)). Let (r_{ij}) ∈ ⟨R⟩^{−}. We must show that the map ϕ : A(R)^{∗}A(R) ⟶ ℂ, defined by ϕ(F_{i}^{∗}F_{j}) = r_{ij}, is completely positive. Note that,

(a_{ij}F_{i}^{∗}F_{j}) ≥ 0 ⟺ (a_{ij}q_{ij}E_{ij}) ≥ 0 for all Q ∈ R
⟺ (a_{ij}q_{ij}) ≥ 0 for all Q ∈ R.

Let ∑_{i,j} a_{ij}F_{i}^{∗}F_{j} ≥ 0. By Lemma 1, ∑_{i,j} a_{ij}F_{i}^{∗}F_{j} ≥ 0 if and only if (a_{ij}F_{i}^{∗}F_{j}) ≥ 0. Consequently ∑_{i,j} a_{ij}F_{i}^{∗}F_{j} ≥ 0 if and only if (a_{ij}q_{ij}) ≥ 0 for all Q ∈ R. Thus (a_{ij}r_{ij}) ≥ 0 which implies that ∑_{i,j} a_{ij}r_{ij} = ϕ(∑_{i,j} a_{ij}F_{i}^{∗}F_{j}) ≥ 0. Hence ϕ is a positive linear functional. Since A(R)^{∗}A(R) is an operator system, ϕ is completely positive and (q_{ij}) ∈ S_{1}(A(R)^{∗}A(R)).

We shall now show that S_{1}(A(R)^{∗}A(R)) ⊆ ⟨R⟩^{−}. Suppose that (q̂_{ij}) ∈ S_{1}(A(R)^{∗}A(R)) and (q̂_{ij}) ∉ ⟨R⟩^{−}. Since ⟨R⟩^{−} is a cone in M_{k} we can pick a bounded linear functional f : M_{k} ⟶ ℂ such that f(⟨R⟩^{−}) ≥ 0 and f((q̂_{ij})) < 0, where f(E_{ij}) = a_{ij}. Thus,

f(R) = ∑_{i,j=1}^{k} a_{ij}r_{ij} ≥ 0 for all R ∈ R ⟺ (a_{ij}r_{ij}) ≥ 0 for all
R ∈ R by Prop.5
⟺ (a_{ij} ⊗ F_{i}^{∗}F_{j}) ≥ 0
⟺ (a_{ij} ⊗ q_{ij}) ≥ 0 for all

(q_{ij}) ∈ S_{1}(A(R)^{∗}A(R))

⟹ ∑_{i,j=1}^{k} a_{ij}q_{ij} ≥ 0 for all

(q_{ij}) ∈ S_{1}(A(R)^{∗}A(R)).

But this contradicts the fact that f((q̂_{ij})) < 0 since f((q̂_{ij})) = ∑_{i,j=1}^{k} a_{ij}q̂_{ij} ≥ 0. Therefore S_{1}(A(R)^{∗}A(R)) ⊆ ⟨R⟩^{−} and S_{1}(A(R)^{∗}A(R)) = ⟨R⟩^{−}.□

Proof: (Theorem A) Recall, that S_{n}(A(P)) = S_{n}(A(P)^{∗}A(P)) for all n ≥ 1 and that S_{1}(A(P)^{∗}A(P)) = ⟨P ⟩ (Section 3). We also have that S_{n}(A(R)^{∗}A(R)) ⊆ S_{n}(A(R)) for all n ≥ 1 via Theorem 4.

Lemma B states that S_{n}(A(P)^{∗}A(P)) ⊆ S_{n}(A(R)^{∗}A(R)) for all n ≥ 1 and Lemma A states that S_{n}(A(P)) = S_{n}(A(R)) for all n ≥ 1. But this implies that S_{n}(A(P)^{∗}A(P)) = S_{n}(A(R)^{∗}A(R)) for all n ≥ 1. Thus by Lemma C we have that

⟨P ⟩ = S_{1}(A(P)^{∗}A(P)) = S_{1}(A(R)^{∗}A(R)) = ⟨R⟩^{−}.

Therefore, I = ⟨(\frac{1}{1−ᾱ_{i}α_{j}} )⟩. □

Acknowledgements

This work is taken from the author’s Ph.D. thesis, done under the supervision of Prof. Vern I. Paulsen. The author wishes to express his gratitude to Prof. Paulsen for his guidance and introducing the idea of studying the “duality” of hyperconvex sets which led to this work.

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Department of Mathematics, University of Georgia, Athens, GA 30602 E-mail address: solazzo@math.uga.edu