Comparing Two Generalized Noncommutative Nevanlinna-Pick Theorems
aa r X i v : . [ m a t h . OA ] J u l COMPARING TWO GENERALIZED NONCOMMUTATIVENEVANLINNA-PICK THEOREMS
RACHAEL M. NORTON
Abstract.
We explore the relationship between two noncommutative generalizations ofthe classical Nevanlinna-Pick theorem: one proved by Constantinescu and Johnson in 2003and the other proved by Muhly and Solel in 2004. To make the comparison, we generalizeConstantinescu and Johnson’s theorem to the context of W ∗ -correspondences and Hardyalgebras. After formulating the so-called displacement equation in this context, we areable to follow Constantinescu and Johnson’s line of reasoning in our proof. Though ourresult is similar in appearance to Muhly and Solel’s, closer inspection reveals differences.Nevertheless, when the given data lie in the center of the dual correspondence, the theoremsare essentially the same. Introduction
In this paper, we explore the relationship between two noncommutative generalizations ofthe famous Nevanlinna-Pick theorem: Constantinescu and Johnson’s Theorem 3.4 in [3] andMuhly and Solel’s Theorem 5.3 in [9]. In Constantinescu and Johnson’s theorem, the givendata are N -tuples of operators on Hilbert space; the interpolating map is an upper triangularmatrix with operator entries; and its existence depends on the positivity of the so-calledPick matrix. Muhly and Solel, however, work in the setting of W ∗ -correspondences. Theyinterpolate points in the dual correspondence; the interpolating map belongs to the Hardyalgebra H ∞ ( E ) of the correspondence; and interpolation occurs when their Pick matrixis a completely positive map. Furthermore, to prove their theorem Constantinescu andJohnson exploit the properties of the displacement equation while Muhly and Solel use thecommutant lifting approach. In order to compare the theorems, we generalize Constantinescuand Johnson’s Theorem 3.4 to the context of W ∗ -correspondences and Hardy algebras. Ourproof follows the trajectory of the original proof of the theorem. Once in this setting, we canconsider the similarities and differences between the theorems. The point evaluation in Muhlyand Solel’s Theorem 5.3 is a homomorphism, while our point evaluation is not; it merely givesrise to an antihomomorphism on the Hardy algebra of the dual correspondence, H ∞ ( E σ ).Furthermore, Muhly and Solel’s interpolating map belongs to H ∞ ( E ) while ours belongs to H ∞ ( E σ ). Nevertheless, in the case when the data lie in the center of the dual correspondence, Z ( E σ ), there exists an interpolating map in H ∞ ( Z ( E σ )) if and only if there exists a map in H ∞ ( Z ( E )) which interpolates the adjoints of the data. Moreover, the interpolating maps arerelated by an isomorphism of the Hardy algebras. Lastly, we give an equivalent conditionfor interpolation in terms of completely bounded maps. Key words and phrases.
Nevanlinna-Pick interpolation, displacement equation, W ∗ -correspondence, non-commutative Hardy algebra. Preliminaries
Throughout this paper, M will be a W ∗ -algebra. We will think of M as a C ∗ -algebra, with-out a preferred representation, that is also a dual space. Let E denote a W ∗ -correspondenceover M in the sense of [8, Definition 2.2]. That is, E is a self-dual Hilbert C ∗ -bimodule over M . The M -valued inner product on E is full, and the left action of M on E is given by afaithful, normal ∗ -homomorphism ϕ : M → L ( E ), where L ( E ) denotes the W ∗ -algebra ofadjointable operators on E . For k ∈ N , we form the tensor power of E , E ⊗ k , balanced over M . E ⊗ k is a W ∗ -correspondence over M with the left action given by ϕ k : M → L ( E ⊗ k ),where ϕ k ( a )( ξ ⊗ ξ ⊗ · · · ⊗ ξ k ) = ( ϕ ( a ) ξ ) ⊗ ξ ⊗ · · · ⊗ ξ k . Let E ⊗ = M , viewed as a bimoduleover itself, and define the Fock space of E to be the ultraweak direct sum F ( E ) = L ∞ k =0 E ⊗ k .The Fock space of E is also a W ∗ -correspondence over M . We denote the left action of M on F ( E ) by ϕ ∞ , defined by the formula ϕ ∞ ( a ) = diag[ a, ϕ ( a ) , ϕ ( a ) , · · · ]. Define the (left)creation operators { T ξ | ξ ∈ E } on F ( E ) by T ξ ( η ) = ξ ⊗ η, η ∈ F ( E ). Matricially,(1) T ξ = T (1) ξ T (2) ξ
0. . . . . . , where T ( k ) ξ : E ⊗ k − → E ⊗ k is given by T ( k ) ξ ( η ⊗ · · · ⊗ η k − ) = ξ ⊗ η ⊗ · · · ⊗ η k − .The tensor algebra over E , denoted T + ( E ), is the norm-closed subalgebra of L ( F ( E ))generated by the left action operators { ϕ ∞ ( a ) | a ∈ M } and the creation operators { T ξ | ξ ∈ E } . The Hardy algebra of E is the ultraweak closure of T + ( E ) in L ( F ( E )) and is denotedby H ∞ ( E ).Let σ : M → B ( H ) be a faithful, normal representation of M on a Hilbert space H . Form E ⊗ σ H , the Hausdorff completion of the algebraic tensor product E ⊗ H in the positivesemidefinite sesquilinear form defined by the formula h ξ ⊗ h, η ⊗ k i = h h, σ ( h ξ, η i ) k i , for ξ ⊗ h, η ⊗ k ∈ E ⊗ σ H . Then σ induces the representation σ E : L ( E ) → B ( E ⊗ σ H ) givenby σ E ( T ) = T ⊗ I H .Define the intertwining space I ( σ, σ E ◦ ϕ ) := { η ∈ B ( H, E ⊗ σ H ) | ησ ( a ) = σ E ◦ ϕ ( a ) η ∀ a ∈ M } . For convenience, we will write E σ instead of I ( σ, σ E ◦ ϕ ), and we will referto this space as the σ -dual of E . E σ is a W ∗ -correspondence over σ ( M ) ′ , the commutant of σ ( M ) in B ( H ), under the following actions and inner product: for a, b ∈ σ ( M ) ′ and η, ξ ∈ E σ , a · η · b := ( I E ⊗ a ) ηb and h η, ξ i := η ∗ ξ. We will denote the left action of σ ( M ) ′ on E σ by ϕ σ .As above, form the tensor powers ( E σ ) ⊗ k , k ∈ N , balanced over σ ( M ) ′ , and the Fock space F ( E σ ). The left action maps are denoted by ϕ σk and ϕ σ ∞ , respectively. Let ι : σ ( M ) ′ → B ( H )be the identity representation of σ ( M ) ′ on H . Form ( E σ ) ⊗ k ⊗ ι H and F ( E σ ) ⊗ ι H . For η ∈ E σ and k ∈ N , define η ( k ) ∈ I ( σ, σ E ⊗ k ◦ ϕ k ) by η ( k ) = ( I E ⊗ k − ⊗ η )( I E ⊗ k − ⊗ η ) · · · ( I E ⊗ η ) η . Notethat η ( k +1) = ( I E ⊗ k ⊗ η ) η ( k ) . Then define the Cauchy Kernel C ( η ) ∈ B ( H, F ( E ) ⊗ σ H ) by C ( η ) = (cid:2) I H η η (2) η (3) · · · (cid:3) T . For ξ, η ∈ E σ , the inner product h C ( ξ ) , C ( η ) i is given bythe formula h C ( ξ ) , C ( η ) i = C ( ξ ) ∗ C ( η ).In order to define a point evaluation for elements in H ∞ ( E σ ), we must first define acouple of maps. As in [9, Lemma 3.8], define U : F ( E σ ) ⊗ ι H → F ( E ) ⊗ σ H to be theHilbert space direct sum U = L ∞ k =0 U k , where U k : ( E σ ) ⊗ k ⊗ ι H → E ⊗ k ⊗ σ H is given bythe formula U k ( η ⊗ η ⊗ · · · ⊗ η k ⊗ h ) = ( I E ⊗ k − ⊗ η )( I E ⊗ k − ⊗ η ) · · · ( I E ⊗ η k − ) η k h. Asin [9, Theorem 3.9], define the ultraweakly continuous, completely isometric representation
OMPARING TWO GENERALIZED NONCOMMUTATIVE NEVANLINNA-PICK THEOREMS 3 ρ : H ∞ ( E σ ) → B ( F ( E ) ⊗ σ H ) by(2) ρ ( X ) = U ( X ⊗ I H ) U ∗ , X ∈ H ∞ ( E σ ) . Then for X ∈ H ∞ ( E σ ) , we define a σ ( M ) ′ -valued point evaluation on E σ by the formula(3) ˆ X ( η ) = h ρ ( X ) C (0) , C ( η ) i , η ∈ E σ , where C (0) = (cid:2) I H · · · (cid:3) T . Note that for X, Y ∈ H ∞ ( E σ ) and λ ∈ C , \ X + λY =ˆ X + λ ˆ Y since ρ is linear. While the point evaluation is not multiplicative, in Section 5 weshow how it gives rise to an antihomomorphism from H ∞ ( E σ ) into the completely boundedmaps on σ ( M ) ′ .We are now ready to state our generalized Nevanlinna-Pick theorem. Theorem 2.1.
Let E be a W ∗ -correspondence over a W ∗ -algebra M , with the left action of M on E given by a faithful, normal ∗ -homomorphism ϕ : M → L ( E ) . Let σ be a faithful,normal representation of M on a Hilbert space H . Let z , . . . , z N ∈ E σ with k z i k < , for i = 1 , . . . , N , and Λ , . . . , Λ N ∈ σ ( M ) ′ . There exists X ∈ H ∞ ( E σ ) with k X k ≤ such that ˆ X ( z i ) = Λ i , i = 1 , . . . , N, if and only if the Pick matrix A = (cid:2) h C ( z i ) , C ( z j ) i − h ( I F ( E ) ⊗ Λ i ) C ( z i ) , ( I F ( E ) ⊗ Λ j ) C ( z j ) i (cid:3) Ni,j =1 (4) is positive semidefinite. Note that if we set N = n, E = C N , M = C , and σ ( a ) = aI H for all a ∈ M , then werecover Constantinescu and Johnson’s Theorem 3.4 in [3]. In fact, we arrived at Theorem2.1 by generalizing [3, Theorem 3.4], and it lends itself most naturally to a comparison withMuhly and Solel’s Theorem 5.3 in [9]. Nevertheless, a statement that avoids E σ may bepreferable in some cases.We can state Theorem 2.1 without reference to the σ -dual as follows. Let F be a W ∗ -correspondence over a W ∗ -algebra P . Let τ : P → B ( H ) be a faithful, normal representationof P on a Hilbert space H . For η ∈ F and X ∈ H ∞ ( F ), define˜ X ( η ) := h ( X ⊗ I H ) ˜ C (0) , ˜ C ( η ) i to be the P -valued point evaluation of X at η , where ˜ C ( η ) = h I H L (1) η L (2) η ⊗ · · · i T and L ( k ) η ⊗ k : H → F ⊗ k ⊗ τ H is given by L ( k ) η ⊗ k ( h ) = η ⊗ k ⊗ h. By Theorem 3.6 in [9], thereexists a W ∗ -correspondence E over a W ∗ -algebra M and a faithful, normal representation σ : M → B ( H ) such that F = E σ , P = σ ( M ) ′ , and τ is the identity map. Let X ∈ H ∞ ( F ) = H ∞ ( E σ ) , η ∈ F = E σ , and Λ ∈ P = σ ( M ) ′ . A simple calculation shows ˜ C ( η ) = U ∗ C ( η ) , and it immediately follows that ˜ X ( η ) = ˆ X ( η ) . Furthermore, Lemma 3.8 in [9] implies that( I F ( E ) ⊗ Λ) C ( η ) = U ( ϕ F ∞ (Λ) ⊗ I H ) ˜ C ( η ). Thus we arrive at the following theorem. Theorem 2.2.
Let F be a W ∗ -correspondence over a W ∗ -algebra P . Let z , . . . , z N ∈ F with k z i k < , i = 1 , . . . , N , and Λ , . . . , Λ N ∈ P . There exists X ∈ H ∞ ( F ) with k X k ≤ suchthat ˜ X ( z i ) = Λ i , i = 1 , . . . , N, if and only if the Pick matrix (cid:2) h ˜ C ( z i ) , ˜ C ( z j ) i − h ( ϕ F ∞ (Λ i ) ⊗ I H ) ˜ C ( z i ) , ( ϕ F ∞ (Λ j ) ⊗ I H ) ˜ C ( z j ) i (cid:3) Ni,j =1 RACHAEL M. NORTON is positive semidefinite.
For convenience, we state the nontangential version of Muhly and Solel’s generalizedNevanlinna-Pick theorem [9, Theorem 5.3] alongside our results.
Theorem 2.3 ([9, Theorem 5.3]) . Let E be a W ∗ -correspondence over a W ∗ -algebra M ,and let σ be a faithful, normal representation of M on H . Given z , . . . , z N ∈ E σ with k z i k < , i = 1 , . . . , N, and Λ , . . . , Λ N ∈ B ( H ) , there exists Y ∈ H ∞ ( E ) with k Y k ≤ suchthat ˆ Y ( z ∗ i ) = Λ i , i = 1 , . . . , N, if and only if the map from M N ( σ ( M ) ′ ) to M N ( B ( H )) defined by [ B ij ] Ni,j =1 [ h C ( z i ) , ( I F ( E ) ⊗ B ij ) C ( z j ) i − h C ( z i )Λ ∗ i , ( I F ( E ) ⊗ B ij ) C ( z j )Λ ∗ j i ] Ni,j =1 is completely positive, where the point evaluation of Y at z ∗ i is given by the formula ˆ Y ( z ∗ i ) = h ( Y ⊗ I H ) C (0) , C ( z i ) i ∗ . In Section 5 we compare Theorems 2.1 and 2.3, and we give a condition for when the twotheorems are equivalent. For now we focus on one difference between the two theorems: If themap in Theorem 2.3 is completely positive, then by setting B ij = I H for all i, j = 1 , . . . , N ,we see that the matrix [ h C ( z i ) , C ( z j ) i − h C ( z i )Λ ∗ i , C ( z j )Λ ∗ j i ] Ni,j =1 is positive. Observe that this matrix is almost identical to the Pick matrix A in equation(4). Nevertheless, its positivity is a neccesary but not sufficient condition for interpolationin Theorem 2.3, while the positivity of (4) is a necessary and sufficient condition for inter-polation in Theorem 2.1. The following simple example, brought to our attention by thereferee, illustrates this point. Example 2.4.
Let Z and Λ be × matrices given by Z = (cid:20) r (cid:21) , < r < , and Λ = (cid:20) ǫ
00 0 (cid:21) , < ǫ ≤ . Now consider two problems: (1)
Find F ( z ) = P ∞ n =0 A n z n in the unit ball of H ∞ ⊗ C × such that F ( Z ) := ∞ X n =0 A n Z n = Λ , where A n Z n is given by multiplication of × matrices. (2) Find f ( z ) = P ∞ n =0 a n z n in the unit ball of H ∞ such that f ( Z ) := ∞ X n =0 a n Z n = Λ , where a n Z n is given by scalar multiplication of a matrix.It can be shown that the first problem is a specific case of Constantinescu and Johnson’sTheorem 3.4 in [3] . Consequently, interpolation occurs if and only if P ∞ n =0 Z ∗ n ( I − Λ ∗ Λ) Z n ≥ , which is the case since k Λ k ≤ . On the other hand, the second problem is a specific case OMPARING TWO GENERALIZED NONCOMMUTATIVE NEVANLINNA-PICK THEOREMS 5 of Theorem 2.3. One can easily check that it has no solution. One can also show that theassociated map in Theorem 2.3 (5) B ∞ X n =0 Z ∗ n BZ n − Λ ∗ Z ∗ n BZ n Λ is not completely positive by applying Choi’s criterion [2, Theorem 2] . Thus in Example 2.4, Constantinescu and Johnson have interpolation but Muhly andSolel do not, despite the fact that we get a positive matrix when we evaluate equation (5)at B = I . 3. Interpolating Maps
Since the proof of Theorem 2.1 is adapted from Constantinescu and Johnson’s proof ofTheorem 3.4 in [3], it will be useful to restate some of their definitions in the context of W ∗ -correspondences. In Theorem 3.4 in [3], the interpolating map is a contraction thatbelongs to an algebra of upper triangular operators (see equation (3.1) in [3]). We take thisopportunity to define this algebra in our setting and examine its relationship to H ∞ ( E σ ).Given a W ∗ -correspondence E over a W ∗ -algebra M and a faithful, normal representation σ of M on a Hilbert space H , define U T ( E, H, σ ) to be the algebra of upper triangularoperators T = (cid:2) T ij (cid:3) ∞ i,j =0 ∈ B ( F ( E ) ⊗ σ H ) such that T j ∈ I ( σ E ⊗ j ◦ ϕ j , σ ) , for j ≥
0, and T ij = I E ⊗ T i − ,j − , for 1 ≤ i ≤ j . That is, T is a bounded, linear operator on F ( E ) ⊗ σ H ofthe form(6) T = T T T T · · · I E ⊗ T I E ⊗ T I E ⊗ T · · · I E ⊗ ⊗ T I E ⊗ ⊗ T · · · , and T j ( σ E ⊗ j ◦ ϕ j ( a )) = σ ( a ) T j for all a ∈ M and j ≥ . The collection of contractions in U T ( E, H, σ ) is called the
Schur class and is denoted by S ( E, H, σ ).The connection between U T ( E, H, σ ) and H ∞ ( E σ ) is made precise by the following lemma. Lemma 3.1.
Define ρ : H ∞ ( E σ ) → B ( F ( E ) ⊗ σ H ) as in equation (2) . Then U T ( E, H, σ ) ∗ = ρ ( H ∞ ( E σ )) .Proof. In [9, Theorem 3.9], Muhly and Solel showed that ρ ( H ∞ ( E σ )) = σ F ( E ) ( H ∞ ( E )) ′ . Toshow that U T ( E, H, σ ) ∗ ⊆ ρ ( H ∞ ( E σ )), it suffices to show that every element of U T ( E, H, σ ) ∗ commutes with the generators of σ F ( E ) ( H ∞ ( E )). That is, we must show that every elementof U T ( E, H, σ ) ∗ commutes with σ F ( E ) ( ϕ ∞ ( a )) , a ∈ M, and with σ F ( E ) ( T ξ ) , ξ ∈ E . RACHAEL M. NORTON
For a ∈ M and T ∈ U T ( E, H, σ ), T ∗ ◦ σ F ( E ) ( ϕ ∞ ( a )) = T ∗ · · · T ∗ I E ⊗ T ∗ · · · T ∗ I E ⊗ T ∗ I E ⊗ ⊗ T ∗ · · · ... ... ... . . . a ⊗ I H ϕ ( a ) ⊗ I H ϕ ( a ) ⊗ I H . . . = T ∗ σ ( a ) 0 0 · · · T ∗ σ ( a ) ϕ ( a ) ⊗ T ∗ · · · T ∗ σ ( a ) ϕ ( a ) ⊗ T ∗ ϕ ( a ) ⊗ T ∗ · · · ... ... ... . . . = σ F ( E ) ( ϕ ∞ ( a )) ◦ T ∗ since T j ∈ I ( σ E ⊗ j ◦ ϕ j , σ ) and a ⊗ I H = σ ( a ).For ξ ∈ E and T ∈ U T ( E, H, σ ), T ∗ ◦ σ F ( E ) ( T ξ ) = T ∗ · · · T ∗ I E ⊗ T ∗ · · · T ∗ I E ⊗ T ∗ I E ⊗ ⊗ T ∗ · · · ... ... ... . . . · · · T (1) ξ ⊗ I H · · · T (2) ξ ⊗ I H · · · ... ... ... . . . = · · · ( I E ⊗ T ∗ )( T (1) ξ ⊗ I H ) 0 0 · · · ( I E ⊗ T ∗ )( T (1) ξ ⊗ I H ) ( I E ⊗ ⊗ T ∗ )( T (2) ξ ⊗ I H ) 0 · · · ( I E ⊗ T ∗ )( T (1) ξ ⊗ I H ) ( I E ⊗ ⊗ T ∗ )( T (2) ξ ⊗ I H ) ( I E ⊗ ⊗ T ∗ )( T (3) ξ ⊗ I H ) · · · ... ... ... . . . = σ F ( E ) ( T ξ ) ◦ T ∗ because( I E ⊗ k ⊗ T ∗ ,i − k )( T ( k ) ξ ⊗ I H )( η ⊗ · · · ⊗ η k − ⊗ h ) = ( I E ⊗ k ⊗ T ∗ ,i − k )( ξ ⊗ η ⊗ · · · ⊗ η k − ⊗ h )= ξ ⊗ η ⊗ · · · ⊗ η k − ⊗ T ∗ ,i − k ( h ) = ( T ( i ) ξ ⊗ I H )( I E ⊗ k − ⊗ T ∗ ,i − k )( η ⊗ · · · η k − ⊗ h ) . For the other inclusion, note that it is a consequence of [9, Theorem 3.9] that ρ ( ϕ σ ∞ ( a )) = I F ( E ) ⊗ a, a ∈ σ ( M ) ′ , and ρ ( T η ) = η I E ⊗ η I E ⊗ ⊗ η . . .. . . , η ∈ E σ . Now it is easy to see that ρ ( ϕ σ ∞ ( a )) and ρ ( T η ) are elements of U T ( E, H, σ ) ∗ . (cid:3) Since U T ( E, H, σ ) ∗ = ρ ( H ∞ ( E σ )), we define the point evaluation of an element in U T ( E, H, σ )at a point in E σ to agree with equation (3). That is, for T ∈ U T ( E, H, σ ) and η ∈ E σ , T ( η ) ∈ σ ( M ) ′ is given by the formula T ( η ) = h C (0) , T C ( η ) i . OMPARING TWO GENERALIZED NONCOMMUTATIVE NEVANLINNA-PICK THEOREMS 7
The following result provides more information about the point evaluation and will beuseful in the proof of Theorem 2.1.
Lemma 3.2. If T ∈ U T ( E, H, σ ) and η ∈ E σ , then T C ( η ) = ( I F ( E ) ⊗ T ( η )) C ( η ) .Proof. Expand the left hand side:
T C ( η ) = T T T · · · I E ⊗ T I E ⊗ T · · · I E ⊗ ⊗ T · · · ... ... ... . . . I H ηη (2) ... = P ∞ r =0 T r η ( r ) P ∞ r =0 ( I E ⊗ T r ) η ( r +1) P ∞ r =0 ( I E ⊗ ⊗ T r ) η ( r +2) ... . Expand the right hand side:( I F ( E ) ⊗ T ( η )) C ( η ) = T ( η ) I E ⊗ T ( η ) I E ⊗ ⊗ T ( η ) . . . I H ηη (2) ... = T ( η )( I E ⊗ T ( η )) η ( I E ⊗ ⊗ T ( η )) η (2) ... . The k th entry of the right hand side is( I E ⊗ k ⊗ T ( η )) η ( k ) = ( I E ⊗ k ⊗ C (0) ∗ T C ( η )) η ( k ) = ( I E ⊗ k ⊗ ∞ X r =0 T r η ( r ) ) η ( k ) = ∞ X r =0 (cid:0) I E ⊗ k ⊗ ( T r ( I E ⊗ r − ⊗ η ) . . . ( I E ⊗ η ) η ) (cid:1) η ( k ) = ∞ X r =0 ( I E ⊗ k ⊗ T r )( I E ⊗ k ⊗ I E ⊗ r − ⊗ η ) . . . ( I E ⊗ k ⊗ I E ⊗ η )( I E ⊗ k ⊗ η ) η ( k ) = ∞ X r =0 ( I E ⊗ k ⊗ T r ) η ( r + k ) which agrees with the k th entry of the left hand side. (cid:3) Displacement Equation
The displacement equation was originally defined by Kailath, Kung, and Morf in [6], andit was used to measure the extent to which a matrix was Toeplitz (see also [7]). We areinterested in a displacement equation of the form A − θ ( A ) = B, where θ is a completely positive, contractive map. In this case, one can solve for the uniquesolution A by computing the resolvent, A = ( I − θ ) − ( B ).In order to apply the displacement theory to our context, we first fix z , z , . . . , z N ∈ E σ with k z i k < , i = 1 , . . . , N, and Λ , . . . , Λ N ∈ σ ( M ) ′ , and we form the matrices(7) U = I H ... I H , V = Λ ∗ ...Λ ∗ N , and z = z . . . z N . For the remainder of this section, we reserve this notation for these specific matrices. Weemphasize that U defined in equation (7) is in accord with [3] and should not be confusedwith the isomorphism in equation (2). RACHAEL M. NORTON
Let H ( N ) denote H ⊗ C N , and let σ ( N ) : M → B ( H ( N ) ) be the representation of M on H ( N ) given by the N × N diagonal matrix σ ( N ) ( a ) = σ ( a ) . . . σ ( a ) , a ∈ M. As in Section 2, define the intertwining space I ( σ ( N ) , ( σ ( N ) ) E ◦ ϕ ) := { η ∈ B ( H ( N ) , E ⊗ σ ( N ) H ( N ) ) | ησ ( N ) ( a ) = ( σ ( N ) ) E ◦ ϕ ( a ) η ∀ a ∈ M } . Also define η ( k ) and the Cauchy Kernel C ( η )for η ∈ I ( σ ( N ) , ( σ ( N ) ) E ◦ ϕ ). Observe that z from equation (7) belongs to I ( σ ( N ) , ( σ ( N ) ) E ◦ ϕ )and k z k <
1. Consider the displacement equation(8) A − z ∗ ( I E ⊗ A ) z = U U ∗ − V V ∗ . Equation (8) admits a unique solution A ∈ σ ( N ) ( M ) ′ . To see this, define θ z : σ ( N ) ( M ) ′ → σ ( N ) ( M ) ′ by θ z ( B ) = z ∗ ( I E ⊗ B ) z . Then for k ∈ N , θ k z ( B ) = ( z ( k ) ) ∗ ( I E ⊗ k ⊗ B ) z ( k ) . Since k θ z k ≤ k z k < , ( I B ( H ( N ) ) − θ z ) − is a completely bounded map on σ ( N ) ( M ) ′ . Consequently,we can solve equation (8) for A : A = ( I B ( H ( N ) ) − θ z ) − ( U U ∗ − V V ∗ ) = ∞ X k =0 θ k z ( U U ∗ − V V ∗ ) = ∞ X k =0 ( z ( k ) ) ∗ ( I E ⊗ k ⊗ ( U U ∗ − V V ∗ )) z ( k ) = ∞ X k =0 ( z ( k ) ) ∗ ( I E ⊗ k ⊗ U U ∗ ) z ( k ) − ∞ X k =0 ( z ( k ) ) ∗ ( I E ⊗ k ⊗ V V ∗ ) z ( k ) = C ( z ) ∗ ( I F ( E ) ⊗ U U ∗ ) C ( z ) − C ( z ) ∗ ( I F ( E ) ⊗ V V ∗ ) C ( z )= (cid:2) h C ( z i ) , C ( z j ) i − h ( I F ( E ) ⊗ Λ i ) C ( z i ) , ( I F ( E ) ⊗ Λ j ) C ( z j ) i (cid:3) Ni,j =1 , which is the Pick matrix from equation (4).In the proof of Theorem 2.1, it will be convenient to write A in terms of different notation.Thus define two maps U ∗∞ and V ∗∞ both from F ( E ) ⊗ H to H ( N ) by U ∗∞ = (cid:2) U z ∗ ( I E ⊗ U ) ( z (2) ) ∗ ( I E ⊗ ⊗ U ) . . . (cid:3) V ∗∞ = (cid:2) V z ∗ ( I E ⊗ V ) ( z (2) ) ∗ ( I E ⊗ ⊗ V ) . . . (cid:3) . Then A = U ∗∞ U ∞ − V ∗∞ V ∞ . Note that we may rewrite U ∞ and V ∞ in terms of the Cauchykernels as follows: U ∞ = (cid:2) C ( z ) · · · C ( z N ) (cid:3) and V ∞ = (cid:2) ( I F ( E ) ⊗ Λ ) C ( z ) · · · ( I F ( E ) ⊗ Λ N ) C ( z N ) (cid:3) . These observations will be useful later, so we summarize them in the following remark.
Remark 4.1.
The Pick matrix (4) is the unique solution to the displacement equation (8) ,and it may be written in the form A = U ∗∞ U ∞ − V ∗∞ V ∞ , where U ∞ = (cid:2) C ( z ) · · · C ( z N ) (cid:3) and V ∞ = (cid:2) ( I F ( E ) ⊗ Λ ) C ( z ) · · · ( I F ( E ) ⊗ Λ N ) C ( z N ) (cid:3) . The following lemma is the crux of the proof of Theorem 2.1. It relates the positivity ofthe Pick matrix to the existence of a special element in the Schur class, S ( E, H, σ ). Recallthat S ( E, H, σ ) is defined to be the collection of contractive upper triangular operators T = [ T ij ] ∞ i,j =0 of the form (6) and satisfying T j ( σ E ⊗ j ◦ ϕ j ( a )) = σ ( a ) T j for all a ∈ M and j ≥ . OMPARING TWO GENERALIZED NONCOMMUTATIVE NEVANLINNA-PICK THEOREMS 9
Lemma 4.2.
The solution to the displacement equation (8) is positive semidefinite if andonly if there exists T ∈ S ( E, H, σ ) such that T U ∞ = V ∞ . In order to prove Lemma 4.2, we will need the following two propositions. Proposition 4.3is a result about transfer maps of time varying systems. The state-space model of a discretetime varying linear system is defined by an equation of the form(9) (cid:20) x ( t ) y ( t ) (cid:21) = (cid:20) A ( t ) B ( t ) C ( t ) D ( t ) (cid:21) (cid:20) x ( t + 1) u ( t ) (cid:21) , t ∈ Z , where { U ( t ) } t ∈ Z , { Y ( t ) } t ∈ Z , and { H ( t ) } t ∈ Z are given families of Hilbert spaces called theinput, output, and state spaces, respectively, and u ( t ) ∈ U ( t ) , y ( t ) ∈ Y ( t ) , and x ( t ) ∈ H ( t )for all t ∈ Z . The operators A ( t ) ∈ B ( H ( t + 1) , H ( t )) , B ( t ) ∈ B ( U ( t ) , H ( t )) , C ( t ) ∈ B ( H ( t +1) , Y ( t )), and D ( t ) ∈ B ( U ( t ) , Y ( t )) are also given. The system (9) is said to be contractive if (cid:13)(cid:13)(cid:13)(cid:13)(cid:20) A ( t ) B ( t ) C ( t ) D ( t ) (cid:21)(cid:13)(cid:13)(cid:13)(cid:13) ≤ t ∈ Z . The operators A ( t ) , B ( t ) , C ( t ) , and D ( t ) uniquelydetermine the so-called transfer map of the system, an operator T : L t ∈ Z U ( t ) → L t ∈ Z Y ( t )that satisfies T ( u ( t )) t ∈ Z = ( y ( t )) t ∈ Z . For more on time varying linear systems, see [4, Section2.3]. The following result is derived from the proof of Lemma 3.1 in [4, Section 2.3]. Proposition 4.3.
The transfer map of a contractive time varying linear system is a con-traction.Proof.
Fix t ∈ Z . Suppose x ( t ) = 0, and let { y ( t ) } t 0, thenthere exists L ∈ σ ( N ) ( M ) ′ such that A = LL ∗ . When we rewrite the displacement equationin terms of L , we get LL ∗ − z ∗ ( I E ⊗ LL ∗ ) z = U U ∗ − V V ∗ . Define ˆ A := (cid:20) L ∗ V ∗ (cid:21) and ˆ B := (cid:20) ( I E ⊗ L ∗ ) z U ∗ (cid:21) . Proposition 4.4. If the solution to the displacement equation (8) is positive semidefinite,then there exists a unique partial isometry Ω : ( E ⊗ H ( N ) ) ⊕ H → H ( N +1) such that ˆ A = Ω ˆ B and ker (Ω) ⊥ ⊆ Range ( ˆ B ) . Moreover, for all a ∈ M , (10) σ ( N +1) ( a )Ω = Ω (cid:20) ( σ ( N ) ) E ◦ ϕ ( a ) 00 σ ( a ) (cid:21) . Proof. If the solution A to the displacement equation is positive semidefinite, then we canrewrite the displacement equation as follows:ˆ A ∗ ˆ A = ˆ B ∗ ˆ B, where ˆ A and ˆ B are defined above. By Douglas’s Lemma [5, Theorem 1], there exists aunique partial isometry Ω : ( E ⊗ H ( N ) ) ⊕ H → H ( N +1) such that ˆ A = Ω ˆ B and ker (Ω) ⊥ ⊆ Range ( ˆ B ). Lastly, we must show that equation (10) holds. Recall that since M is a W ∗ -algebra, it is generated by its unitaries. Thus it suffices to prove equation (10) forall unitary elements of M . Let u ∈ M be unitary, and define the partial isometry ˆΩ = σ ( N +1) ( u ∗ )Ω (cid:20) ( σ ( N ) ) E ◦ ϕ ( u ) 00 σ ( u ) (cid:21) . We will show ˆΩ = Ω.Note that the intertwining relations satisfied by the entries of ˆ A and ˆ B imply(11) σ ( N +1) ( a ) ˆ A = ˆ Aσ ( N ) ( a ) and (cid:20) ( σ ( N ) ) E ◦ ϕ ( a ) 00 σ ( a ) (cid:21) ˆ B = ˆ Bσ ( N ) ( a )for all a ∈ M . Then ˆ A = ˆΩ ˆ B since ˆ A = Ω ˆ B and equation (11) holds. By the uniqueness ofΩ, it remains to show that ker ( ˆΩ) ⊥ ⊆ Range ( ˆ B ). That is, we must show P ≤ Q , where P is projection onto ker ( ˆΩ) ⊥ and Q is projection onto Range ( ˆ B ).Observe that Q commutes with (cid:20) ( σ ( N ) ) E ◦ ϕ ( a ) 00 σ ( a ) (cid:21) for all a ∈ M since (11) holds.Thus P = ˆΩ ∗ ˆΩ = (cid:20) ( σ ( N ) ) E ◦ ϕ ( u ∗ ) 00 σ ( u ∗ ) (cid:21) Ω ∗ Ω (cid:20) ( σ ( N ) ) E ◦ ϕ ( u ) 00 σ ( u ) (cid:21) ≤ (cid:20) ( σ ( N ) ) E ◦ ϕ ( u ∗ ) 00 σ ( u ∗ ) (cid:21) Q (cid:20) ( σ ( N ) ) E ◦ ϕ ( u ) 00 σ ( u ) (cid:21) = Q, where the inequality follows from the fact that ker (Ω) ⊥ ⊆ Range ( ˆ B ). (cid:3) Proof of Lemma 4.2. Suppose the solution A to the displacement equation (8) is positivesemidefinite, and let Ω be as in Proposition 4.4. Then we may write Ω = (cid:20) X ZY W (cid:21) , forsome X ∈ B ( E ⊗ H ( N ) , H ( N ) ) , Z ∈ B ( H, H ( N ) ) , Y ∈ B ( E ⊗ H ( N ) , H ) , and W ∈ B ( H ). Thefollowing intertwining relations are a consequence of equation (10):(12) X ∈ I (( σ ( N ) ) E ◦ ϕ, σ ( N ) ) , Z ∈ I ( σ, σ ( N ) ) , Y ∈ I (( σ ( N ) ) E ◦ ϕ, σ ) , W ∈ σ ( M ) ′ . OMPARING TWO GENERALIZED NONCOMMUTATIVE NEVANLINNA-PICK THEOREMS 11 Writing ˆ A = Ω ˆ B in terms of the entries of ˆ A, Ω , and ˆ B , we get the system of equations(13) L ∗ = X ( I E ⊗ L ∗ ) z + ZU ∗ V ∗ = Y ( I E ⊗ L ∗ ) z + W U ∗ . After substituting the first equation into the second K times, we get(14) V ∗ = W U ∗ + K − X k =0 Y ( I E ⊗ (( X ∗ ) ( k ) ) ∗ )( I E ⊗ k +1 ⊗ ZU ∗ ) z ( k +1) + Y ( I E ⊗ (( X ∗ ) ( K ) ) ∗ )( I E ⊗ K +1 ⊗ L ∗ ) z ( K +1) . We can bound the last term in equation (14) by k Y ( I E ⊗ (( X ∗ ) ( K ) ) ∗ )( I E ⊗ K +1 ⊗ L ∗ ) z ( K +1) k ≤ k Y kk (( X ∗ ) ( K ) ) ∗ kk L ∗ kk z ( K +1) k≤ k Y kk X k K k L ∗ kk z k K +1 . Since k z k < k X k ≤ 1, the last term goes to 0 as K goes to infinity, which shows V ∗ = W U ∗ + ∞ X k =0 Y ( I E ⊗ (( X ∗ ) ( k ) ) ∗ )( I E ⊗ k +1 ⊗ ZU ∗ ) z ( k +1) . Form the infinite upper triangular matrix T = [ T ij ] ∞ i,j =0 defined as follows: T ij = j < iI E ⊗ i ⊗ W j = iI E ⊗ i ⊗ Y ( I E ⊗ Z ) j = i + 1 I E ⊗ i ⊗ Y ( I E ⊗ (( X ∗ ) ( j − i − ) ∗ )( I E ⊗ j − i ⊗ Z ) j > i + 1That is, T = W Y ( I E ⊗ Z ) Y ( I E ⊗ X )( I E ⊗ ⊗ Z ) Y ( I E ⊗ (( X ∗ ) (2) ) ∗ )( I E ⊗ ⊗ Z ) · · · I E ⊗ W I E ⊗ Y ( I E ⊗ Z ) I E ⊗ Y ( I E ⊗ X )( I E ⊗ ⊗ Z ) · · · I E ⊗ ⊗ W I E ⊗ ⊗ Y ( I E ⊗ Z ) · · · ... ... ... ... . . . . Note that T U ∞ = V ∞ . We want to show that T extends to an element of S ( E, H, σ ). It iseasy to check that T j ∈ I ( σ E ⊗ j ◦ ϕ j , σ ) , j ≥ , because of the intertwining relations (12)satisfied by X, Z, Y, and W . To show that k T k ≤ 1, we show that T is the transfer map ofa contractive time varying linear system.From the system of equations (13) we have that, for all t ∈ N and for all h ∈ H ( N ) ,(15) ( I E ⊗ t ⊗ L ∗ ) z ( t ) h = ( I E ⊗ t ⊗ X )( I E ⊗ t +1 ⊗ L ∗ ) z ( t +1) h + ( I E ⊗ t ⊗ Z )( I E ⊗ t ⊗ U ∗ ) z ( t ) h ( I E ⊗ t ⊗ V ∗ ) z ( t ) h = ( I E ⊗ t ⊗ Y )( I E ⊗ t +1 ⊗ L ∗ ) z ( t +1) h + ( I E ⊗ t ⊗ W )( I E ⊗ t ⊗ U ∗ ) z ( t ) h. Fix h ∈ H ( N ) . For t ∈ N , define x ( t ) = ( I E ⊗ t ⊗ L ∗ ) z ( t ) h, u ( t ) = ( I E ⊗ t ⊗ U ∗ ) z ( t ) h, and y ( t ) = ( I E ⊗ t ⊗ V ∗ ) z ( t ) h. Also define A ( t ) = I E ⊗ t ⊗ X, B ( t ) = I E ⊗ t ⊗ Z, C ( t ) = I E ⊗ t ⊗ Y, and D ( t ) = I E ⊗ t ⊗ W . Then U ∞ h = u (0) u (1) u (2)... , V ∞ h = y (0) y (1) y (2)... , and the system (15) may be rewritten as x ( t ) = A ( t ) x ( t + 1) + B ( t ) u ( t ) y ( t ) = C ( t ) x ( t + 1) + D ( t ) u ( t ) , t ∈ N , where U ( t ) = E ⊗ t ⊗ H, Y ( t ) = E ⊗ t ⊗ H , and H ( t ) = E ⊗ t ⊗ H ( N ) . Since T U ∞ h = V ∞ h, T isthe transfer map of the system. The matrices (cid:20) A ( t ) B ( t ) C ( t ) D ( t ) (cid:21) = (cid:20) I E ⊗ t ⊗ X I E ⊗ t ⊗ ZI E ⊗ t ⊗ Y I E ⊗ t ⊗ W (cid:21) = I E ⊗ t ⊗ (cid:20) X ZY W (cid:21) have norm equal to 1 for all t , since Ω = (cid:20) X ZY W (cid:21) is of norm 1. Thus T is the transfer mapof a contractive system. Proposition 4.3 implies that T ∈ S ( E, H, σ ) . Conversely, if there exists T ∈ S ( E, H, σ ) such that T U ∞ = V ∞ , then the solution to thedisplacement equation may be written as follows: A = U ∗∞ U ∞ − V ∗∞ V ∞ = U ∗∞ U ∞ − U ∗∞ T ∗ T U ∞ = U ∗∞ ( I − T ∗ T ) U ∞ ≥ , since k T k ≤ (cid:3) Finally, we prove our generalized Nevanlinna-Pick Theorem. Proof of Theorem 2.1. We have already noted in Remark 4.1 that the Pick matrix A inequation (4) is the unique solution to the displacement equation, and we may write A = U ∗∞ U ∞ − V ∗∞ V ∞ .If A ≥ 0, then by Lemma 4.2, there exists T ∈ S ( E, H, σ ) such that T U ∞ = V ∞ . ByRemark 4.1, we rewrite U ∞ and V ∞ in terms of the Cauchy kernels to get T (cid:2) C ( z ) · · · C ( z N ) (cid:3) = (cid:2) ( I F ( E ) ⊗ Λ ) C ( z ) · · · ( I F ( E ) ⊗ Λ N ) C ( z N ) (cid:3) Comparing the matrices entrywise, we see that(16) T C ( z i ) = ( I F ( E ) ⊗ Λ i ) C ( z i ) , i = 1 , . . . , N. By Lemma 3.2, we can rewrite the left hand side of equation (16) to get( I F ( E ) ⊗ T ( z i )) C ( z i ) = ( I F ( E ) ⊗ Λ i ) C ( z i ) . It follows that T ( z i ) = Λ i for all i = 1 , . . . , N . Together with Lemma 3.1, this implies thatthere exists X ∈ H ∞ ( E σ ) with k X k ≤ X ( z i ) = Λ i for all i = 1 , . . . , N .Conversely, suppose there exists X ∈ H ∞ ( E σ ) with k X k ≤ X ( z i ) = Λ i for all i = 1 , . . . , N . Then by Lemma 3.1, there exists T ∈ S ( E, H, σ ) such that T ( z i ) = Λ i for all i = 1 , . . . , N . By the above calculations, T U ∞ = V ∞ , and by Lemma 4.2, A ≥ (cid:3) OMPARING TWO GENERALIZED NONCOMMUTATIVE NEVANLINNA-PICK THEOREMS 13 Remarks on Point Evaluations As we noted after equation (3), the point evaluation defined by it is not multiplicative.Nevertheless, thanks to [11, Theorem 19], point evaluations may be viewed as giving rise toantihomomorphisms from H ∞ ( E σ ) into the completely bounded maps on σ ( M ) ′ . The proofof this assertion is simply a matter of shifting emphasis. We let E σ play the role of E in [11,Theorem 19] and use Lemma 3.2. Here are the details.First recall that for a pair of points z and w in E σ with norm less than 1, the innerproduct h C ( z ) , C ( w ) i is given by the formula h C ( z ) , C ( w ) i = C ( z ) ∗ C ( w ). Consequently, themap a 7→ h C ( z ) , ρ ( ϕ σ ∞ ( a )) C ( w ) i = h C ( z ) , ( I F ( E ) ⊗ a ) C ( w ) i is a completely bounded mapfrom σ ( M ) ′ into itself. So if z ∈ E σ with k z k < X ∈ H ∞ ( E σ ), we may define the mapΦ z X on σ ( M ) ′ by the formula(17) Φ z X ( a ) := h C ( z ) , ρ ( ϕ σ ∞ ( a )) ρ ( X ) C (0) i , a ∈ σ ( M ) ′ . Note that C (0) = (cid:2) I H · · · (cid:3) T , so our definition is precisely that in [11, Theorem 19]with E σ replacing E , ρ ( ϕ σ ∞ ( a )) replacing ϕ ∞ ( a ), ρ ( X ) replacing X , and z replacing ξ in thenotation of that theorem.Now note that formula (2) of [11, Theorem 19] translated to our context allows us to write ρ ( X ) ∗ ρ ( ϕ σ ∞ ( a ∗ )) C ( z ) = ρ ( ϕ σ ∞ (Φ z X ( a ) ∗ )) C ( z ) . Consequently, for X, Z ∈ H ∞ ( E σ ) , a ∈ σ ( M ) ′ , and z ∈ E σ with k z k < z XZ ( a ) = h C ( z ) , ρ ( ϕ σ ∞ ( a )) ρ ( XZ ) C (0) i = h ρ ( X ) ∗ ρ ( ϕ σ ∞ ( a ∗ )) C ( z ) , ρ ( Z ) C (0) i = h ρ ( ϕ σ ∞ (Φ z X ( a ) ∗ )) C ( z ) , ρ ( Z ) C (0) i = h C ( z ) , ρ ( ϕ σ ∞ (Φ z X ( a ))) ρ ( Z ) C (0) i = Φ z Z (Φ z X ( a )) , which shows that Φ z XZ = Φ z Z ◦ Φ z X . Thus we arrive at the following theorem. Theorem 5.1. Let z ∈ E σ with k z k < and X ∈ H ∞ ( E σ ) . Define Φ z X : σ ( M ) ′ → σ ( M ) ′ byformula (17) . Then the map X Φ z X is an algebra antihomomorphism from H ∞ ( E σ ) intothe completely bounded maps on σ ( M ) ′ . We conclude with a theorem that relates our generalized Nevanlinna-Pick theorem, Theo-rem 2.1, to Muhly and Solel’s, Theorem 2.3, and gives a new characterization for interpolationin terms of completely bounded maps. First, we define the center of a W ∗ -correspondenceas in [10, Definition 4.11]. Definition 5.2. If E is a W ∗ -correspondence over a W ∗ -algebra M, then the center of E ,denoted Z ( E ) , is the collection of points ξ ∈ E such that a · ξ = ξ · a for all a ∈ M . In [10, Lemma 4.12], Muhly and Solel proved that if E is a W ∗ -correspondence over a W ∗ -algebra M , then Z ( E ) is a W ∗ -correspondence over the commutative W ∗ -algebra Z ( M ). Ingeneral, we will say that a W ∗ -correspondence E over a commutative W ∗ -algebra M is central if E equals its center. Note that in Theorem 2.1 (respectively, Theorem 2.3), the Pick matrix,and thus its positivity (resp., complete positivity), does not change if the correspondence( σ ( M ) ′ , E σ ) (resp., ( M, E )) is replaced by the correspondence ( Z ( σ ( M ) ′ ) , Z ( E σ )) (resp.,( Z ( M ) , Z ( E ))). Thus there exists an interpolating map in H ∞ ( E σ ) (resp., H ∞ ( E )) if andonly if there exists an interpolating map in H ∞ ( Z ( E σ )) (resp., H ∞ ( Z ( E ))). Consequently,for the final theorem we restrict our attention to the correspondences ( Z ( σ ( M ) ′ ) , Z ( E σ )) and( Z ( M ) , Z ( E )). We choose to do this because the centers are isomorphic as correspondencesin the following sense. Definition 5.3 ([10, Definition 2.2]) . An isomorphism of a W ∗ -correspondence E over M and a W ∗ -correspondence E over M is a pair ( σ, Ψ) where σ : M → M is an isomorphismof W ∗ -algebras, Ψ : E → E is a vector space isomorphism, and for e, f ∈ E and a, b ∈ M ,we have Ψ( a · e · b ) = σ ( a ) · Ψ( e ) · σ ( b ) and h Ψ( e ) , Ψ( f ) i = σ ( h e, f i ) . Define γ : Z ( E ) → Z ( E σ ) by γ ( ξ ) = L ξ , where L ξ : H → E ⊗ σ H is given by L ξ ( h ) = ξ ⊗ h .In [10, Lemma 4.12], Muhly and Solel proved that the pair ( σ, γ ) is an isomorphism of thecorrespondences ( Z ( M ) , Z ( E )) and ( Z ( σ ( M ) ′ ) , Z ( E σ )). Proposition 5.4. For k ∈ N , define the map γ k : Z ( E ) ⊗ k → Z ( E σ ) ⊗ k by γ k ( ξ ⊗ · · · ⊗ ξ k ) = L ξ ⊗· · ·⊗ L ξ k . The pair ( σ, γ k ) is an isomorphism of ( Z ( M ) , Z ( E ) ⊗ k ) onto ( Z ( σ ( M ) ′ ) , Z ( E σ ) ⊗ k ) .Proof. Let ξ ⊗ · · · ⊗ ξ k , η ⊗ · · · ⊗ η k ∈ Z ( E ) ⊗ k . Since ξ i , η i ∈ Z ( E ), L ξ j , L η i ∈ Z ( E σ ), and( σ, γ ) is an isomorphism of correspondences, we have h γ k ( ξ ⊗ · · · ⊗ ξ k ) , γ k ( η ⊗ · · · ⊗ η k ) i = h L ξ ⊗ · · · ⊗ L ξ k , L η ⊗ · · · ⊗ L η k i = h L ξ ⊗ · · · ⊗ L ξ k , h L ξ , L η i · L η ⊗ · · · ⊗ L η k i = h L ξ ⊗ · · · ⊗ L ξ k , L η ⊗ · · · ⊗ L η k ih L ξ , L η i = . . . = h L ξ k , L η k i · · · h L ξ , L η i = σ ( h ξ k , η k i ) · · · σ ( h ξ , η i ) = σ ( h ξ ⊗ · · · ⊗ ξ k , η ⊗ · · · ⊗ η k i ) . Thus h γ k ( ξ ) , γ k ( η ) i = σ ( h ξ, η i ) for all ξ, η ∈ Z ( E ) ⊗ k . Furthermore, since σ is an isomorphism, k σ ( h ξ, ξ i ) k = kh ξ, ξ ik for all ξ ∈ Z ( E ) ⊗ k . Thus γ k is an isometry.For ξ ⊗ ξ ⊗ · · · ⊗ ξ k ∈ Z ( E ) ⊗ k , a, b ∈ Z ( M ), and h ∈ H we have γ k ( a · ( ξ ⊗ ξ ⊗ · · · ⊗ ξ k ) · b )( h ) = γ k (( a · ξ ) ⊗ ξ ⊗ · · · ⊗ ( ξ k · b ))( h ) = L a · ξ ⊗ L ξ ⊗ · · · ⊗ L ξ k · b ( h )= L ξ σ ( a ) ⊗ L ξ ⊗ · · · ⊗ L ξ k σ ( b )( h ) = L ξ ⊗ L ξ ⊗ · · · ⊗ L ξ k σ ( a ) σ ( b )( h )= ( I E ⊗ k ⊗ σ ( a )) γ k ( ξ ⊗ · · · ξ k ) σ ( b ) h. Hence γ k ( a · ξ ⊗ ξ ⊗ · · · ⊗ ξ k · b ) = σ ( a ) · γ k ( ξ ⊗ ξ ⊗ · · · ⊗ ξ k ) · σ ( b ) . (cid:3) Define γ ∞ : F ( Z ( E )) → F ( Z ( E σ )) by γ ∞ = diag [ σ, γ, γ , . . . ]. Since γ k is an isomorphismof correspondences for each k ∈ N , it follows that γ ∞ is an isomorphism of correspondencesas well. Proposition 5.5. For ξ ∈ Z ( E ) , γ ∞ T ξ γ − ∞ = T γ ( ξ ) , where T ξ is the left creation operator in H ∞ ( Z ( E )) determined by ξ , and T γ ( ξ ) is the left creation operator in H ∞ ( Z ( E σ )) determinedby γ ( ξ ) . For a ∈ Z ( M ) , γ ∞ ϕ ∞ ( a ) γ − ∞ = ϕ σ ∞ ( σ ( a )) , where ϕ ∞ ( a ) is the left action operatorin H ∞ ( Z ( E )) determined by a , and ϕ σ ∞ ( σ ( a )) is the left action operator in H ∞ ( Z ( E σ )) determined by σ ( a ) .Proof. Let (cid:2) η η η · · · (cid:3) T ∈ F ( Z ( E σ )). Since F ( Z ( E σ )) is isomorphic to F ( Z ( E )) via γ ∞ ,there exist α ij ∈ Z ( E ) such that η i = L α i ⊗ · · · ⊗ L α ii , for i ≥ , and α ∈ Z ( M ) such that OMPARING TWO GENERALIZED NONCOMMUTATIVE NEVANLINNA-PICK THEOREMS 15 σ ( α ) = η . Thus for ξ ∈ Z ( E ), we have γ ∞ T ξ γ − ∞ η η η ... = γ ∞ T ξ α α α ⊗ α ... = γ ∞ ξ · α ξ ⊗ α ξ ⊗ α ⊗ α ... = L ξ σ ( α ) L ξ ⊗ L α L ξ ⊗ L α ⊗ L α ... = T γ ( ξ ) η η η ... . For a ∈ Z ( M ), we have γ ∞ ϕ ∞ ( a ) γ − ∞ η η η ... = γ ∞ ϕ ∞ ( a ) α α α ⊗ α ... = γ ∞ aα ϕ ( a )( α ) ϕ ( a )( α ⊗ α )... = σ ( aα ) L a · α L a · α ⊗ L α ... = σ ( a ) σ ( α ) σ ( a ) · L α σ ( a ) · L α ⊗ L α ... = ϕ σ ∞ ( σ ( a )) η η η ... . (cid:3) Thus we arrive at the following isomorphism from H ∞ ( Z ( E )) onto H ∞ ( Z ( E σ )). Proposition 5.6. The map defined on the generators of H ∞ ( Z ( E )) by T ξ γ ∞ T ξ γ − ∞ , ξ ∈ Z ( E ) , and ϕ ∞ ( a ) γ ∞ ϕ ∞ ( a ) γ − ∞ , a ∈ Z ( M ) , extends to an isomorphism Γ from H ∞ ( Z ( E )) onto H ∞ ( Z ( E σ )) . With Γ in hand, we are finally able to prove that Theorems 2.1 and 2.3 are essentially thesame when we restrict to the centers of the correspondences. Theorem 5.7. Let z , . . . , z N ∈ Z ( E σ ) with k z i k < , i = 1 , . . . , N, and Λ , . . . , Λ N ∈ Z ( σ ( M ) ′ ) . Define Ψ z i Λ i : σ ( M ) ′ → σ ( M ) ′ by Ψ z i Λ i ( a ) = h C ( z i ) , ( I F ( E ) ⊗ a Λ ∗ i ) C (0) i . For X ∈ H ∞ ( Z ( E σ )) , define Φ z i X : σ ( M ) ′ → σ ( M ) ′ by Φ z i X ( a ) = h C ( z i ) , ρ ( ϕ σ ∞ ( a )) ρ ( X ) C (0) i .Then the following are equivalent: (1) there exists Y ∈ H ∞ ( Z ( E )) with k Y k ≤ such that ˆ Y ( z ∗ i ) = Λ ∗ i , i = 1 , . . . , N, in thesense of Theorem 2.3. (2) there exists X = Γ( Y ) ∈ H ∞ ( Z ( E σ )) with k X k ≤ such that ˆ X ( z i ) = Λ i , i =1 , . . . , N , in the sense of Theorem 2.1. (3) Φ z i X = Ψ z i Λ i , i = 1 , . . . , N .Proof. First we prove (1) is equivalent to (2). Suppose ξ ∈ Z ( E ) and Y = T ξ ∈ H ∞ ( Z ( E )).By definition of point evaluation in Theorem 2.3, we haveˆ Y ( z ∗ i ) ∗ = h ( Y ⊗ I H ) C (0) , C ( z i ) i = h ( T ξ ⊗ I H ) C (0) , C ( z i ) i = ( T (1) ξ ∗ ⊗ I H ) z i , where T (1) ξ is defined after equation (1). Since Z ( E ) is isomorphic to Z ( E σ ) via γ, thereexists α i ∈ Z ( E ) such that z i = L α i . Thus for h ∈ H ,( T (1) ξ ∗ ⊗ I H ) z i h = T (1) ξ ∗ ( α i ) ⊗ h = h ξ, α i i ⊗ h = σ ( h ξ, α i i ) h. Now for X = T γ ( ξ ) ∈ H ∞ ( Z ( E σ )), by definition of point evaluation in equation (3) we seethatˆ X ( z i ) h = h U ( X ⊗ I H ) U ∗ C (0) , C ( z i ) i h = h ( T γ ( ξ ) ⊗ I H ) U ∗ C (0) , U ∗ C ( z i ) i h = ( T (1) γ ( ξ ) ∗ ⊗ I H ) L z i h = T (1) γ ( ξ ) ∗ ( z i ) ⊗ h = h γ ( ξ ) , z i i ⊗ h = L ∗ ξ ( α i ⊗ h )= σ ( h ξ, α i i ) h = ˆ Y ( z ∗ i ) ∗ h. Thus ˆ Y ( z ∗ i ) = Λ ∗ i if and only if ˆ X ( z i ) = Λ i , where X = Γ( Y ) ∈ H ∞ ( Z ( E σ )). Similarly, if a ∈ Z ( M ) and Y = ϕ ∞ ( a ) ∈ H ∞ ( Z ( E )), then a quick calculation shows that ˆ Y ( z ∗ i ) ∗ = σ ( a ) ∗ .Moreover, if X = ϕ σ ∞ ( σ ( a )) ∈ H ∞ ( Z ( E σ )), then ˆ X ( z i ) = σ ( a ) ∗ as well. So again we havethat ˆ Y ( z ∗ i ) = Λ ∗ i if and only if ˆ X ( z i ) = Λ i , where X = Γ( Y ) ∈ H ∞ ( Z ( E σ )). Since theequivalence holds for the generators of H ∞ ( Z ( E )) and H ∞ ( Z ( E σ )), we conclude that (1) isequivalent to (2).The following calculation shows that (2) and (3) are equivalent. Note that the proof ofLemma 3.2 shows that for all X ∈ H ∞ ( E σ ), all z ∈ Z ( E σ ) with k z k < 1, and all a ∈ σ ( M ) ′ , ρ ( X ) ∗ ρ ( ϕ σ ∞ ( a ∗ )) C ( z ) = ρ ( X ) ∗ ( I F ( E ) ⊗ a ∗ ) C ( z ) = ( I F ( E ) ⊗ ˆ X ( z ))( I F ( E ) ⊗ a ∗ ) C ( z ) . Thus we haveΦ z i X ( a ) = h C ( z i ) , ρ ( ϕ σ ∞ ( a )) ρ ( X ) C (0) i = h ρ ( X ) ∗ ρ ( ϕ σ ∞ ( a ∗ )) C ( z i ) , C (0) i = h ( I F ( E ) ⊗ ˆ X ( z i ))( I F ( E ) ⊗ a ∗ ) C ( z i ) , C (0) i = h C ( z i ) , ( I F ( E ) ⊗ a ˆ X ( z i ) ∗ C (0) i , which is equal to Ψ z i Λ i ( a ) if and only if ˆ X ( z i ) = Λ i for i = 1 , . . . , N . (cid:3) We note that in [1], Ball and ter Horst also addressed the issues we discussed here. Theircontext is not quite as general as ours and it is formulated somewhat differently, but it maybe possible to extend their arguments to our setting. The first condition in [1, Theorem3.3] seems related to Muhly and Solel’s theorem via an application of Choi’s criterion [2,Theorem 2]. Consequently, the relationship between Muhly and Solel’s Theorem 5.3 in [9]and Constantinescu and Johnson’s Theorem 3.4 in [3] is illuminated further by their theorem. acknowledgements I would like to thank Joe Ball for his insight into the connection between Constantinescuand Johnson’s result and Muhly and Solel’s; Jenni Good for her meticulous explanationsand observations; and Paul Muhly and Baruch Solel for their astute feedback. References [1] Joseph A. Ball and Sanne ter Horst. Multivariable operator-valued Nevanlinna-Pick interpolation: asurvey. In J. J. Grobler, L. E. Labuschagne, and M. M¨oller, editors, Operator Algebras, Operator Theoryand Applications: 18th International Workshop on Operator Theory and Applications, Potchefstroom,July 2007 , pages 1–72. Birkh¨auser, Basel, 2010.[2] Man-Duen Choi. Completely positive linear maps on complex matrices. Linear Algebra and its Appli-cations , 10(3):285 – 290, 1975. OMPARING TWO GENERALIZED NONCOMMUTATIVE NEVANLINNA-PICK THEOREMS 17 [3] T. Constantinescu and J. L. Johnson. A note on noncommutative interpolation. Canad. Math. Bull. ,46(1):59–70, 2003.[4] Tiberiu Constantinescu. Schur parameters, factorization and dilation problems , volume 82 of OperatorTheory: Advances and Applications . Birkh¨auser Verlag, Basel, 1996.[5] R. G. Douglas. On majorization, factorization, and range inclusion of operators on Hilbert space. Proc.Amer. Math. Soc. , 17:413–415, 1966.[6] Thomas Kailath, Sun-Yuan Kung, and Martin Morf. Displacement ranks of matrices and linear equa-tions. Journal of Mathematical Analysis and Applications , 68(2):395 – 407, 1979.[7] Thomas Kailath and Ali H. Sayed. Displacement structure: Theory and applications. SIAM Review ,37(3):297–386, 1995.[8] Paul S. Muhly and Baruch Solel. Quantum Markov processes (correspondences and dilations). Internat.J. Math. , 13(8):863–906, 2002.[9] Paul S. Muhly and Baruch Solel. Hardy algebras, W ∗ -correspondences and interpolation theory. Math.Ann. , 330(2):353–415, 2004.[10] Paul S. Muhly and Baruch Solel. Schur class operator functions and automorphisms of Hardy algebras. Doc. Math. , 13:365–411, 2008.[11] Paul S. Muhly and Baruch Solel. The Poisson kernel for Hardy algebras. Complex Anal. Oper. Theory ,3(1):221–242, 2009. Department of Mathematics, University of Iowa, Iowa City, IA 52242 E-mail address ::