Absolute continuity, Interpolation and the Lyapunov order
aa r X i v : . [ m a t h . OA ] J u l ABSOLUTE CONTINUITY, INTERPOLATION AND THELYAPUNOV ORDER
PAUL S. MUHLY AND BARUCH SOLEL
Abstract.
We extend our Nevanlinna-Pick theorem for Hardy algebras andtheir representations to cover interpolation at the absolutely continuous pointsof the boundaries of their discs of representations. The Lyapunov order playsa crucial role in our analysis. Introduction
The celebrated theorem of Nevanlinna and Pick asserts that if n distinct points, z , z , . . . , z n , are given in the open unit disc D and if n other complex numbers arealso given, w , w , . . . , w n , then there is a function f in the Hardy algebra H ∞ ( T ) ,with norm at most one, such that f ( z i ) = w i , i = 1 , , . . . , n , if and only if the Pickmatrix (cid:18) − w i w j − z i z j (cid:19) ni,j =1 is positive semidefinite. In [4, Theorem 5.3], we generalized this Nevanlinna-Picktheorem to the setting of Hardy algebras over W ∗ -correspondences. Here we intendto push the work in [4] further, using tools developed in [2]. In a sense that weshall make precise, we show that there is a condition similar to the positivity ofthe Pick matrix that allows one to interpolate at “absolutely continuous” points ofthe boundaries of the domains considered in [4]. Before stating our main theorem,we want to see how one might try to extend the Nevanlinna-Pick theorem to coverfamilies of absolutely continuous contraction operators along the lines suggested by[4, Theorem 6.1].For this purpose, suppose H is a Hilbert space and Z := ( Z , Z , . . . , Z n ) is an n -tuple of operators in B ( H ) . Then Z defines a completely positive operator Φ Z on the n × n matrices over B ( H ) via the formula Φ Z (( a ij )) : = Z Z . . . Z n a a · · · a n a a ... . . . a n a n · · · a nn Z Z . . . Z n ∗ =( Z i a ij Z ∗ j ) . Mathematics Subject Classification.
Primary: 15A24, 46H25, 47L30, 47L55, Secondary:46H25, 47L65.
Key words and phrases.
Nevanlinna-Pick interpolation, representations, Hardy algebra, abso-lute continuity, Lyapunov order.The research of both authors was supported in part by a U.S.-Israel Binational Science Foun-dation grant. The second author was also supported by the Technion V.P.R. Fund.
If all the Z i ’s have norm less than , then I − Φ Z is an invertible map on M n ( B ( H )) and ( I − Φ Z ) − is also completely positive. The following theorem, then, is a specialcase of [4, Theorem 6.1]. Theorem 1.1.
Suppose Z , Z , · · · , Z n are n distinct operators in B ( H ) , eachof norm less than , and suppose W , W , · · · , W n are n operators in B ( H ) . Thenthere is an function f in H ∞ ( T ) , with supremum norm at most , such that f ( Z i ) = W i , i = 1 , , . . . , n , where f ( Z i ) is defined through the Riesz functional calculus, ifand only if the Pick map (1) ( I − Φ W ) ◦ ( I − Φ Z ) − defined on M n ( B ( H )) is completely positive. Observe that when H is one-dimensional this theorem recovers the classical the-orem of Nevanlinna and Pick that we stated at the outset. Now Sz.-Nagy and Foiaşhave shown that the proper domain for their H ∞ -functional calculus is the collec-tion of all absolutely continuous contractions . One way to say that a contraction T is absolutely continuous is to say that when T is decomposed as T = T cnu + U ,where T cnu is completely non-unitary and U is unitary, then the spectral measureof U is absolutely continuous with respect to Lebesgue measure on the circle. Thecontent of [10, Theorems III.2.1 and III.2.3] is that a contraction T is absolutelycontinuous if and only if the H ∞ ( T ) -functional calculus may be evaluated on T . Itis therefore of interest to modify the hypothesis of Theorem 1.1 and ask for con-ditions that allow one to interpolate in the wider context where the variables Z i are assumed to be merely absolutely continuous contractions. In that setting themap I − Φ Z need no longer be invertible, and so it may not be possible to formthe generalized Pick operator ( I − Φ W ) ◦ ( I − Φ Z ) − , let alone determine whetheror not it is completely positive. However, there is a notion from matrix analysis,called the “Lyapunov order”, which suggests a replacement for the condition thatthe Pick operator ( I − Φ W ) ◦ ( I − Φ Z ) − be completely positive. To formulate itwe require an idea from the theory of completely positive maps that we analyzedin [2]. Definition 1.2.
Let Φ be a completely positive map on a W ∗ -algebra A . Anelement a ∈ A is called superharmonic for Φ in case a ≥ and Φ( a ) ≤ a ; a is called pure superharmonic in case a is superharmonic and Φ n ( a ) ց as n → ∞ .The superharmonic elements for a completely positive map evidently form aconvex subset in the cone of all non-negative elements in the W ∗ -algebra A . Definition 1.3.
Let B be a W ∗ -algebra and suppose A is a sub- W ∗ -algebra of B .Suppose Φ : A → A is a completely positive map and that Ψ : B → B is also com-pletely positive. Then we say Ψ completely dominates Φ in the sense of Lyapunov in case every pure superharmonic element of M n ( A ) for Φ n is superharmonic for Ψ n , where Φ n (resp. Ψ n ) is the usual promotion of Φ (resp. Ψ ) to M n ( A ) (resp. M n ( B ) ).The following proposition links the notion of complete Lyapunov domination tothe complete positivity of (1). Proposition 1.4.
Suppose that A is a sub- W ∗ -algebra of a W ∗ -algebra B andsuppose Φ and Ψ are completely positive maps on A and B , respectively. Assumethat k Φ k < , so I − Φ is invertible. Then the Pick operator, P := ( I − Ψ) ◦ ( I − BSOLUTE CONTINUITY, INTERPOLATION AND THE LYAPUNOV ORDER 3 Φ) − , is completely positive if and only if Ψ completely dominates Φ in the senseof Lyapunov.Proof. Note that the hypothesis that k Φ k < implies that every superharmonicelement of A is pure superharmonic. Also note that it suffices to prove that P ispositive if and only if { a ∈ A | a ≥ , Φ( a ) ≤ a } ⊆ { b ∈ B | b ≥ , Ψ( b ) ≤ b } ,since the same argument will work for every n . Suppose, then, that P is positiveand suppose that a ≥ and Φ( a ) ≤ a . Then ( I − Φ)( a ) ≥ . Consequently, ≤ P (( I − Φ)( a )) = ( I − Ψ)( a ) , showing that a ≥ Ψ( a ) . Suppose, conversely, that b ≥ . Then since k Φ k < and Φ is positive, ( I − Φ) − = P n ≥ Φ n is positive.Consequently, a = ( I − Φ) − ( b ) is positive. But also, since ( I − Φ)( a ) = b is positive, a ≥ Ψ( a ) , by hypothesis. That is, P ( b ) = a − Ψ( a ) ≥ , which is what we want toshow. (cid:3) Our extension of Theorem 1.1 can now be formulated as
Theorem 1.5.
Suppose Z , Z , · · · , Z n are n distinct absolutely continuous con-tractions on a Hilbert space H and suppose W , W , · · · , W n are n contractions on H , then there is a function f ∈ H ∞ ( T ) , of norm at most , such that f ( Z i ) = W i , i = 1 , , . . . , n , if and only if Φ W completely dominates Φ Z in the sense of Lya-punov. The technology we use to prove Theorem 1.5 works in the more general context ofHardy algebras over W ∗ -correspondences, as we mentioned earlier. This is the arenain which our analysis takes place. But first, we must provide some background from[4, 2]. We shall follow terminology and most of the notation from [2]. In particular,we shall cite the second section of [2] for further background because it gives a fairlydetailed birds-eye view of the theory as of 2010.AcknowledgmentWe are very grateful to Nir Cohen for introducing us to the Lyapunov order.2. Background and the Main Theorem
Throughout this note, M will denote a fixed W ∗ -algebra. We will treat M asan abstract C ∗ -algebra that is a dual space and we will not think of it as actingconcretely on Hilbert space except through representations that we will specify.Also, E will denote a W ∗ -correspondence over M . This means first that E is aright Hilbert C ∗ -module over M which is self-dual. Consequently, the algebra ofall bounded adjointable M -module maps on E , L ( E ) , is all the bounded modulemaps and L ( E ) is a W ∗ -algebra. To say that E is a W ∗ -correspondence over M means, then, that there is a normal representation ϕ : M → L ( E ) , making E aleft M -module [2, Paragraph 2.2]. To eliminate technical digressions we assumethat ϕ is faithful and unital. The tensor powers of E , E ⊗ n , will be the self-dualcompletions of the usual C ∗ -Hilbert module tensor powers, and the Fock space F ( E ) will be the self-dual completion of the C ∗ -direct sum of the E ⊗ n . Then F ( E ) is a W ∗ -correspondence over M and we denote by ϕ ∞ the left action of M on F ( E ) [2, Paragraph 2.7]. If ξ ∈ E , then T ξ will denote the creation operatorit determines: T ξ η := ξ ⊗ η , η ∈ F ( E ) . The norm-closed subalgebra generated ϕ ∞ ( M ) and { T ξ | ξ ∈ E } is called the tensor algebra of E and will be denoted by PAUL S. MUHLY AND BARUCH SOLEL T + ( E ) [2, Paragraph 2.7]. The ultra weak closure of T + ( E ) in L ( F ( E )) is calledthe Hardy algebra of E and is denoted H ∞ ( E ) [2, Definition 2.1].Suppose σ : M → B ( H σ ) is a normal representation and let σ E : L ( E ) → B ( E ⊗ σ H σ ) be the induced representation of L ( E ) in the sense of Rieffel [7, 8]: σ E ( T ) := T ⊗ I , T ∈ L ( E ) . We write I ( σ E ◦ ϕ, σ ) for the set of all operators C : E ⊗ σ H σ → H σ that satisfy the equation Cσ ◦ ϕ ( a ) = σ ( a ) C for all a ∈ M ; i.e., I ( σ E ◦ ϕ, σ ) denotes all the intertwiners of σ E ◦ ϕ and σ . Also, we write D ( E, σ ) for the set of all elements of I ( σ E ◦ ϕ, σ ) that have norm less than , and we write D ( E, σ ) for its norm closure. In [6] we proved Lemma 2.1. (See [2, Paragraph 2.8] .) Given z ∈ D ( E, σ ) , define z × σ by z × σ ( ϕ ∞ ( a )) := σ ( a ) and z × σ ( T ξ )( h ) := z ( ξ ⊗ h ) , a ∈ M , ξ ∈ E , and h ∈ H σ .Then z × σ extends to a completely contractive (c.c.) representation of T + ( E ) on H σ . Conversely, given a c.c. representation ρ of T + ( E ) , then ρ = z × σ , where σ := ρ ◦ ϕ ∞ and z ( ξ ⊗ h ) := ρ ( T ξ ) h . Further, for F ∈ H ∞ ( E ) , the B ( H σ ) -valuedfunction b F σ , defined on D ( E, σ ) by b F σ ( z ) := z × σ ( F ) , is bounded analytic and itextends to be continuous on D ( E, σ ) when F ∈ T + ( E ) .Remark . We note here that our D ( E, σ ) is denoted D ( E σ ) ∗ in [2, Paragraph2.8], where E σ := I ( σ E ◦ ϕ, σ ) ∗ = I ( σ, σ E ◦ ϕ ) is the σ -dual of E [2, Paragraph2.6]. This dual space plays an important role in our theory, as we shall see, but wehave opted for the change of notation in order to eliminate numerous unnecessaryand often confusing adjoints from our formulas. Definition 2.3.
A point z ∈ D ( E, σ ) and the representation z × σ are called abso-lutely continuous in case z × σ extends to be an ultra weakly continuous represen-tation of H ∞ ( E ) in B ( H σ ) . We write AC ( E, σ ) for all the absolutely continuouspoints of D ( E, σ ) .Our choice of terminology is inspired by the fact that when M = E = C , then z is absolutely continuous in our sense if and only if z , which is just an ordinarycontraction operator on H σ , is absolutely continuous in the sense described in theIntroduction.In general, D ( E, σ ) ⊆ AC ( E, σ ) ⊆ D ( E, σ ) , and both inclusions are proper. If M = E = C , and if σ is the one-dimensional representation of C on C , then D ( E, σ ) is the open unit disc in the complex plane and D ( E, σ ) = AC ( E, σ ) . In every othersetting of which we are aware, D ( E, σ ) ( AC ( E, σ ) . Also, we know of no situationwhere D ( E, σ ) = AC ( E, σ ) . We have been able to identify AC ( E, σ ) explicitly innumerous instances [2, Sections 4 and 5] and we know a lot about this space, butthere is still much that remains mysterious.The σ -dual of E , E σ := I ( σ, σ E ◦ ϕ ) , is important in this study for severalreasons. The first is that it is a W ∗ -correspondence over σ ( M ) ′ in a very naturalway. For ξ, η ∈ E σ , h ξ, η i is defined to be ξ ∗ η - the product being the ordinaryoperator product, which makes sense as an operator on H σ since ξ and η both mapfrom H σ to E ⊗ σ H σ . The actions of σ ( M ) ′ on E σ are given by the formula: a · ξ · b := ( I E ⊗ a ) ξb, a, b ∈ σ ( M ) ′ , ξ ∈ E σ . Again, the products on the right hand side of the equation are ordinary operatorproducts. The concept of the σ - dual of a W ∗ -correspondence was formalized in[4], but it appeared, implicitly, in a number of places. A key role that it will play BSOLUTE CONTINUITY, INTERPOLATION AND THE LYAPUNOV ORDER 5 here is in the identification of the commutant of an induced representation, whichwe will describe in the next section. But here we can already see its relevance forthe present considerations by virtue of the following observation: Let z , z , . . . , z n be points in D ( E, σ ) . Then they define a map Φ z on the n × n matrices over σ ( M ) ′ by the formula(2) Φ z (( a ij )) := ( h z i , a ij · z j i ) ( a ij ) ∈ M n ( σ ( M ) ′ ) . A moment’s reflection reveals that Φ z is completely positive, as it is the compositionof manifestly completely positive maps.Our objective in this note is the proof of the following theorem, which will occupythe next section. Theorem 2.4.
Suppose E is a W ∗ -correspondence over a W ∗ -algebra M and that σ is a faithful normal representation of M on the Hilbert space H σ . Suppose, too,that n distinct points z , z , . . . , z n ∈ AC ( E, σ ) are given and that n operators in B ( H σ ) , W , W , · · · , W n , are given. Define the map Φ z on M n ( σ ( M ) ′ ) by theformula Φ z (( a ij )) = ( h z i , a ij · z j i ) and define the map Φ W on M n ( B ( H σ )) by theformula Φ W (( T ij )) := (cid:0) W i T ij W ∗ j (cid:1) . Then there is an element F in H ∞ ( E ) , with k F k ≤ , such that b F ( z i ) = W i , i = 1 , , . . . , n , if and only if Φ W completelydominates Φ z in the sense of Lyapunov.Proof of Theorem 1.5. That theorem is an immediate consequence of Theorem 2.4.Indeed, in the setting of the former, M = E = C , and σ is just a multiple of theidentity representation, the multiple being the Hilbert space dimension of H σ . Sincewe may safely identify C ⊗ σ H σ with H σ , D ( E, σ ) may be identified with the closedunit ball in B ( H σ ) , i.e., with all contractions on H σ . As we noted, the celebratedtheorems of Sz.-Nagy and Foiaş identify AC ( C , σ ) with the absolutely continuouscontractions on H σ in the classical sense. When these identifications are made, Φ z of Theorem 2.4 becomes the Φ Z of Theorem 1.5 and, of course, the two Φ W ’s arethe same. (cid:3) The Proof of Theorem 2.4
We will first show that if Φ W completely dominates Φ z in the sense of Lyapunov,then we can find an interpolating F ∈ H ∞ ( E ) of norm at most . The route weshall follow is similar, in certain respects, to the route followed in the proof of [4,Theorem 5.3] and is based, ultimately, on the commutant lifting approach to theclassical Nevanlinna-Pick theorem pioneered by Sarason [9]. For this purpose, weneed another way to express Lyapunov dominance that reflects the fact that the z i ’s involved all lie in A C ( E, σ ) . The key tool in our approach is the notion of aninduced representation for T + ( E ) and the connection such representations have withthe concept of absolute continuity. They are defined as follows: Let τ be a normalrepresentation of M on the Hilbert space H τ . Then we may induce τ to F ( E ) ,obtaining a normal representation τ F ( E ) of L ( F ( E )) on the Hilbert space F ( E ) ⊗ τ H τ . The restriction of τ F ( E ) to T + ( E ) , then, is called the representation of T + ( E ) induced by τ . It is clearly an absolutely continuous representation of T + ( E ) , since H ∞ ( E ) is contained in L ( F ( E )) by definition and τ F ( E ) is ultraweakly continuous.As we showed in [2], and will discuss in a moment, induced representations are thearchitypical absolutely continuous representations. We continue to use the notation τ F ( E ) for its restrictions to T + ( E ) and H ∞ ( E ) . PAUL S. MUHLY AND BARUCH SOLEL
In [5] we develop at length the analogies between induced representations of T + ( E ) and H ∞ ( E ) and unilateral shifts. Indeed, a unilateral shift arises from aninduced representation of T + ( E ) , where M = E = C . Definition 3.1.
Let π be a faithful representation of M on H π and assume that π has infinite multiplicity. Then π F ( E ) is called the universal induced representation of T + ( E ) and H ∞ ( E ) determined by π .Any two faithful π ’s with infinite multiplicity give unitarily equivalent inducedrepresentations. Further, every induced representation of T + ( E ) is unitarily equiv-alent to a subrepresentation of π F ( E ) obtained by restricting π F ( E ) to a subspaceof the form F ( E ) ⊗ π K , where K is a subspace of H π that reduces π [2, Paragraphs2.5 and 2.11]. This explains the terminology, allowing us to use the definite article.The representation π and the induced representation π F ( E ) will be fixed for theremainder of this note.We also shall extend the notation I ( σ E ◦ ϕ, σ ) , and write I ( ρ , ρ ) for the set ofoperators C : H ρ → H ρ that intertwine ρ and ρ , where ρ and ρ are any twocompletely contractive representations of T + ( E ) . If ρ i is written as z i × σ i , i = 1 , ,then it is easy to see that an operator C : H ρ → H ρ lies in I ( ρ , ρ ) if and onlyif C ∈ I ( σ , σ ) and z ( I E ⊗ C ) = C z .An important linkage among the universal induced representation, intertwiners,and absolute continuity is the following theorem. Theorem 3.2. [2, Theorem 4.7]
A point z ∈ D ( E, σ ) is absolutely continuous ifand only if _ { Ran ( c ) | c ∈ I ( π F ( E ) , z × σ ) } = H σ . Further, for F ∈ H ∞ ( E ) , z ∈ AC ( E, σ ) , and c ∈ I ( π F ( E ) , z × σ ) , (3) b F ( z ) c = cπ F ( E ) ( F ) . Proof.
The first assertion is explicitly in [2] as Theorem 4.7. The second assertionis easily checked on generators of H ∞ ( E ) of the form ϕ ∞ ( a ) , a ∈ M , and T ξ , ξ ∈ E .That is all that is necessary to check. (cid:3) Recall that if z ∈ D ( E, σ ) , then z ∗ lies in the W ∗ -correspondence E σ over σ ( M ) ′ .It therefore defines a completely positive map Θ z on σ ( M ) ′ by the formula Θ z ( a ) = h z ∗ , a · z ∗ i = z ( I E ⊗ a ) z ∗ , a ∈ σ ( M ) ′ . Indeed, Θ z is just a special case of the map Φ z in the statement of Theorem 2.4. Weare going to use the following theorem from [2] to obtain an alternate formulationof the complete Lyapunov dominance assertion in that theorem. Theorem 3.3. [2, Theorem 4.6] If z ∈ D ( E, σ ) , then an operator q ∈ σ ( M ) ′ is apure superharmonic operator for Θ z if and only if q can be written as q = cc ∗ foran element c ∈ I ( π F ( E ) , z × σ ) . Corollary 3.4.
We adopt the notation of Theorem 2.4. The map Φ W completelydominates Φ z in the sense of Lyapunov if and only if the following condition issatisfied: For every integer m ≥ , for every choice of function l : { , , . . . , m } →{ , , . . . , n } , and for any choice of m operators c j ∈ I ( π F ( E ) , z l ( j ) × σ ) the operatormatrix inequality (4) ( W l ( i ) c i c ∗ j W ∗ l ( j ) ) mi,j =1 ≤ ( c i c ∗ j ) mi,j =1 BSOLUTE CONTINUITY, INTERPOLATION AND THE LYAPUNOV ORDER 7 is satisfied.Proof.
Fix m and a function l : { , , . . . , m } → { , , . . . , n } . Write H ( m ) σ for thedirect sum of m copies of H σ and let σ m be the inflated representation of M on H ( m ) σ , i.e., σ m ( a ) := diag { a, a, · · · , a } . Then σ m ( M ) ′ = M m ( σ ( M ) ′ ) . Also, write E ( m ) for the direct sum of copies of E , which is also a W ∗ -correspondence over M in the obvious way, and set ˜ z := ( z l (1) , z l (2) , · · · , z l ( m ) ) . Then we may view ˜ z as a map from E ( m ) ⊗ σ m H ( m ) σ = ( E ⊗ σ H σ ) ( m ) to H ( m ) σ , which clearly belongsto I ( σ E ( m ) m ◦ ϕ, σ m ) . Consequently, ˜ z defines a completely positive map Θ ˜ z on σ m ( M ) ′ = M m ( σ ( M ) ′ ) , and it is easy to see that Θ ˜ z (( b l ( i ) ,l ( j ) )) mi,j =1 = (cid:16) z l ( i ) ( I E ⊗ b l ( i ) ,l ( j ) ) z ∗ l ( j ) (cid:17) , ( b l ( i ) ,l ( j ) ) mi,j =1 ∈ M m ( σ ( M ) ′ ) . Moreover, by Theorem 3.3, the pure superharmonic elements M m ( σ ( M ) ′ ) for Θ ˜ z are of the form ( c i c ∗ j ) mi,j =1 , where c i ∈ I ( π F ( E ) , ( z l ( i ) × σ )) . On the other hand, the W i ’s may be used to define the completely positive map Ψ W,l on M m ( σ ( M ) ′ ) bythe formula Ψ W,l (( b l ( i ) ,l ( j ) )) := (cid:16) W l ( i ) b l ( i ) ,l ( j ) W ∗ l ( j ) (cid:17) , ( b l ( i ) ,l ( j ) ) mi,j =1 ∈ M m ( σ ( M ) ′ ) . The inequality 4 is the statement that the condition of the corollary is equivalentto the assertion that Ψ W,l dominates Θ ˜ z in the sense of Lyapunov for each choiceof m and l . It is now evident that the condition of the corollary implies that Φ W completely dominates Φ z in the sense of Lyapunov by choosing m and l judiciously.On the other hand, if Φ W completely dominates Φ z , then given m and l , one canclearly choose a k so that the domination of (Φ z ) k by (Φ W ) k in the sense of Lyapunovgives the desired inequalities of the condition for that m and l . (cid:3) In order to follow the commutant lifting approach pioneered by Sarason, werequire the description of the commutant of π F ( E ) ( H ∞ ( E )) that we developed in[4]. The description there works for any induced representation, but we formulateit here specifically for π F ( E ) . Theorem 3.5. [4, Theorem 3.9]
Write ι for the identity representation of π ( M ) ′ on H π , and let τ be the induced representation of L ( E π ) acting on F ( E π ) ⊗ ι H π ,i.e., let τ = ι F ( E π ) . Then the map U : F ( E π ) ⊗ ι H π → F ( E ) ⊗ π H π , defined bythe formula (5) U ( ξ ⊗ ξ ⊗ · · · ⊗ ξ n ⊗ h ) := ( I E ⊗ ( n − ⊗ ξ )( I E ⊗ ( n − ⊗ ξ ) · · · ( I E ⊗ ξ n − ) ξ n ( h ) ,ξ ⊗ ξ ⊗ · · · ⊗ ξ n ⊗ h ∈ ( E π ) ⊗ n ⊗ ι H π , is a Hilbert space isomorphism and U τ ( H ∞ ( E π )) U ∗ = π F ( E ) ( H ∞ ( E )) ′ . Likewise, U ∗ π F ( E ) ( H ∞ ( E )) U = τ ( H ∞ ( E π )) ′ , and the double commutant relationshold: π F ( E ) ( H ∞ ( E )) ′′ = π F ( E ) ( H ∞ ( E )) , and τ ( H ∞ ( E π )) ′′ = τ ( H ∞ ( E π )) . We are now ready to show how the complete domination of Φ z in the sense ofLyapunov by Φ W implies that we can interpolate the W ’s at the z ’s in Theorem2.4. PAUL S. MUHLY AND BARUCH SOLEL
Lemma 3.6.
Let M = span { U ∗ c ∗ h | h ∈ H σ , c ∈ I ( π F ( E ) , z i × σ ) , ≤ i ≤ n } . Then M is a closed subspace of F ( E π ) ⊗ ι H π that is invariant under τ ( H ∞ ( E π )) ∗ .Proof. For X ∈ τ ( H ∞ ( E π )) , U τ ( X ) U ∗ lies in the commutant of π F ( E ) ( H ∞ ( E )) by Theorem (3.5). Consequently cU τ ( X ) U ∗ ∈ I ( π F ( E ) , z i × σ ) for every c ∈I ( π F ( E ) , z i × σ ) . But then τ ( X ) ∗ U ∗ c ∗ h = U ∗ ( U τ ( X ) ∗ U ∗ ) c ∗ h = U ∗ ( cU τ ( X ) U ∗ ) ∗ h lies in M for all U ∗ c ∗ h ∈ M . (cid:3) Lemma 3.7.
The correspondence, U ∗ c ∗ h → U ∗ c ∗ W ∗ i , c ∈ I ( π F ( E ) , z i × σ ) , de-fined on the generators of M extends to a well-defined contraction operator on M , say R , if and only if for every integer m ≥ , for every choice of func-tion l : { , , . . . , m } → { , , . . . , n } , and for every choice of m operators c j ∈I ( π F ( E ) , z l ( j ) × σ ) the operator matrix inequality ( W l ( i ) c i c ∗ j W ∗ l ( j ) ) mi,j =1 ≤ ( c i c ∗ j ) mi,j =1 is satisfied. In this event, R commutes with the restriction of τ ( H ∞ ( E π )) ∗ to M .Proof. A linear combination of generators of M is a vector of the form k = P mj =1 U ∗ c ∗ j h j , where c j ∈ I ( π F ( E ) , z l ( j ) × σ ) for some m and function l : { , , . . . , m } →{ , , . . . n } . Since k k k = X j,i h c i c ∗ j h j , h i i , while k m X j =1 U ∗ c ∗ j W ∗ l ( j ) h j k = X i,j h W i ( l ) c i c ∗ j W ∗ i ( j ) h j , h l i , the first assertion is immediate. But the second is also immediate since R is “rightmultiplication” by W ∗ i on a generator of the form U ∗ c ∗ h , c ∈ I ( π F ( E ) , z i × σ ) , i.e., RU ∗ c ∗ h = U ∗ c ∗ W ∗ i h , while the restriction of τ ( X ) ∗ to M acts by left multiplicationfor all X ∈ H ∞ ( E π ) : τ ( X ) ∗ U ∗ c ∗ h = U ∗ ( U τ ( X ) ∗ U ∗ ) c ∗ h . (cid:3) Since M is invariant for τ ( H ∞ ( E π )) ∗ , we obtain an ultra weakly continuous com-pletely contractive representation ρ of H ∞ ( E π ) on M by compressing τ ( H ∞ ( E π )) to M , i.e., ρ ( X ) := P M τ ( X ) |M , X ∈ H ∞ ( E π ) . Since τ is isometric in the sense of [6] and since R ∗ commutes with ρ ( H ∞ ( E π )) , wemay apply our commutant lifting theorem [6, Theorem 4.4] to conclude that thereis an operator Y ∈ B ( F ( E π ) ⊗ ι H π ) of norm at most one such that P M Y |M = R ∗ , Y M ⊥ ⊆ M ⊥ , and Y commutes with τ ( H ∞ ( E π )) (see [2, Theorem 2.6], also). ByTheorem 3.5, there is an F ∈ H ∞ ( E ) , k F k ≤ , such that Y = U ∗ π F ( E ) ( F ) U . Weconclude from the properties of Y and the definition of R that U ∗ π F ( E ) ( F ) ∗ c ∗ h = ( U ∗ π F ( E ) ( F ) ∗ U ) U ∗ c ∗ h = Y ∗ U ∗ c ∗ h = RU ∗ c ∗ h = U ∗ c ∗ W ∗ i h for all c ∈ I ( π F ( E ) , z i × σ ) . This, in turn, implies that cπ F ( E ) ( F ) = W i c for all such c . But cπ F ( E ) ( F ) = b F ( z i ) c for all c ∈ I ( π F ( E ) , z i × σ ) , by equation (3)in Theorem 3.2. Therefore, b F ( z i ) c = W i c BSOLUTE CONTINUITY, INTERPOLATION AND THE LYAPUNOV ORDER 9 for all i and all c ∈ I ( π F ( E ) , z i × σ ) . However, by hypothesis, all the z i lie in AC ( E, σ ) . Consequently, by the first assertion of Theorem 3.2, the closed span ofthe ranges of the c ’s in I ( π F ( E ) , z i × σ ) is all of H σ , for every i . We conclude that b F ( z i ) = W i for every i . This completes the proof that if Φ W completely dominates Φ z in the sense of Lyapunov, then there is an F ∈ H ∞ ( E ) that interpolates W i at z i . Proof of the Converse.
Part of the argument just given is reversible. Suppose F isan element of H ∞ ( E ) of norm at most one such that b F ( z i ) = W i , i = 1 , , · · · , n .Then for each c ∈ I ( π F ( E ) , z i × σ ) cπ F ( E ) ( F ) = b F ( z i ) c = W i c, by equation (3). But then ( U ∗ π F ( E ) ( F ) ∗ U ) U ∗ c ∗ = U ∗ c ∗ W ∗ i for all c ∈ I ( π F ( E ) , z i × σ ) . Since the norm of F is at most we concludefrom Lemma 3.7 that for every integer m ≥ , for every choice of function l : { , , . . . , m } → { , , . . . , n } , and for every choice of m operators c j ∈ I ( π F ( E ) , z l ( j ) × σ ) the operator matrix inequality ( W l ( i ) c i c ∗ j W ∗ l ( j ) ) mi,j =1 ≤ ( c i c ∗ j ) mi,j =1 is satisfied. So by Corollary 3.4, we conclude that Φ W completely dominates Φ z inthe sense of Lyapunov. (cid:3) References [1] N. Cohen and I. Lewkowicz,
The Lyapunov order for real matrices , Linear Algebra and ItsApplications, , (2009), 1849-1866.[2] P. S. Muhly and B. Solel,
Representations of Hardy algebras: absolute continuity, inter-twiners, and superharmonic operators , Integral Equations and Operator Theory (2011),151-203 (arXiv:1006.1398).[3] P. S. Muhly and B. Solel, Schur Class Operator Functions and Automorphisms of HardyAlgebras , Documenta Math. (2008), 365–411.[4] P. S. Muhly and B. Solel, Hardy algebras, W ∗ -correspondences and interpolation theory ,Math. Ann. (2004), 353-415.[5] P. S. Muhly and B. Solel , Tensor algebras, induced representations, and the Wold decompo-sition , Canad. J. Math. (1999), 850-880.[6] P. S. Muhly and B. Solel, Tensor algebras over C ∗ - correspondences (Representations, dila-tions, and C ∗ - envelopes ), J. Functional Anal. (1998), 389–457.[7] M. Rieffel, Morita equivalence for C ∗ -algebras and W ∗ -algebras , J. Pure Appl. Alg. (1974),51–96.[8] M. Rieffel, Induced representations of C ∗ -algebras , Adv. in Math. (1974), 176–257.[9] D. Sarason, Generalized interpolation in H ∞ , Trans. Amer. Math. Soc. (1967), 179–203.[10] B. Sz.-Nagy, C. Foiaş, H. Bercovici, and L. Kérchy, Harmonic Analysis of Operators onHilbert Space , Springer, New York, 2010.
Department of Mathematics, University of Iowa, Iowa City, IA 52242
E-mail address : [email protected] Department of Mathematics, Technion, 32000 Haifa, Israel
E-mail address ::