aa r X i v : . [ m a t h . OA ] O c t Reflexivity of non commutative Hardy Algebras
Leonid Helmer
Abstract
Let H ∞ ( E ) be a non commutative Hardy algebra, associated with a W ∗ -correspondence E . These algebras were introduced in 2004, [16], byP. Muhly and B. Solel, and generalize the classical Hardy algebra of theunit disc H ∞ ( D ). As a special case one obtains also the algebra F ∞ ofPopescu, which is H ∞ ( C n ) in our setting.In this paper we view the algebra H ∞ ( E ) as acting on a Hilbert spacevia an induced representation. We write it ρ ( H ∞ ( E )) and we study thereflexivity of ρ ( H ∞ ( E )). This question was studied by A. Arias and G.Popescu in the context of the algebra F ∞ , and by other authors in severalother special cases. As it will be clear from our work, the extension tothe case of a general W ∗ -correspondence E over a general W ∗ -algebra M requires new techniques and approach.We obtain some partial results in the general case and we turn to thecase of a correspondence over factor. Under some additional assumptionson the representation π : M → B ( H ) we show that ρ π ( H ∞ ( E )) is reflex-ive. Then we apply these results to analytic crossed products ρ ( H ∞ ( α M ))and obtain their reflexivity for any automorphism α ∈ Aut ( M ) whenever M is a factor. Finally, we show also the reflexivity of the compression ofthe Hardy algebra to a suitable coinvariant subspace M , which may bethought of as a generalized symmetric Fock space. In this paper we consider the question of reflexivity of the non commutativeHardy algebras ρ ( H ∞ )( E ). These algebras, that were introduced by Muhlyand Solel in [15] and [16], can be viewed as a far reaching generalizations of theclassical Hardy algebra H ∞ ( D ).Remember that H ∞ ( D ) is the algebra of all the bounded analytic functionson the unit disc D ⊂ C , and may be identified with the WOT closed algebra,generated by the unilateral shift on the Hilbert space l ( Z + ). In [21] G. Popescugeneralized it to WOT-closed algebras that are generated by d shifts and denotedby F ∞ d . Note that the free semigroup algebra L n studied in the late 90-th byK. Davidson and D. Pitts coincides with F ∞ , (see [6]).These algebras were further generalized by Muhly and Solel who introducedin [15] and [16] the non commutative tensor algebras T + ( E ) and Hardy algebras H ∞ ( E ), associated with a C ∗ - or a W ∗ -correspondence E . By a right Hilbert C ∗ -module over the C ∗ -algebra A we mean a right A -module E , equipped withan A -valued inner product, that is a function E × E → A such that ( ξ, ζ ) ξ, ζ i ∈ A , for ξ, ζ ∈ A , a ∈ A , and1) h ξ, ζa i = h ξ, ζ i a ,2) h ξ, ζ + ζ i = h ξ, ζ i + h ξ, ζ i ,3) h ξ, ζ i = h ζ, ξ i ∗ , 1) h ξ, ξ i ≥
0, and h ξ, ξ i = 0 if and only if ξ = 0.This inner product defines a norm on E by the formula k ξ k := kh ξ, ξ ik .The completion of ( E, k · k ) with respect to this norm is called a (right) Hilbert C ∗ -module.By a bounded operator T we mean a right module map T : E → E , which isbounded in the norm defined above. Let T be a map from Hilbert C ∗ -module E into a Hilbert C ∗ -module F . We say that T is adjointable if there exists amap T ∗ : F → E such that h T x, y i = h x, T ∗ y i ,x ∈ E , y ∈ F . Such T ∗ is called the adjoint map or the adjoint operatorfor T . Every adjointable map is A -linear and bounded, ( [12]). The set of alladjointable maps from E to F we denote by L ( E, F ) = L A ( E, F ), and if E = F we write L ( E ). It is known that L ( E ) is a C ∗ -algebra. The standard referenceto the theory of Hilbert C ∗ -modules is [12].Every C ∗ -algebra A has a natural structure of a Hilbert C ∗ -module overitself. The right action is the multiplication in A and for the inner product set h a, b i = a ∗ b . The Hilbert C ∗ -module over the C ∗ -algebra A is called self-dualif every bounded A -linear map T : E → A is given by an inner product, whichmeans that there is x ∈ E such that for every y ∈ E , T y = h x, y i . It is knownthat if E is self-dual then B ( E ) = L ( E ) ( [12]).In this paper we are interested in the Hilbert modules over W ∗ -algebras.Recall that by a W ∗ -algebra we mean a C ∗ -algebra M that admits a faithfulrepresentations π : M → B ( H ) such that π ( M ) is von Neumann algebra onthe Hilbert space H . According to the abstract characterization given by Sakai,[26], C ∗ -algebra M is a W ∗ -algebra if and only if it is isomorphic to a dualBanach space.A detailed study of Hilbert modules over W ∗ -algebras was made by Paschkein [19].A C ∗ -Hilbert module E over the W ∗ -algebra M which is also self-dual willbe called a Hilbert W ∗ -module. In [19] Paschke proved that every Hilbert C ∗ -module over the W ∗ -algebra M admits the so called self-dual completion. Inthis case both E and L ( E ) are dual spaces in the sense of Banach space theory,[19]. In particular, L ( E ) is a W ∗ -algebra. The weak ∗ -topology on E , thatcomes from the structure of the dual space on E will be called the σ -topology(following the terminology of [3]).By a W ∗ -correspondence we mean a right Hilbert W ∗ -module which is madeinto a bimodule over M by some normal ∗ -homomorphism φ of M into the W ∗ -algebra of adjointable operators L ( E ). Associated with a W ∗ -correspondence E we have another W ∗ -correspondence F ( E ) over the same algebra M , that isdefined to be the direct sum M ⊕ E ⊕ E ⊗ ⊕ ... of the internal tensor powers of E . An exact definitions will be given in the next section. F ( E ) is called the fullFock correspondence and, in fact, it is a W ∗ -correspondence with the left actionof M denoted by φ ∞ , which is a natural extension of φ to a representation of M in the algebra of adjointable operators on F ( E ). The non commutative Hardy2lgebra of a correspondence E is, by definition, the weak ∗ -closure in L ( F ( E )) ofthe algebra spanned by operators of the form T ξ , ξ ∈ E , where T ξ ( η ) := ξ ⊗ η , η ∈F ( E ), and φ ∞ ( a ), a ∈ M . In fact, Muhly and Solel defined this Hardy algebraas the weak ∗ closure of the noncommutative tensor algebra T + ( E ). The algebra T + ( E ) was defined first in [15] as the norm closed (nonselfadjoint) algebraspanned by the same set of generators, and it generalizes the noncommutativedisc algebra A n of Popescu, which in turn is a noncommutative generalizationof the classical disc algebra. Finally, the C ∗ -algebra, generated by the same setof operators is called the Toeplitz C ∗ -algebra of the given correspondence. Inthis context the algebra F ∞ coincides with the Hardy algebra H ∞ ( C n ), where C n is considered as the Hilbert W ∗ -module over C .These non commutative Hardy algebras, or simply Hardy algebras, general-ize a wide class of known nonselfadjoint operator algebras. If we take M = E = C then F ( E ) is the Hilbert space l ( Z + ), and the associated Hardy algebra is theclassical algebra H ∞ ( D ). Various choices of W ∗ -correspondence E give us suchalgebras as the algebra F ∞ of G. Popescu, the free semigroup algebras, quiveralgebras and analytic crossed products.In this paper we view the algebra H ∞ ( E ) as acting on a Hilbert spacevia an induced representation (induced in sense of Rieffel, see [23]). Given arepresentation π of M on H , the induced representation, written ρ π (or simply ρ ,) is a representation of H ∞ ( E ) on the Hilbert space F ( E ) ⊗ π H defined bysending X ∈ H ∞ ( E ) to X ⊗ I H . Thus we consider the question of reflexivityof the algebra ρ ( H ∞ ( E )) and our results may be viewed as an extension of theresults of A.Arias and G.Popescu in [2]. It will be clear from our work that theextension to more general von Neumann algebra M requires new techniques andapproach. A key tool that we will need and use here is the concept of dualityfor W ∗ -correspondences, developed in [16].Let A be any algebra acting on a Hilbert space H . Then the algebra A issaid to be reflexive if it is defined by its invariant subspaces. In a more details,with every subset A ⊆ B ( H ) let Lat A be the lattice of all A -invariant closedsubspaces in H . Thus, Lat A := {M ⊆ H : A M ⊆ M , ∀ A ∈ A , M is a closed subspace in H } . Let L be a set of closed subspaces in H . Then the operator Alg associatesto L an algebra Alg
L ⊆ B ( H ) as follows. Alg L := { A ∈ B ( H ) : L ⊆
Lat A } , i.e. Alg L = { A ∈ B ( H ) : A M ⊆ M , ∀M ∈ L} . Clearly, Alg L is a unitalWOT-closed (hence ultraweakly closed) subalgebra in B ( H ).Let A be a subalgebra in B ( H ). Then always A ⊆
Alg Lat A , and bydefinition, A is reflexive if A = Alg Lat A . Thus, every reflexive algebra A is necessarily unital and ultraweakly closed.3 single operator T is called reflexive if the WOT closed algebra, generatedby T and identity I , is reflexive. We denote this algebra by W ( T ). The uni-lateral shift S is reflexive. In this case W ( S ) is the algebra of analytic Toeplitzoperators, and the weak topology and the ultraweak topology coincide whenrestricted to W ( S ). Thus the Hardy algebra H ∞ ( T ) ∼ = ρ ( H ∞ ( C )) is reflexive.The following simple example shows that not every WOT closed algebrais reflexive. Let A be an algebra of all 2 × C of the form (cid:18) a b a (cid:19) , a, b ∈ C . Clearly, A is WOT closed. But it is easy to see that Alg Lat A = { (cid:18) a b c (cid:19) : a, b, c ∈ C } Thus, A is not reflexive. The notion of reflexivity was introduced first by H.Radjavi and P. Rosenthal in [24] (the terminology was suggested by P. Halmos).It is easy to see that every von Neumann algebra is reflexive, which is equivalentto the von Neumann bicommutant theorem. Since the paper [24] appears, thesubject of reflexivity was a subject of intensive study and generalizations. Agood general overview of reflexivity is given by D. Hadwin in [7].The works which are closely connected to our theme are [2], [22], [4], [5],[6], [10], [11] [9]. In particular, in [2], A. Arias and G. Popescu proved reflex-ivity of the algebra F ∞ , which, as we shall see later, coincides with ρ ( H ∞ ( C n )in our setting. Later, in [22] Popescu proved reflexivity of this algebra whenit is compressed to the symmetric Fock space. This algebra also was studiedby Davidson and Pitts and by Davidson in the context of the free semigroupalgebra L n , see [4], [5], [6]. In the recent work ‘[10], M. Kennedy showed thereflexivity of all the free semigroup algebras and the hyperreflexivity of some ofthem (the definition of hyperreflexivity will be given later). In the work [11],D. Kribs and S. Power initiated the study of the free semigroupoid algebras L G and in particular proved their reflexivity. In [9] E. Kakariadis showed thereflexivity of the analytic crossed product ρ ( H ∞ ( α M )) in the special case whenthe α is a unitary implemented automorphism of von Neumann algebra.This paper is organized as follows. After introducing preliminaries we startwith some simple observations about hyperreflexivity and then turn to our mainresults. One of the our central result is Theorem 3 . ρ ( H ∞ ( E )) we need to show thatfor every Y ⊗ I H ∈ Alg Lat ρ ( H ∞ ( E )) each Fourier coefficient { Φ j ( Y ) ⊗ I H } , j = 0 , , ... is in ρ ( H ∞ ( E )). This approach generalizes the approach of Popescuand Arias in their proof of the reflexivity of F ∞ , [2]. To this end, we introducethe coinvariant subspace M that may be thought of as a generalization of thesymmetric Fock space and seems to be of particular interest. Using this subspacewe are able to show that Φ ( Y ) is in ρ ( H ∞ ( E )). For the Fourier coefficientsΦ j ( Y ) with j ≥ Y ∈ Alg Lat ( H ∞ ( E )), this method doesnot work for a general W ∗ -algebra. But if M is assumed to be a factor, thenunder an additional assumption on the representation π we are able to showthe reflexivity of our algebra. As an example we consider the analytic crossed4roducts ρ ( H ∞ ( α M )). Finally, we consider the compression of the Hardy alge-bra to the subspace M , i.e. we consider the algebra Qρ ( H ∞ ( E )) | M , where Q isthe projection onto M , and show that it is reflexive.This work is based on part of the authors Ph.D. thesis. W ∗ -correspondences and Hardy algebras. We start by recalling the definition of a W ∗ -correspondence. Definition 2.1.
Let E be a (right) Hilbert W ∗ -module over a W ∗ -algebra M ,that is a self-dual C ∗ -module over the W ∗ -algebra M , and let φ : N → L ( E ) bea normal ∗ -homomorphism of the W ∗ -algebra N into the algebra of adjointableoperators L ( E ) . Then φ defines on E the structure of the left module over N .This N - M -bimodule is called the W ∗ -correspondence from N to M . If N = M we speak about a W ∗ -correspondence over M . In what follows we always assume that our W ∗ -correspondence over M isessential as a left M -module, meaning that φ ( M ) E is dense in E in the σ -topology.The obvious example of W ∗ -correspondence over M = C is the ordinaryHilbert space H with the inner product is taken to be linear in the secondvariable.Let M be a W ∗ -algebra which we view as the Hilbert C ∗ -module over it-self. The self dual completion gives rise to a W ∗ -module. If α is some normalautomorphism of M we set φ ( a ) b := α ( a ) b for the left action. Then M turnsout to be a W ∗ -correspondence over itself and is denoted by α M . More gener-ally, if α : M → M is a normal ∗ -homomorphism, that is α ∈ End ( M ), then E := α ( M ) M turns out to be a M ∗ -correspondence with α as the left action.Note that α (1) = p - some projection of M . Thus, we conclude that E is acyclic right module of the form pM . Later, we will return to this module inmore details.Let M , N and Q be three W ∗ -algebras, E be a M - N W ∗ - correspondence,and F be a N - Q W ∗ -correspondence. Write π for the left action of N on F ,and write E ⊗ alg F for the algebraic tensor product of C -vector spaces E and F . By the internal C ∗ -tensor product (balanced over π ) of these modules wemean the Hausdorff completion of E ⊗ alg F by the inner product defined by h ξ ⊗ η , ξ ⊗ η i = h η , π ( h ξ , ξ i ) η i . This tensor product will be denoted by E ⊗ π F . For the right action of Q we set ( ξ ⊗ η ) b = ξ ⊗ ( ηb ), b ∈ Q . The self-dualcompletion of E ⊗ π F gives rise to a Hilbert W ∗ -module over the W ∗ -algebra Q . For the left action we set φ E ⊗ π F ( a )( ξ ⊗ η ) = ( φ E ( a ) ξ ) ⊗ η . Thus, we obtainon E ⊗ π F a structure of a W ∗ -correspondence from M to Q .It is easy to see that if E is a Hilbert module over a W ∗ -algebra M and π : M → B ( H ) a normal representation of M on the Hilbert space H then E ⊗ π H is Hilbert space. 5et E be a W ∗ -correspondence over a W ∗ -algebra M with a left actiondefined as usual by a normal ∗ -homomorphism φ . For each n ≥ E ⊗ n bethe self-dual internal tensor power (balanced over φ ). So, E ⊗ n itself turns outto be a W ∗ -correspondence in a natural way, with the left action ξ φ n ( a ) ξ =( φ ( a ) ξ ) ⊗ ... ⊗ ξ n , ξ = ξ ⊗ ... ⊗ ξ n ∈ E ⊗ n , and with an M -valued inner productwhich comes from the construction of the internal tensor product.Let { E i : i ∈ I } be a family of W ∗ -correspondences over M . By P ⊕ i E i wedenote the set of all sequences x = ( x i ), x i ∈ E i , such that P i h x i , x i i convergesin M (considered as a C ∗ -algebra). For x = ( x i ) and y = ( y i ) in P ⊕ i E i , wedefine h x, y i = P i h x i , y i i . This defines an inner product on P ⊕ i E i and in fact P ⊕ i E i is complete in the norm defined by this inner product. With the obviousright action of M on P ⊕ i E i , this module is a Hilbert module over M in the C ∗ -sense. Then the self-dual completion of P ⊕ i E i will be called the ultraweakdirect sum of the family { E i : i ∈ I } (for the explicit description see [19].For the left action we set φ = P ⊕ i φ E i , where φ E i is a left action of M on E i .Clearly, we obtain on P ⊕ i E i a structure of the W ∗ -correspondence over M .We form the full Fock space F ( E ) = P ⊕ n ≥ E ⊗ n , where E ⊗ = M and thedirect sum taken in the ultraweak sense. This is a W ∗ -correspondence withleft action given by φ ∞ : M → L ( F ( E )), where φ ∞ ( a ) = P n ≥ φ n ( a ). The M -valued inner product on F ( E ) is defined in an obvious way.For each ξ ∈ E and each η ∈ F ( E ), let T ξ : η ξ ⊗ η be a creation operatoron F ( E ). Clearly, T ξ ∈ L ( F ( E )). Definition 2.2.
Given a W ∗ -correspondence E over a W ∗ -algebra M .1) The norm closed subalgebra of L ( F ( E )) , generated by all creation opera-tors T ξ , ξ ∈ E , and all operators φ ∞ ( a ) , a ∈ M , is called the tensor algebra of E . It is denoted by T + ( E ) .2) The Hardy algebra H ∞ ( E ) is the ultraweak closure of T + ( E ) in the W ∗ -algebra L ( F ( E )) . Examples 2.3. (1) Let A = E = C . Then F ( E ) = l ( Z + ) and the algebra T + ( E ) is the algebra of analytic Toeplitz operators with continuous symbols, soit can be identified with the disc algebra A ( D ). The algebra H ∞ ( E ), in thiscase, is H ∞ ( D ).(2) Let A = C and let E be an n -dimensional Hilbert space over C , i.e., H = C n . In this case T + ( E ) is the non commutative disc algebra A n , studiedby Popescu and others. The algebra H ∞ ( C n ) was denoted F ∞ n by Popescu.This algebra can be identified with the free semigroup algebra L n studied byDavidson and Pitts.Let π : M → B ( H ) be a normal representation of a W ∗ -algebra M on aHilbert space H and let E be a W ∗ -correspondence over M . As we alreadynoted, the W ∗ -internal tensor product E ⊗ π H is a Hilbert space. The repre-sentation π E : L ( E ) → B ( E ⊗ π H ) defined by π E : S S ⊗ I H , ∀ S ∈ L ( E ) .
6s called the induced representation (in the sense of Rieffel). If π is a faithfulnormal representation then π E maps L ( E ) into B ( E ⊗ π H ) homeomorphicallywith respect to the ultraweak topologies, [16, Lemma 2.1].In this work, we consider the image of H ∞ ( E ) under an induced representa-tion, defined as follows. Let π : M → B ( H ) be a faithful normal representation.For a W ∗ -correspondence E over M let π F ( E ) be the induced representationof L ( F ( E )) in B ( F ( E ) ⊗ π H ). Then the induced representation of the Hardyalgebra H ∞ ( E ) is the restriction ρ := π F ( E ) | H ∞ ( E ) : H ∞ ( E ) → B ( F ( E ) ⊗ π H ) . (1)This restriction is an ultraweakly continuous representation of H ∞ ( E ) and theimage ρ ( H ∞ ( E )) is an ultraweakly closed subalgebra of B ( F ( E ) ⊗ π H ). Weshall refer to ρ as the representation induced by π . Later, when we discussseveral representation of H ∞ ( E ) that are induced by different representations π , σ etc. of M , we shall write ρ π , ρ σ etc.So, ρ ( H ∞ ( E )) acts on F ( E ) ⊗ π H and ρ is defined by ρ : X X ⊗ I H , ∀ X ∈ H ∞ ( E ) . Note that the notion of the induced representation generalizes the notion ofpure isometry in the theory of a single operator.We will frequently use the following result of Rieffel [23, Theorem 6.23].The formulation here is in a form convenient for us ( [14, p. 853]).
Theorem 2.4. . Let E be a W ∗ -correspondence over the algebra M and π : M → B ( H ) be a normal faithful representation of M on the Hilbert space H .Then the operator R in B ( E ⊗ π H ) commutes with π E ( L ( E )) if and only if R is of the form I E ⊗ X , where X ∈ π ( M ) ′ , i.e., π E ( L ( E )) ′ = I E ⊗ π ( M ) ′ . Definition 2.5.
Let E be a W ∗ -correspondence over a W ∗ -algebra M .(1) By a covariant representation of E , or of the pair ( E, M ) , on a Hilbertspace H , we mean a pair ( T, σ ) , where σ : M → B ( H ) is a nondegeneratenormal ∗ -homomorphism, and T is a bimodule (with respect to σ ) map T : E → B ( H ) , that is a linear map such that T ( ξa ) = T ( ξ ) σ ( a ) and T ( φ ( a ) ξ ) = σ ( a ) T ( ξ ) , ξ ∈ E and a ∈ M . We require also that T will be continuous withrespect to the σ -topology on E and the ultraweak topology on B ( H ) .(2) The representation ( T, σ ) is called (completely) bounded, (completely)contractive, if so is the map T . For a completely contractive covariant repre-sentation we write also c.c.c.r.(3) The covariant representation ( T, σ ) is called isometric covariant repre-sentation (i.c.r.) if T ( ξ ) ∗ T ( η ) = σ ( h ξ, η i ) . The operator space structure on E to which this definition refers is the onewhich comes from the embedding of E into its so-called linking algebra L ( E ),see [15]. 7very isometric covariant representation ( V, π ) of E is completely contractive(see [15, Corollary 2.13]).As an important example let ρ = π F ( E ) | H ∞ ( E ) be an induced representationof the Hardy algebra H ∞ ( E ). For the representation σ set σ = π F ( E ) ◦ φ ∞ , and set V ( ξ ) = π F ( E ) ( T ξ ) , ξ ∈ E. Definition 2.6.
The pair ( V, σ ) is called the covariant representation inducedby π , or simply the induced covariant representation (associated with ρ ). It is easy to check that (
V, σ ) in the above Definition is isometric, hence, iscompletely contractive.Let (
T, σ ) be a c.c.c.r. of (
E, M ) on the Hilbert space H as above. Witheach such representation we associate the operator ˜ T : E ⊗ σ H → H , that onthe elementary tensors is defined by˜ T ( ξ ⊗ h ) := T ( ξ )( h ) . ˜ T is well defined since T ( ξa ) = T ( ξ ) σ ( a ). In [15] Muhly and Solel show thatthe properties of ˜ T reflect the properties of the covariant representation ( T, σ ).They proved that ( α ) ˜ T is bounded iff T is completely bounded, and in thiscase k T k cb = k ˜ T k ; ( β ) ˜ T is contractive iff T is completely contractive; and ( γ )˜ T is an isometry iff ( T, σ ) is an isometric representation. A simple calculationgives us the intertwining relation˜
T σ E ◦ φ ( a ) = ˜ T ( φ ( a ) ⊗ I H ) = σ ( a ) ˜ T , ∀ a ∈ A. (2)We have the following lemma, taken from [16, Lemma 2.5]. Lemma 2.7.
There exists a bijective correspondence ( T, σ ) ↔ ˜ T , between allcompletely contractive representations ( T, σ ) of E on a Hilbert space H , and allcontractive operators ˜ T : E ⊗ σ H → H , that satisfy the relation ˜ T σ E ◦ φ ( a ) = σ ( a ) ˜ T , ∀ a ∈ A . Given a contractive operator ˜ T : E ⊗ σ H → H that satisfies theabove intertwining relation, then the associated covariant representation ( T, σ ) ,is defined by T ( ξ ) h := ˜ T ( ξ ⊗ h ) , h ∈ H and ξ ∈ E . Remark 2.8.
Let E be a W ∗ -correspondence over the algebra M and let ( T, σ )be a c.c.c.r. of (
E, M ) on a Hilbert space H . It is shown in [16] that for everysuch c.c.c.r. there exists a completely contractive representation ρ : T + ( E ) → B ( H ) such that ρ ( T ξ ) = T ( ξ ) for every ξ ∈ E and ρ ( φ ∞ ( a )) = σ ( a ) for every a ∈ M . Moreover, the correspondence ( T, σ ) ↔ ρ is a bijection between the setof all c.c.c.r. of E and all completely contractive representations of T + ( E ) whoserestrictions to φ ∞ ( M ) are continuous with respect to the ultraweak topologyon L ( F ( E )).The representation ρ of T + ( E ) that corresponds to the c.c.c.r. ( T, σ ) is calledthe integrated form of (
T, σ ) and denoted by σ × T . In its turn, the c.c.c.r. ( T, σ )8s called the desintegrated form of ρ . Preceding results show that, given a normalrepresentation σ of M , the set of all completely contractive representations of T + ( E ) whose restrictions to φ ∞ ( M ) is given by σ can be parameterized by thecontractions ˜ T ∈ B ( E ⊗ σ H, H ), that satisfy the relation (2).In [16] it was shown that, if the c.c.c.r. (
T, σ ) of (
E, M ) is such that k ˜ T k < σ × T extends from T + ( E ) to an ultraweakly continuousrepresentation of H ∞ ( E ). For a general c.c.c.r. ( T, σ ), the question when suchextention is possible σ × T is more delicate, see about this [18].In the above notations, the induced representation ρ π is an integrated formof the ( V, σ ), the covariant induced representation of E from Definition 2 . V, σ ) be an isometric covariantrepresentation of a general W ∗ -correspondence E on a Hilbert space G . Forevery n ≥ V ⊗ n , σ ) for the isometric covariant representation of E ⊗ n onthe same space G defined by the formula V ⊗ n ( ξ ⊗ ... ⊗ ξ n ) = V ( ξ ) · · · V ( ξ n ), n ≥
1. The associated isometric operator ˜ V n : E ⊗ n ⊗ σ G → G (which is calledthe generalized power of ˜ V ), satisfies the identity ˜ V n σ E ⊗ n ◦ φ n = ˜ V n ( φ n ⊗ I G ) = σ ˜ V n . In this notation ˜ V = ˜ V .With ( V, σ ) we may associate the ”shift” L , that acts on the lattice of σ ( M )-invariant subspaces of G , and is defined as follows. Let M ∈
Lat ( σ ( M )), thenwe set L ( M ) := _ { V ( ξ ) k : ξ ∈ E, k ∈ M} . (3)The s -power L s ( M ) is defined in the obvious way (with L ( M ) = M )).The subspace M ∈
Lat ( σ ( M )), as well as its projection P M ∈ σ ( M ) ′ , iscalled wandering with respect to ( V, σ ), if the subspaces L s ( M ), s = 0 , , ... ,are mutually orthogonal. Write σ ′ for the restriction σ | M , where M is wander-ing. Then the Hilbert space E ⊗ s ⊗ σ ′ M is isometrically isomorphic (under thegeneralised power ˜ V s ) to L s ( M ). Hence, we obtain an isometric isomorphism F ( E ) ⊗ σ ′ M ∼ = ⊕ X s ≥ L s ( M ) . W ∗ -correspondences and commutant. The principal tool that will be used often in this work is the duality of W ∗ -correspondences, that was developed in [16, Section 3]. We shall need the notionof isomorphic W ∗ -correspondences. Let E and F be W ∗ -correspondences over W ∗ -algebras M and N respectively. The left action of M on E will be denotedas usual by φ and the left action of N on F by ψ , thus, ψ : N → L ( F ) is anormal ∗ -homomorphism. Definition 2.9.
An isomorphism of E and F is a pair ( σ, Φ) where1) σ : M → N is an isomorphism of W ∗ -algebras and ) Φ : E → F is a vector space isomorphism preserving the σ -topology,and which is also (a) a bimodule map, Φ( φ ( a ) xb ) = ψ ( σ ( a ))Φ( x ) σ ( b ) , x ∈ Ea, b ∈ M , and(b) Φ ”preserves” the inner product, h Φ( x ) , Φ( y ) i = σ ( h x, y i ) , x, y ∈ E . Let π : M → B ( H ) be a normal representation of M on a Hilbert space H .We put E π := { η : H → E ⊗ π H : ηπ ( a ) = ( φ ( a ) ⊗ I H ) η, a ∈ M } . (4)On the set E π we define the structure of a W ∗ -correspondence over thevon Neumann algebra π ( M ) ′ putting h η, ζ i := η ∗ ζ for the π ( M ) ′ -valued innerproduct, η, ζ ∈ E π . It is easy to check that h η, ζ i ∈ π ( M ) ′ . For the bimoduleoperations: X · η = ( I ⊗ X ) η , and η · Y = ηY , where X, Y ∈ π ( M ) ′ . Definition 2.10.
The W ∗ -correspondence E π is called the π -dual of E . Write ( E π ) ∗ for the space of adjoints of the operators in E π . By D ( E π ) and D ( E π ) we denote the (norm) closed and open unit balls in E π respectively. Let η ∗ ∈ D ( E π ) ∗ and let ( T, π ) the associated c.c.c.r. of (
E, M ) on H such that η ∗ =˜ T . Since every ( T, π ) satisfies relation (2), we obtain that all the representations(
T, π ) of (
E, M ) are parameterized by the points of D ( E π ) ∗ . Hence, all thecompletely contractive representations ρ of T + ( E ) such that ρ ◦ φ ∞ = π areparameterized bijectively by D ( E π ) ∗ .Let ι : π ( M ) ′ → B ( H ) be the identity representation. Then we can form E π,ι := ( E π ) ι . So, E π,ι = { S : H → E π ⊗ ι H : Sι ( a ) = ι E π ◦ φ E π ( a ) S, a ∈ π ( M ) ′ } . This is a W ∗ -correspondence over π ( M ) ′′ = π ( M ).In [16] it was proved that for every faithful normal representation π of a W ∗ -algebra M , every W ∗ -correspondence E over M is isomorphic to E π,ι . Wegive a short description of this isomorphism.Let L ξ : h ξ ⊗ h , h ∈ H , ξ ∈ E . L ξ is a bounded linear map since k L ξ h k ≤ k ξ k k h k and L ∗ ξ ( ζ ⊗ h ) = π ( h ξ, ζ i ) h . For each ξ ∈ E we define themap ˆ ξ : H → E π ⊗ ι H by means of its adjoint:ˆ ξ ∗ ( η ⊗ h ) = L ∗ ξ ( η ( h )) ,η ⊗ h ∈ E π ⊗ ι H . Theorem 2.11. ( [16, Theorem 3.6]) If the representation π of M on H is faithful, then the map ξ ˆ ξ just defined, is an isomorphism of the W ∗ -correspondences E and E π,ι . For every k ≥
0, let U k : E ⊗ k ⊗ π H → ( E π ) ⊗ k ⊗ ι H be the map defined interms of its adjoint by U ∗ k ( η ⊗ ... ⊗ η n ⊗ h ) = ( I E ⊗ k − ⊗ η ) ... ( I E ⊗ η k − ) η k ( h ).It is proved in [16] that U k is a Hilbert space isomorphism from E ⊗ k ⊗ π H onto( E π ) ⊗ k ⊗ ι H .By Theorem 2 .
11, for every k ≥ W ∗ -correspondence E ⊗ k over M isisomorphic to the W ∗ -correspondence ( E ⊗ k ) π,ι ∼ = ( E π,ι ) ⊗ k . If ξ ∈ E ⊗ k then10he corresponding element b ξ ∈ ( E ⊗ k ) π,ι is defined now by the formula b ξ ∗ ( η ⊗ ... ⊗ η k ⊗ h ) = L ∗ ξ U ∗ k ( η ⊗ ... ⊗ η k ⊗ h ) , where L ξ : h ξ ⊗ h is a bounded linear map from H to E ⊗ k ⊗ π H . Thus, weobtain ˆ ξ = U k L ξ , for ξ ∈ E ⊗ k . (5)For the dual correspondence ( π -dual to E ) we can form the (dual) Fockspace F ( E π ), which is a W ∗ -correspondence over π ( M ) ′ , and the Hilbert space F ( E π ) ⊗ ι H . Let us define U := P ⊕ k ≥ U k . It follows that the map U := P ⊕ k ≥ U k is a Hilbert space isomorphism from F ( E ) ⊗ π H onto F ( E π ) ⊗ ι H ,and its adjoint acts on decomposable tensors by U ∗ ( η ⊗ ... ⊗ η n ⊗ h ) = ( I E ⊗ n − ⊗ η ) ... ( I E ⊗ η n − ) η n h . Definition 2.12.
The map U π = U : F ( E ) ⊗ π H → F ( E π ) ⊗ ι H will be calledthe Fourier transform determined by π . Let π : M → B ( H ) be a faithful normal representation. Then there existsa natural canonical isometric representation of E π on F ( E ) ⊗ π H induced by π . Let ν : π ( M ) ′ → B ( F ( E ) ⊗ π H ) be a ∗ -representation defined by ν ( b ) = I F ( E ) ⊗ b . Then ν is a faithful normal representation of the von Neumannalgebra π ( M ) ′ . By Rieffel’s Theorem 2 . π F ( E ) ( L ( F ( E ))) ′ = ν ( π ( M ) ′ ) = { I F ( E ) ⊗ b : b ∈ π ( M ) ′ } . Let η ∈ E π . Then for every n ≥
0, the operators L η,n : E ⊗ n ⊗ π H → E ⊗ n +1 ⊗ π H are defined by L η,n ( ξ ⊗ h ) = ξ ⊗ ηh , where wehave identified E ⊗ n +1 ⊗ π H with E ⊗ n ⊗ π E ◦ φ ( E ⊗ π H ). Since k L η,n k ≤ k η k , wemay define the operator Ψ( η ) : F ( E ) ⊗ π H → F ( E ) ⊗ π H by Ψ( η ) = P ⊕ k ≥ L η,k .Thus we may think of Ψ( η ) as I F ( E ) ⊗ η on F ( E ) ⊗ π H . It is easy to see that Ψis a bimodule map, and not hard to check that (Ψ , ν ) is an isometric covariantrepresentation of ( E π , π ( M ) ′ ) on the Hilbert space F ( E ) ⊗ π H , (for more detailssee [16]).Now, combining the integrated form ν × Ψ of (Ψ , ν ) with the definition ofthe Fourier transform U = U π we obtain the formulas U ∗ ι F ( E π ) ( T η ) U = Ψ( η ) , (6)where η ∈ E π and T η is the corresponding creation operator in H ∞ ( E π ), and U ∗ ι F ( E π ) ( φ E π , ∞ ( b )) U = ν ( b ) , (7)where b ∈ π ( M ) ′ and φ E π , ∞ is the left action of π ( M ) ′ on F ( E π ). This equalitycan be rewritten as U ( I F ( E ) ⊗ b ) = ( φ E π , ∞ ( b ) ⊗ I H ) U. (8)Thus, the Fourier transform U intertwines the actions of π ( M ) ′ on F ( E ) ⊗ π H and on F ( E π ) ⊗ ι H respectively.The following theorem identifies the commutant of the Hardy algebra rep-resented by an induced representation.11 heorem 2.13. ( [16], Theorem 3.9) Let E be a W ∗ -correspondence over M ,and let π : M → B ( H ) be a faithful normal representation of M on a Hilbertspace H . Write ρ π for the representation π F ( E ) of H ∞ ( E ) on F ( E ) ⊗ π H induced by π , and write ρ π for the representation of H ∞ ( E π ) on F ( E ) ⊗ π H defined by ρ π ( X ) = U ∗ ι F ( E π ) ( X ) U, (9) with X ∈ H ∞ ( E π ) . Then ρ π is an ultraweakly continuous, completely isometricrepresentation of H ∞ ( E π ) that extends the representation ν × Ψ of T + ( E π ) , and ρ π ( H ∞ ( E π )) is the commutant of ρ π ( H ∞ ( E )) , i.e. ρ π ( H ∞ ( E π )) = ρ π ( H ∞ ( E )) ′ . Corollary 2.14. ( [16], Corollary 3.10) In the preceding notation, ρ π ( H ∞ ( E )) ′′ = ρ π ( H ∞ ( E )) . Combining this corollary with the well known fact that the commutant A ′ ofevery operator algebra A is WOT-closed, we obtain that ρ π ( H ∞ ( E )) is WOT-closed. In this section we consider the reflexivity of the Hardy algebra ρ π ( H ∞ ( E )).First we introduce the notions of a quantitative analog of reflexivity, the hyper-reflexivity for operator algebras, and obtain some elementary consequences inour setting. After this observations we present our main results. Let H be a Hilbert and let A be some operator algebra, acting on it. We write H ( ∞ )0 for the direct sum of countable number of copies of a Hilbert space H .It can be naturally identified with the tensor product H ⊗ l , were l = l ( Z + ).The operator A ⊗ I ∈ A ⊗ I can be viewed as the infinite ampliation of A ,( A ∈ A ). Then the algebra A ⊗ I l is unitarily equivalent to the infiniteampliation A ( ∞ )0 . In [24] Radjavi and Rosenthal proved that if the algebra A is unital and WOT-closed then the algebra A ⊗ I is reflexive. In fact, thistheorem may be strengthened by requiring that A will be only ultraweaklyclosed.The operator algebra of the form A⊗ I , where I is the identity operator on aninfinite dimensional Hilbert space K , is called an algebra of infinite multiplicity.Thus every WOT-closed or even ultraweakly closed operator algebra of infinitemultiplicity is reflexive.Returning to our general setting, let M be a general W ∗ -algebra, E be anarbitrary W ∗ -correspondence over M and π be a faithful normal representationof M on a Hilbert space H . Recall that we showed in previous section that thealgebra ρ π ( H ∞ ( E )) is WOT-closed. Assume that π has an infinite multiplicity.This means that there is a separable Hilbert space K such that H = H ⊗ K and there is a normal representation π : M → H such that π = π ⊗ I K .12et us show that in this case the algebra ρ π ( H ( ∞ ) ( E )) has an infinite multi-plicity. The space F ( E ) ⊗ π H may be written as ( F ( E ) ⊗ π H ) ⊗ K . Thenthe induced representation π F ( E ) of the von Neumann algebra L ( F ( E )) hasthe form ( π ⊗ I K ) F ( E ) : Z Z ⊗ I H = ( Z ⊗ I H ) ⊗ I K = π F ( E )0 ( Z ) ⊗ I K .Hence F ( E ) ⊗ π H and ( F ( E ) ⊗ π H ) ⊗ K are identified as L ( F ( E ))-modulesand π F ( E ) ( · ) = π F ( E )0 ( · ) ⊗ I K has an infinite multiplicity. Conversely, for everyfaithful normal representation π of M on a Hilbert space H , the infinite ampli-ation ρ π ( H ∞ ( E )) ( ∞ ) is acting on the Hilbert space H ( ∞ )0 ∼ = H ⊗ l , and maybe identified with the algebra ρ π ( H ∞ ( E )) ⊗ I l . Since the algebra ρ π ( H ∞ ( E ))is WOT-closed, we get that ρ π ( H ∞ ( E )) = H ∞ ( E ) ⊗ I H = ( H ∞ ( E ) ⊗ I H ) ⊗ I K is reflexive.In fact, every unital WOT-closed operator algebra of infinite multiplicity ishyperreflexive. This notion is a quantitative version of reflexivity.Let A ⊆ B ( H ) be an operator algebra acting on a Hilbert space H . Forevery T ∈ B ( H ) set β ( T, A ) := sup {k P ⊥ T P k : P ∈ Lat A} . Then, for every T ∈ B ( H ), β ( T, A ) ≤ dist ( T, A ), since if P ∈ Lat A then, forevery S ∈ A , k P ⊥ T P k = k P ⊥ ( T − S ) P k ≤ k T − S k . Then, for every S ∈ A , β ( T, A ) ≤ k T − S k . Clearly, β ( T, A ) defines a seminorm on B ( H ). Clearly also, β ( T, A ) = 0 if and only if T ∈ Alg Lat A .In these terms we may redefine the reflexivity of the algebra A as follows: A is reflexive if and only if A = { T ∈ B ( H ) : β ( T, A ) = 0 } By definition, the algebra A is hyperreflexive if there is a constant C ≥
0, whichis independent of T , such that β ( T, A ) ≤ dist ( A , T ) ≤ Cβ ( T, A ) . The infimum C T of all such numbers C is called the constant of hyperreflexivityor the distance constant. Clearly, each hyperreflexive algebra is reflexive. It isknown that the following classes of algebras are hyperreflexive: the nest algebras(Arveson, [1]), the algebra L n (Davidson in [5] for n = 1 and Davidson and Pittsin [6] for n > L G for some special case of graphs, and in [8] F. Jaeck and S. Power showsthe hyperreflexivity of L G for any finite graph G . Note also that the problemof characterization of von Neumann algebras which are hyperreflexive still openand equivalent to numerous long standing unsolved classical problems. But somepartial results are known, for example, every injective von Neumann algebra ishyperreflexive.The fact that every WOT-closed algebra A of infinite multiplicity is hy-perreflexive is folklore (see [6]), and can be found, for example, in [6] (withdistance constant at most 9), and with a a very short proof in [8] (with distanceconstant 3). 13 heorem 3.1. Let π : M → B ( H ) be a faithful normal representation ofinfinite multiplicity. Then the algebra ρ π ( H ∞ ( E )) is hyperreflexive with distanceconstant at most 3. Proof. Indeed, we saw ρ π ( H ∞ ( E )) is WOT-closed and in the assumptions ofthe theorem, our algebra ρ π ( H ∞ ( E )) has an infinite multiplicity. Hence it ishypereflexive with the distance constant 3. (cid:3) Let us consider the special case when the algebra M is a factor of type III .Since M is a factor, then for every two projections p and q in M one has either p (cid:22) q or q (cid:22) p (in the sense of Murray-von Neumann) and since M is of type III all nonzero projections in M are infinite and equivalent. We shall use thefact that every nonzero projection p in such a factor can be “divided by ℵ ”,i.e. there is a sequence { p i } i ≥ ⊂ M of pairwise orthogonal subprojections of p ,such that p = P p i and p i ∼ p for every i . Corollary 3.2.
Let M be a factor of type III. Then the algebra ρ π ( H ∞ ( E )) ishyperreflexive. Proof. Since M is a factor of type III , so is π ( M ) ′ . It follows that π is ofinfinite multiplicity. To see this, write I = P i ≥ p i where { p i } ∞ i =0 are pairwiseorthogonal, equivalent projections in π ( M ) ′ . Thus, there are partial isometries { u j } ∞ j =0 in π ( M ) ′ with u ∗ j u j = p and u j u ∗ j = p j . Writing H for p H and π for π | H we easily see that π is unitarily equivalent to π ⊗ I l . Hence, π is of infinitemultiplicity. It follows from Theorem 3 . ρ π ( H ∞ ( E )) is hyperreflexive andthe distance constant is at most 3. (cid:3) Consider the algebra
Alg Lat ρ ( H ∞ ( E )) where ρ = ρ π is an induced represen-tation of H ∞ ( E ) defined by the faithful normal representation π .First we shall show that every Z ∈ Alg Lat ρ ( H ∞ ( E )) lies in ρ ( L ( F ( E ))),hence, Z has the form Z = Y ⊗ I H , where Y is some element in L ( F ( E )). Weneed the following simple auxiliary lemma. Lemma 3.3.
Let B be a von Neumann algebra acting on a Hilbert space H ,and let A ⊂ B ( H ) be some operator algebra. Assume that B ⊂ A ′ . Then Alg Lat
A ⊂ B ′ . Proof. For each b ∈ B and each a ∈ A , ba = ab . In particular, for eachprojection p ∈ B we have pa = ap , and the range of such projection is in Lat A .Let M ∈
Lat A and let p M be its projection. Then for each c ∈ Alg Lat A wehave cp M = p M cp M . In particular (1 − p ) cp = 0 for every projection p ∈ B .Since B is a von Neumann algebra we have also pc (1 − p ) = 0 for all p ∈ B .Hence cp = pcp = pc . So, every c ∈ Alg Lat A commutes with every projection p ∈ B . It follows that c ∈ B ′ . 14 Let ι be a identity representation of π ( M ) ′ on the Hilbert space H . Recallfrom the Preliminary section that for b ∈ π ( M ) ′ the formula b I F ( E ) ⊗ ι ( b )defines a faithful normal representation of π ( M ) ′ on F ( E ) ⊗ π H . We write I ∞ ⊗ ι for this representation. Similarly, we write I n ⊗ ι for subsrepresentationof π ( M ) ′ on E ⊗ n ⊗ π H . Thus, if ξ ∈ E ⊗ n , then ( I n ⊗ ι ( b ))( ξ ⊗ h ) = ξ ⊗ bh , n ≥
0. Frequently we shall drop the letter ι , writing I n ⊗ b for I n ⊗ ι ( b ). Corollary 3.4.
Alg Lat ρ ( H ∞ ( E )) ⊂ ( I ∞ ⊗ π ( M ) ′ ) ′ = L ( F ( E )) ⊗ I H .In particular, every Z ∈ Alg Lat ρ π ( H ∞ ( E )) is of the form Y ⊗ I H , forsome Y ∈ L ( F ( E )) . Proof. Let us take in the previous lemma B := ( I ∞ ⊗ π ( M ) ′ ) and A := ρ ( H ∞ ( E )) = H ∞ ( E ) ⊗ I H . Clearly, I ∞ ⊗ π ( M ) ′ ⊂ ρ ( H ∞ ( E )) ′ . Hence Alg Lat ρ ( H ∞ ( E )) ⊂ ( I ∞ ⊗ π ( M ) ′ ) ′ . By Rieffel’s Theorem 2 . I ∞ ⊗ π ( M ) ′ ) ′ = L ( F ( E )) ⊗ I H . Thus, Alg Lat ρ π ( H ∞ ( E )) ⊂ L ( F ( E )) ⊗ I H , and the corollary follows. (cid:3) In our attempts to prove the reflexivity of ρ ( H ∞ ( E )) we try to generalize theproof of Arias and Popescu from [2]. Thus, with each Y ⊗ I H ∈ Alg Lat ρ ( H ∞ ( E ))we associate the series of its Fourier coefficients.Namely, according to [16], with every T ∈ L ( F ( E )) we associate the seriesof operators Φ j ( T ), that will be called Fourier coefficients, as follows.Let { W t : t ∈ R } be the one-parameter unitary group in L ( F ( E )), definedby W t := ∞ X n =0 e int P n . Here P n is the projection on the n-th summand in F ( E ). One can check thatthis series converges in the w ∗ -topology in L ( F ( E )). Further, let γ t ( Y ) = AdW t ( Y ) = W t Y W ∗ t . Then { γ t : t ∈ R } , is a w ∗ -continuous action of R on L ( F ( E )), called the gauge automorphism group.The j -th Fourier coefficient Φ j ( T ) is defined byΦ j ( T ) = (1 / π ) Z π e − ijt γ t ( T ) dt. where the integral converges in the w ∗ -topology in L ( F ( E )). Simple calculationgives us Φ j ( T ) = X k P k + j T P k . (10) Lemma 3.5.
Let Y ⊗ I H ∈ Alg Lat ρ ( H ∞ ( E )) , then for each j = 0 , , , ... ,the operator Φ j ( Y ) ⊗ I H is in Alg Lat ρ ( H ∞ ( E )) . W t T ξ = e it T ξ W t , for ξ ∈ E , t ∈ R .Note that for every t , W ∗ t = W − t and that W t has a closed range . It followsthat if M ∈
Lat ρ ( H ∞ ( E )) then also M t := ( W t ⊗ I H ) M is in Lat ρ ( H ∞ ( E )).Let Y ⊗ I H ∈ Alg Lat ρ ( H ∞ ( E )). Clearly, it is enough to show that also γ t ( Y ) ⊗ I H ∈ Alg Lat ρ ( H ∞ ( E )). We have ( W t Y W ∗ t ⊗ I H ) M = ( W t Y ⊗ I H ) M − t ⊆ ( W t ⊗ I H ) M − t = ( W t ⊗ I H )( W ∗ t ⊗ I H ) M = M . So, ( γ t ( Y ) ⊗ I H ) M ⊆M . (cid:3) The operator Φ j ( Y ) ⊗ I H will be called the j -th Fourier coefficient of Y ⊗ I H ∈ Alg Lat ρ ( H ∞ ( E )). Note that if Y ⊗ I H ∈ Alg Lat ρ ( H ∞ ( E )), then from theformula Φ j ( Y ) = P k P k + j Y P k it follows that Φ j ( Y ) = 0 for every j < ρ ( H ∞ ( E )) it is enough to show that, given Y ⊗ I H ∈ Alg Lat ρ ( H ∞ ( E )), then every Φ j ( Y ) ⊗ I H , j = 0 , , , ... , is in ρ ( H ∞ ( E )). Lemma 3.6.
Let Y ⊗ I H ∈ Alg Lat H ∞ ( E ) ⊗ I H and assume that each Φ j ( Y ) ⊗ I H ∈ ρ ( H ∞ ( E )) , j ≥ . Then Y ⊗ I H ∈ ρ ( H ∞ ( E )) . Proof. It follows from [16, p.366], that the k -th arithmetic mean operators σ k ( Y ) := P | j | 0, tend to Y in the weak ∗ -topology in L ( F ( E )) as k → ∞ . But then the operators σ k ( Y ) ⊗ I H tend ultraweakly to Y ⊗ I H ∈ B ( F ( E ) ⊗ H ). Hence, if all Φ j ( Y ) ⊗ I H are in ρ ( H ∞ ( E )) then so is Y ⊗ I H . (cid:3) Let ξ ∈ E , and let L ξ be the operator on H defined by L ξ : h ξ ⊗ h . By L ( k ) ξ we denote the operator L ( k ) ξ : E ⊗ k ⊗ π H → E ⊗ k +1 ⊗ π H , that on the elementarytensors is defined by η ⊗ h ξ ⊗ η ⊗ h . So, L ξ = L (0) ξ . Analogously, for arbitrary x ∈ E ⊗ k , we let L ( n ) x denote the operator from E ⊗ n ⊗ π H to E ⊗ n + k ⊗ π H definedby ζ ⊗ h x ⊗ ( ζ ⊗ h ), where ζ ∈ E ⊗ n . For simplicity we often write L (0) x = L x .In the following theorem, which is a main result in this subsection, we agreeto write ξ ( n ) for a general element of the correspondence E ⊗ n , when n ≥ Theorem 3.7. Let Y ⊗ I H ∈ Alg Lat ρ ( H ∞ ( E )) . Then1) For j = 0 there exists a sequence { a s } s ≥ ⊂ M such that (Φ ( Y ) ⊗ I H ) | H = π ( a ) for s = 0 , and (Φ ( Y ) ⊗ I H ) | E ⊗ s ⊗ π H = φ s ( a s ) ⊗ I H for every s > ,. Hence, with respect to the decomposition F ( E ) ⊗ π H = P ⊕ s ≥ E ⊗ s ⊗ π H the operator Φ ( Y ) ⊗ I H is represented by the diagonal matrixdiag ( π ( a ) , φ ( a ) ⊗ I H , ... ) . (11) 2) For j ≥ there exists a sequence { ξ ( j ) s } s ≥ ⊂ E ⊗ j such that (Φ j ( Y ) ⊗ I H ) | E ⊗ s ⊗ π H = ( T ξ ( j ) s ⊗ I H ) | E ⊗ s ⊗ π H = L ( s ) ξ ( j ) s . Thus, with respect to the decompo-sition F ( E ) ⊗ π H = P ⊕ s ≥ E ⊗ s ⊗ π H , the operator Φ j ( Y ) ⊗ I H is representedby the ”j”-th subdiagonal matrix ... ...... ... ... ... ... ... ... ... ... ... ...L (0) ξ ( j )0 ... ... L (1) ξ ( j )1 ... ... L (2) ξ ( j )2 ... ...... ... ... ... ... ... (12)Proof. Recall first from Lemma 3 . Y ⊗ I H ∈ Alg Lat ρ ( H ∞ ) so isΦ j ( Y ) ⊗ I H for every j ≥ 0. (1) Since Φ ( Y ) = P k P k Y P k , we obtain that(Φ ( Y ) ⊗ I H ) | E ⊗ k ⊗ π H = P k Y P k , i.e. each summand E ⊗ k ⊗ π H of F ( E ) ⊗ π H is Φ ( Y ) ⊗ I H - invariant. Consider the restriction (Φ ( Y ) ⊗ I H ) | H where H ∼ = M ⊗ π H . The representation I ∞ ⊗ ι of π ( M ) ′ restricted to H is simply ι ,the identity representation of π ( M ) ′ on H . Hence, by Corollary 3 . j ( Y ) ⊗ I H ∈ Alg Lat ρ ( H ∞ ), we get the identity (Φ ( Y ) ⊗ I H )( I ∞ ⊗ ι ( b )) =( I ∞ ⊗ ι ( b ))(Φ ( Y ) ⊗ I H ) , b ∈ π ( M ) ′ , which when restricted to H , yields(Φ ( Y ) ⊗ I H ) b = b (Φ ( Y ) ⊗ I H ) , b ∈ π ( M ) ′ . Thus, (Φ ( Y ) ⊗ I H ) | H = π ( a ), for some a ∈ M .Let n ≥ 1. For the restriction (Φ ( Y ) ⊗ I H ) | E ⊗ n ⊗ π H we have(Φ ( Y ) ⊗ I H )( I ∞ ⊗ b ) | E ⊗ n ⊗ π H = ( P n Y P n ⊗ I H )( I n ⊗ b ) = ( I n ⊗ b )( P n Y P n ⊗ I H ) , where I n is the identity operator I E ⊗ n . We write S n ⊗ I H for the restriction(Φ ( Y ) ⊗ I H ) | E ⊗ n ⊗ π H , where S n = P n Y P n ∈ L ( E ⊗ n ).We want to show that for every n ≥ S n = φ n ( a n ) for some a n ∈ M . Take x ∈ E ⊗ n ⊗ π H be arbitrary and set M x := ρ ( H ∞ ( E )) x. So, M x is a ρ ( H ∞ ( E ))-invariant subspace in F ( E ) ⊗ π H and can be writtenas M x = ( φ n ( M ) ⊗ I H ) x ⊕ E ⊗ φ n ⊗ I H x ⊕ ... We see that M x ∩ ( E ⊗ n ⊗ π H ) = ( φ n ( M ) ⊗ I H ) x and is invariant underΦ ( Y ) ⊗ I H because Φ ⊗ I H ∈ Alg Lat ρ ( H ∞ ( E )). Thus, ( S n ⊗ I H )( x ) ∈ ( φ n ( M ) ⊗ I H ) x and we obtain that there is a net ( a α ) ⊂ M such that( S n ⊗ I H ) x = lim α ( φ n ( a α ) ⊗ I H ) x, and the net ( a α ) depends on the choice of x ∈ E ⊗ n ⊗ π H .Fix any projection p ∈ ( φ n ( M ) ⊗ I H ) ′ . Replacing x by px and by p ⊥ x (where p ⊥ = I − p ), we get two nets ( b α ) and ( c α ) in M such that( S n ⊗ I H )( px ) = lim α ( φ n ( b α ) ⊗ I H )( px ) , S n ⊗ I H )( p ⊥ x ) = lim α ( φ n ( c α ) ⊗ I H )( p ⊥ x ) , Then( S n ⊗ I H ) x = lim α ( φ n ( b α ) ⊗ I H )( px ) + lim α φ n ( c α ) ⊗ I H )( p ⊥ x ) . Now applying p on both sides and using the facts that p ∈ ( φ n ( M ) ⊗ I H ) ′ and pp ⊥ = 0 we obtain p ( S n ⊗ I H ) x = lim α ( φ n ( b α ) ⊗ I H )( px ) = ( S n ⊗ I H ) px. Hence for every projection p ∈ ( φ n ( M ) ⊗ I H ) ′ we have p ( S n ⊗ I H ) = ( S n ⊗ I H ) p at x ∈ E ⊗ n ⊗ π H . Since the choice of x is arbitrary we get p ( S n ⊗ I H ) = ( S n ⊗ I H ) p, on E ⊗ n ⊗ π H . Since p ∈ ( φ n ( M ) ⊗ I H ) ′ is an arbitrary projection we get: S n ⊗ I H ∈ ( φ n ( M ) ⊗ I H ) ′′ = φ n ( M ) ⊗ I H , and this implies that there exists a n ∈ M such that S n ⊗ I H = φ ( a n ) ⊗ I H . We proved that (Φ ⊗ I H ) | E ⊗ n ⊗ π H = φ n ( a n ) ⊗ I H for some a n ∈ M .Hence, Φ ( Y ) = diag ( π ( a ) , φ ( a ) ⊗ I H , φ ( a ) ⊗ I H , ... ).(2) Let j ≥ 1. From the formula Φ j ( Y ) = P k P k + j Y P k we see that for every s ≥ j ( Y ) ⊗ I H : E ⊗ s ⊗ π H → E ⊗ s + j ⊗ π H. Consider the restriction (Φ j ( Y ) ⊗ I H ) | H . Thus, Φ j ( Y ) ⊗ I H | H acts from H to E ⊗ j ⊗ π H .By Corollary 3 . . j ( Y ) ⊗ I H )( I ∞ ⊗ b ) x = ( I ∞ ⊗ b )(Φ j ( Y ) ⊗ I H ) x, where x ∈ F ( E ) ⊗ π H and b ∈ π ( M ) ′ .In particular, for every h ∈ H (Φ j ( Y ) ⊗ I H )( bh ) = ( I j ⊗ b )(Φ j ( Y ) ⊗ I H ) h. Let U = U π : F ( E ) ⊗ π H → F ( E π ) ⊗ ι H be the Fourier transform definedby π (see Definition 2 . U is a Hilbertspace isomorphism as well as each restriction U s := U | E ⊗ s ⊗ π H : E ⊗ s ⊗ π H → ( E π ) ⊗ s ⊗ ι H . From formula (8), the operator U intertwines the representations I ∞ ⊗ ι and ι F ( E π ) of π ( M ) ′ on F ( E ) ⊗ π H and F ( E π ) ⊗ ι H respectively.18egarding H and ( E π ) ⊗ j ⊗ ι H as subspaces in F ( E ) ⊗ π H and F ( E π ) ⊗ ι H respectively, we consider the operator U (Φ j ( Y ) ⊗ I H ) | H : H → ( E π ) ⊗ j ⊗ ι H. Hence, U (Φ j ( Y ) ⊗ I H ) | H intertwines the actions ι and ι ( E π ) ⊗ j of π ( M ) ′ . Itfollows that U (Φ j ( Y ) ⊗ I H ) | H is contained in the second dual ( E ⊗ j ) π,ι and, bythe duality theory ( E ⊗ j ) π,ι ∼ = E ⊗ j . Hence, there exists a unique ξ ( j ) ∈ E ⊗ j ,that corresponds to U (Φ j ( Y ) ⊗ I H ) | H . We write d ξ ( j ) = U (Φ j ( Y ) ⊗ I H ) | H .We will show that (Φ j ( Y ) ⊗ I H ) | H = ( T ξ ( j ) ⊗ I H ) | H = L ξ ( j ) . To this endrecall that by formula (5) form Chapter 2, we have d ξ ( j ) = U j ◦ L ξ ( j ) . Thus d ξ ( j ) = U j (Φ j ( Y ) ⊗ I H ) | H = U j ◦ L ξ ( j ) , and we get L (0) ξ ( j ) = ( T ξ ( j ) ⊗ I H ) | H = (Φ j ( Y ) ⊗ I H ) | H . (13)We write ξ ( j )0 for the ξ ( j ) obtained above.Now we consider the restriction S n ⊗ I H := (Φ j ( Y ) ⊗ I H ) | E ⊗ n ⊗ π H for n ≥ S n ⊗ I H = P n + j Y P n ⊗ I H acts from E ⊗ n ⊗ π H into E ⊗ j + n ⊗ π H .Put K n := E ⊗ n ⊗ π H and recall that E ⊗ j ⊗ φ n ⊗ I H K n = E ⊗ j ⊗ φ n ⊗ I H ( E ⊗ n ⊗ π H ) ∼ = E ⊗ j + n ⊗ π H . Thus, S n ⊗ I H acts from K n into E ⊗ j ⊗ φ n ⊗ I H K n .Take x ∈ K n arbitrary and form the cyclic ρ ( H ∞ ( E ))-invariant subspace M x = ρ ( H ∞ ( E )) x . The subspace M x has the representation M x = ( φ n ( M ) ⊗ I H ) x ⊕ E ⊗ φ n ⊗ I H x ⊕ ... ⊕ E ⊗ j ⊗ φ n ⊗ I H x ⊕ ... Since Φ j ⊗ I H ∈ Alg Lat ρ ( H ∞ ( E )) we get (Φ j ⊗ I H ) M x ⊆ M x . Then ( S n ⊗ I H ) x ∈ ( E ⊗ j ⊗ K n ) ∩ M x = E ⊗ j ⊗ φ n ⊗ I H x , and there exists some net ( θ ( j ) α ) ⊂ E ⊗ j such that ( S n ⊗ I H ) x = lim α θ ( j ) α ⊗ x = lim α ( T θ ( j ) α ⊗ I H ) x. Let p ∈ ( φ n ( M ) ⊗ I H ) ′ be any projection, then for every α ( T θ ( j ) α ⊗ I H ) px = θ ( j ) α ⊗ px = ( I j ⊗ p )( θ ( j ) α ⊗ x ) = ( I j ⊗ p )( T θ ( j ) α ⊗ I H ) x, where I j denotes the identity operator I E ⊗ j . Now take px and p ⊥ x = ( I − p ) x in K n , one can find a nets ( ζ ( j ) α ) and ( ϑ ( j ) α ) in E ⊗ j such that ( S n ⊗ I H ) px =lim α ( T ζ ( j ) α ⊗ I H ) px and ( S n ⊗ I H ) p ⊥ x = lim α ( T ϑ ( j ) α ⊗ I H ) p ⊥ x . So,lim α ( T θ ( j ) α ⊗ I H ) x = lim α ( T ζ ( j ) α ⊗ I H ) px + lim α ( T ϑ ( j ) α ⊗ I H ) p ⊥ x. Applying I j ⊗ p on both sides, we obtain19 I j ⊗ p )( S n ⊗ I H ) x = ( I j ⊗ p ) lim α ( T θ ( j ) α ⊗ I H ) x == lim α ( T ζ ( j ) α ⊗ I H ) px = ( S n ⊗ I H ) px. Since p is an arbitrary projection in ( φ n ( M ) ⊗ I H ) ′ and this holds for every x ∈ K n , we obtain that S n ⊗ I H intertwines the action of ( φ n ( M ) ⊗ I H ) ′ :( I j ⊗ b )( S n ⊗ I H ) = ( S n ⊗ I H ) b, ∀ b ∈ ( φ n ( M ) ⊗ I H ) ′ . (14)Denote ι n := id ( φ n ( M ) ⊗ I H ) ′ the identity representation of ( φ n ( M ) ⊗ I H ) ′ on K n and let U φ n ⊗ I H : E ⊗ j ⊗ φ n ⊗ I H K n → ( E φ n ⊗ I H ) ⊗ j ⊗ ι n K n , be the Fourier transform defined by φ n ( · ) ⊗ I H . By formula (8) from Prelimi-naries, U φ n ⊗ I H intertwines the actions of ( φ n ( M ) ⊗ I H ) ′ : U φ n ⊗ I H ( I j ⊗ b ) = ( φ E φn ⊗ IH ( b ) ⊗ I H ) U φ n ⊗ I H , for every b ∈ ( φ n ( M ) ⊗ I H ) ′ .Thus, the composition U φ n ⊗ I H ( S n ⊗ I H ) is in the second dual ( E ⊗ j ) φ n ⊗ I H ,ι n .By duality there exists a unique ξ ( j ) ∈ E ⊗ j such that d ξ ( j ) ∗ (˜ η ⊗ ... ⊗ ˜ η j ⊗ k n ) = ˜ L ∗ ξ ( j ) (( I j − ⊗ ˜ η ) ... ( I ⊗ ˜ η j − )˜ η j ( k n )) , where ˜ L ξ ( j ) = ( T ξ ( j ) ⊗ I H ) | K n , ˜ η ∈ E φ n ⊗ I H and k n ∈ K n . Thus, d ξ ( j ) = ˜ U j ◦ ˜ L ξ ( j ) ,where we denote ˜ U j = U φ n ⊗ I H | E ⊗ j ⊗ φn ⊗ IH K n . Hence˜ U ∗ j d ξ ( j ) = ˜ L ξ ( j ) . We write ξ ( j ) n for ξ ( j ) obtained above.We obtain that(Φ j ( Y ) ⊗ I H ) | E ⊗ n ⊗ π H = ( T ξ ( j ) n ⊗ I H ) | E ⊗ n ⊗ π H = L ( n ) ξ ( j ) n , and this proves the matrix representation (12) and the proof of the theorem iscomplete. (cid:3) Our next step is to consider the action of every Φ j ( Y ) ⊗ I H and of its adjoint(Φ j ( Y ) ⊗ I H ) ∗ on a suitable ρ ( H ∞ ( E ))-coinvariant subspaces M . Observe that(Φ j ( Y ) ⊗ I H ) ∗ = Φ j ( Y ) ∗ ⊗ I H .Our considerations are based on the following simple facts. Let A be anoperator algebra, acting in Hilbert space H . Then Lat A ∗ = ( Lat A ) ⊥ , where as usual A ∗ is the algebra of adjoint of elements of A and ( Lat A ) ⊥ isthe lattice of orthogonal complements of subspaces from Lat A .20 emma 3.8. Let A be an operator algebra, acting in Hilbert space H . Then ( Alg Lat A ) ∗ ⊆ Alg Lat A ∗ . Proof. Let Y ∈ Alg Lat A . From Lat A ∗ = ( Lat A ) ⊥ we get (1 − p M ⊥ ) Y ∗ p M ⊥ =0, for every M ∈ Lat A . So, Y ∗ M ⊥ ⊆ M ⊥ . Finally, Y ∗ ˜ M ⊆ ˜ M for every Y ∈ Alg Lat A and every ˜ M ∈ Lat A ∗ . (cid:3) For every s ≥ 0, let H ( s ) := W { η ( s ) ( h ) : η ∈ E π , h ∈ H } , s = 0 , , , ... ,where, clearly, H (0) = H . Put M = ⊕ X s ≥ H ( s ) . (15)Recalling the definition of the Fourier transform U π we see that H ( s ) = U ∗ π ˜ H ( s ), where ˜ H ( s ) = W { η ⊗ s ⊗ h : η ∈ E π , h ∈ H } . So, if we set f M := P ⊕ s ≥ ˜ H ( s ), then f M = U π ( M ).Notice that H (0) = H and H (1) = E ⊗ π H = U ∗ π ( ˜ H ( s )) = U ∗ π ( E π ⊗ ι H ),which follows from [16, Lemma 3.5]. But when s ≥ H ( s ) ingeneral is a proper subspace of E ⊗ s ⊗ π H . Take for example E = C n and M = C . In this case E π is all bounded operators B ( H, C n ⊗ π H ) and canbe identified with the n -fold column space C n ( B ( H )) over the algebra B ( H )(see [16, Example 4.2]). If we take H = C then C n ( B ( C )) ∼ = C n . Hence,( C n ) ⊗ s ⊗ ι C ∼ = ( C n ) ⊗ s and the subspace ˜ H ( s ) now is W { η ⊗ ... ⊗ η : η ∈ C n } ,that is, the symmetric tensor power of C n . Thus, the subspace f M is the ordinarysymmetric Fock space of C n and ˜ H ( s ) $ ( C n ) ⊗ s , for s ≥ f M , as well asits Fourier image M , the symmetric part (subspace) of the corresponding fullFock space. Theorem 3.9. The subspace M ⊂ F ( E ) ⊗ π H is ρ ( H ∞ ( E )) -coinvariant. Let,further, Y ⊗ I H ∈ Alg Lat ρ ( H ∞ ( E )) and Φ j ( Y ) ⊗ I H be it’s j -th Fouriercoefficient. Then, in the notation of Theorem . ,a) if j = 0 then (Φ ( Y ) ∗ ⊗ I H ) | M = ( φ ∞ ( a ) ∗ ⊗ I H ) | M ; b) if j ≥ , then (Φ j ( Y ) ∗ ⊗ I H ) | M = ( T ∗ ξ ( j )0 ⊗ I H ) | M ,ξ ( j )0 ∈ E ⊗ j . For the proof of the theorem we shall need a couple of lemmas. Lemma 3.10. The subspace M is ρ ( H ∞ ( E )) -coinvariant. a ∈ M . For every η ∈ E π and every s ≥ 0, one have( φ ∞ ( a ∗ ) ⊗ I H ) η ( s ) ( h ) = η ( s ) ( π ( a ∗ ) h ) . Thus, every operator φ ∞ ( a ∗ ) ⊗ I H leaves invariant every subspace H ( s ).We shall use by the following simple formula (in fact, it was used in [16]).Let η ∈ E π and s ≥ j such that 1 ≤ j ≤ s and every x ∈ E ⊗ j , L ( s − j ) ∗ x η ( s ) ( h ) = η ( s − j ) L (0) ∗ x η ( j ) ( h ) , x ∈ E ⊗ j , ≤ j ≤ s, (16)where L ( n ) x are operators that were defined before Theorem 3 . 7. In particular,for ξ ∈ E , L ∗ ξ η ( s ) ( h ) = η ( s − ( L ∗ ξ η ( h )) , s ≥ . Hence, for every s ≥ η ( s ) ( h ) ∈ H ( s ),( T ∗ ξ ⊗ I H ) η ( s ) ( h ) = η ( s − ( L ∗ ξ ( h )) ∈ H ( s − , thus, for every ξ ∈ E , ( T ∗ ξ ⊗ I H ) H ( s ) ⊂ H ( s − M is ρ ( H ∞ ( E ))-coinvariant. (cid:3) Now let us recall the definition of the Cauchy transform from [17] (see also[16]).Let η ∈ E π , with k η k < 1. The Cauchy transform C η is the operator from H ∼ = M ⊗ π H into F ( E ) ⊗ π H defined by C η : h h + η ( h ) + η (2) ( h ) + η (3) ( h ) + ..., where η ( k ) ( h ) = ( I E ⊗ k − ⊗ η ) ... ( I E ⊗ η ) η ( h ). It is pointed out in [17] that C η is bounded with k ˜ C η k ≤ −k η k . Thus, C η = column ( I, η, η (2) , η (3) , ... ).It is easy to see that for every k ≥ a ∈ M we have the equality( I E ⊗ k ⊗ η )( φ k ( a ) ⊗ I H ) = ( φ k +1 ( a ) ⊗ I H )( I E ⊗ k ⊗ η ) . Hence ( φ k ( a ) ⊗ I H ) η ( k ) ( h ) = η ( k ) ( π ( a ) h ) for every h ∈ H , and we get C η π ( a ) = ( φ ∞ ( a ) ⊗ I H ) C η , a ∈ M. So, C η ∈ F ( E π ). Let us show that C η has a closed range. To this end let K η = ( I F ( E ) ⊗ ∆ ∗ ( η )) C η be the Poisson kernel associated with η as definedin [17, Definition 8]. By Proposition 10 of that paper, K η is an isometrymapping from H to F ( E ) ⊗ π H and K ∗ η K η = C ∗ η ( I F ( E ) ⊗ (∆ ∗ ( η )) ) C η = I H .So, C ∗ η ( I F ( E ) ⊗ (∆ ∗ ( η )) ) is the left inverse of C η , hence C η has a closed range.For a given η ∈ D ( E π ) let M η := _ { C η h : h ∈ H } . (17)22ince C η has a closed range we get M η = C η ( H ) and every element of M η has the form h + η ( h ) + η (2) ( h ) + ... , for some h ∈ H .Clearly, M η ⊂ M for every η ∈ D ( E π ). Moreover, each subspace M η is ρ ( H ∞ ( E ))-coinvariant, which may be shown in the same way as ρ ( H ∞ ( E ))-coinvariance of M . Lemma 3.11. 1) Let Y ⊗ I H ∈ Alg Lat ρ ( H ∞ ( E )) . Then, in the notations ofTheorem . ,(i) for j = 0 , (Φ ( Y ) ∗ ⊗ I H ) | M η = ( φ ∞ ( a ) ∗ ⊗ I H ) | M η . and(ii) for j ≥ , (Φ j ( Y ) ∗ ⊗ I H ) | M η = ( T ∗ ξ ( j )0 ⊗ I H ) | M η . Proof. M η is ρ ( H ∞ ( E ))-coinvariant, and hence (Φ j ( Y ) ⊗ I H ) ∗ M η ⊆ M η forevery j ≥ 0. Take x = P s ≥ η ( s ) ( h ) ∈ M η .1) If j = 0, then(Φ ( Y ) ⊗ I H ) ∗ x = X s ≥ (Φ ( Y ) ∗ ⊗ I H ) η ( s ) ( h ) = X s ≥ η ( s ) ( k ) , for some k ∈ H . So,(Φ ( Y ) ∗ ⊗ I H ) η ( s ) ( h ) = η ( s ) ( k ) , s ≥ . By part 1) of Theorem 3 . 7, (Φ ( Y ) ∗ ⊗ I H ) | H = π ( a ∗ ). Hence π ( a ∗ ) h = k andfor every s ≥ η ( s ) , η ( s ) ( k ) = η ( s ) ( π ( a ∗ ) h ) = ( φ s ( a ∗ ) ⊗ I H ) η ( s ) ( h ) . So, (Φ ( Y ) ∗ ⊗ I H ) η ( s ) ( h ) = ( φ s ( a ∗ ) ⊗ I H ) η ( s ) ( h ) , s ≥ . Hence, (Φ ( Y ) ⊗ I H ) ∗ | M η = ( φ ∞ ( a ∗ ) ⊗ I H ) | M η .2) Let j ≥ x = P s ≥ η ( s ) ( h ) ∈ M η as in 1). Since M η , is (Φ j ( Y ) ⊗ I H ) ∗ -invariant we have(Φ j ( Y ) ∗ ⊗ I H ) x = X s ≥ η ( s ) ( k ) ∈ M η , for some k ∈ H .From the inclusion (Φ j ( Y ) ⊗ I H )( E ⊗ s ⊗ π H ) ⊆ E ⊗ s + j ⊗ π H we get (Φ j ( Y ) ∗ ⊗ I H ) η ( l ) ( h ) = 0 for l < j . Further,(Φ j ( Y ) ∗ ⊗ I H ) η ( s + j ) ( h ) = η ( s ) ( k ) , s ≥ , s = 0: (Φ j ( Y ) ∗ ⊗ I H ) η ( j ) ( h ) = k. By Theorem 3 . 7, 2), (Φ j ( Y ) ∗ ⊗ I H ) | E ⊗ j ⊗ π H = ( T ∗ ξ ( j )0 ⊗ I H ) | E ⊗ j ⊗ π H = L (0) ∗ ξ ( j )0 .Hence, k = L (0) ∗ ξ ( j )0 η ( j ) ( h ) . So, for s ≥ j ( Y ) ∗ ⊗ I H ) η ( s + j ) ( h ) = η ( s ) ( k ) = η ( s ) ( L (0) ∗ ξ ( j )0 η ( j ) ( h )) . By identity (16) (Φ j ( Y ) ∗ ⊗ I H ) η ( s + j ) ( h ) = L ( s ) ∗ ξ ( j )0 η ( s + j ) ( h ) . Hence, for every x ∈ M η ,(Φ j ( Y ) ∗ ⊗ I H ) x = ( T ∗ ξ ( j )0 ⊗ I H ) x, and lemma follows. (cid:3) Proof of Theorem . 9. By Lemma 3 . M is ρ ( H ∞ ( E ))-coinvariant. Bythe last lemma the restriction (Φ j ( Y ) ⊗ I H ) ∗ | M η is of the form ( X ∗ ⊗ I H ) | M η ,where X is an element of H ∞ ( E ) and is either of the form φ ∞ ( a ) with a ∈ M ,or T ξ ( j ) with ξ ( j ) ∈ E ⊗ j , j ≥ 1. Note also that X is independent of η and, thus,this holds also for W { M η : η ∈ E η , k η k < } . Clearly, this is also true for everysubspace H ( s ), s = 0 , , , ... . Thus, (Φ j ( Y ) ⊗ I H ) ∗ | H ( s ) = ( X ∗ ⊗ I H ) | H ( s ) , s ≥ j ( Y ) ⊗ I H ) ∗ | M = ( X ∗ ⊗ I H ) | M , (18)where X ∈ H ∞ ( E ) of the form pointed above. (cid:3) Proposition 3.12. Let Y ⊗ I H ∈ Alg Lat ρ ( H ∞ ( E )) as above and let Φ j ( Y ) ⊗ I H be its j -th Fourier coefficient, represented as in Theorem . . Let Q = P M the projection onto M .1) If j = 0 then Q (Φ ( Y ) ⊗ I H ) | M = ( φ ∞ ( a ) ⊗ I H ) | M . 2) If j ≥ then Q (Φ j ( Y ) ⊗ I H ) | M = Q ( T ξ ( j )0 ⊗ I H ) | M . Write Q s for the projection onto H ( s ) , such that Q = P s Q s . Then Q s (Φ j ( Y ) ⊗ I H ) | E ⊗ s ⊗ π H = Q s ( T ξ ( j )0 ⊗ I H ) | E ⊗ s ⊗ π H . ( Y ) ⊗ I H = diag ( π ( a ) , φ ( a ) ⊗ I H , ... ). For every s ≥ H ( s ) is Φ ( Y ) ⊗ I H -invariant and so is M . Since (Φ ( Y ) ∗ ⊗ I H ) | M =( φ ∞ ( a ∗ ) ⊗ I H ) | M we get(Φ ( Y ) ⊗ I H ) | M = ( φ ∞ ( a ) ⊗ I H ) | M . 2) Set S ∗ ⊗ I H = Φ j ( Y ) ∗ ⊗ I H − T ∗ ξ j ⊗ I H . Then ( S ∗ ⊗ I H ) | M = 0, that is Q ( S ∗ ⊗ I H ) Q = 0. Hence, Q ( S ⊗ I H ) Q = 0 and, in particular, Q s + j ( S ⊗ I H ) Q s =0 for every s ≥ (cid:3) Corollary 3.13. Let Z ⊗ I H be an operator in Alg Lat ρ π ( H ∞ ( E )) , admittinga representation Z ⊗ I H = diag ( π ( a ) , φ ( a ) ⊗ I H , ... ) , a s ∈ M. Then ( Z ⊗ I H ) | M = ( φ ∞ ( a ) ⊗ I H ) | M . Hence, for all s ≥ Z ⊗ I H ) | H ( s ) = ( φ ∞ ( a ) ⊗ I H ) | H ( s ) , and a = a . Proof. From the previous proposition we get ( Z ⊗ I H ) | M = ( φ ∞ ( a ) ⊗ I H ) | M and, for every s ≥ 0, ( Z ⊗ I H ) | H ( s ) = ( φ ∞ ( a ) ⊗ I H ) | H ( s ) . As we saw H (0) = H and H (1) = E ⊗ π H , and the equality a = a follows. (cid:3) At this point we are able to show that for every Y ⊗ I H ∈ Alg Lat ρ π ( H ∞ ( E )),its “0”-th Fourier coefficient Φ ( Y ) ⊗ I H is in ρ π ( H ∞ ( E )) (Theorem 3 . A be an operator al-gebra acting on Hilbert space H , N ∈ Lat A and T ∈ Alg Lat A . Then T | N ∈ Alg Lat ( A| N ). If B another operator algebra acting on a Hilbert space K such that there is a unitary W : H → K with W A W ∗ = B , then for every T ∈ Alg Lat A , W T W ∗ ∈ Alg Lat B .The first claim is evident. For the second note that if A = W ∗ BW for some B ∈ B and if M ∈ Lat B , then A ( W ∗ M ) = W ∗ BW ( W ∗ M ) = W ∗ B M ⊂ W ∗ M ⊂ M ∗ . Thus, W ∗ M ∈ Lat A . Now if T ∈ Alg Lat A then T W ∗ M ⊂ W ∗ M , hence( W T W ∗ ) M ⊂ M .Fix s ≥ K s := E ⊗ s ⊗ π H and π s := φ s ⊗ I H . Then π s is afaithful normal representation of M on K s . Then the space F ( E ) ⊗ π s K s maybe identified with the subspace G s := P ⊕ l ≥ s E ⊗ l ⊗ π H ⊂ F ( E ) ⊗ π H as follows.25et k s = ζ ⊗ h ∈ K s , with ζ ∈ E ⊗ s . Then the formula W s,l : ξ ⊗ ... ⊗ ξ l ⊗ ζ ⊗ h ξ ⊗ ... ⊗ ξ l ⊗ k s defines the identification E ⊗ l + s ⊗ π H ∼ = E ⊗ l ⊗ π s K s , l, s ≥ . Hence, we obtain a unitary operator W s : G s = ⊕ X l ≥ s E ⊗ l ⊗ π H → F ( E ) ⊗ π s K s . Write ρ for the induced representation π F ( E ) and ρ s for the induced rep-resentation π F ( E ) s of H ∞ ( E ) on F ( E ) ⊗ π s K s . Thus, ρ s ( X ) = X ⊗ I K s when X ∈ H ∞ ( E ).Further, by ρ | G s we denote the representation obtained by the restrictionof ρ to G s . Thus, ρ | G s ( X ) = ( X ⊗ I H ) | G s when X ∈ H ∞ ( E ). Lemma 3.14. Fix s ≥ and let W s be as above. Then1) W ∗ s ρ s ( X ) W s = ρ ( X ) | G s , X ∈ H ∞ ( E ) ;2) for every Y ⊗ I H ∈ Alg Lat ρ ( H ∞ ( E )) , the restriction ( Y ⊗ I H ) | G s isin Alg Lat ( ρ ( H ∞ ( E )) | G s ) , and W ( Y ⊗ I H ) | G s W ∗ ∈ Alg Lat ρ s ( H ∞ ( E )) . Proof. 1) Enough to check it for the generators of the Hardy algebra. So, take φ ∞ ( a ) ∈ H ∞ ( E ), a ∈ M . For z ⊗ ζ ⊗ h ∈ E ⊗ l + s ⊗ π H , where we write z for ξ ⊗ ... ⊗ ξ l and ζ ∈ E ⊗ s , we have W ∗ s ρ ( φ ∞ ( a )) W s ( z ⊗ ζ ⊗ h ) = W ∗ s ( φ ∞ ( a ) ⊗ I K s )( z ⊗ k s ) =( φ l ( a ) z ) ⊗ ζ ⊗ h = ( φ ∞ ( a ) ⊗ I H )( z ⊗ ζ ⊗ h ) = ρ ( φ ∞ ( a ))( z ⊗ ζ ⊗ h ) . Now take T ξ ∈ H ∞ ( E ), ξ ∈ E . Then W ∗ s ρ s ( T ξ ) W s ( z ⊗ ζ ⊗ h ) = W ∗ s ( T ξ ⊗ I K s )( z ⊗ k s ) = ξ ⊗ z ⊗ ζ ⊗ h = ( T ξ ⊗ I H )( z ⊗ ζ ⊗ h ) = ρ ( T ξ )( z ⊗ ζ ⊗ h ) , Hence, W ∗ s ρ s ( X ) W s = ρ ( X ) | G s , X ∈ H ∞ ( E ) . 2) Let Y ⊗ I H ∈ Alg Lat ρ ( H ∞ ( E )). Then using by the observations donebefore lemma, ( Y ⊗ I H ) | G s ∈ Alg Lat ( ρ ( H ∞ ( E )) | G s ) and W ( Y ⊗ I H ) | G s W ∗ ∈ Alg Lat ρ s ( H ∞ ( E )). (cid:3) heorem 3.15. Let Y ⊗ I H ∈ Alg Lat ρ ( H ∞ ( E )) . Then Φ ( Y ) ⊗ I H ∈ ρ ( H ∞ ( E )) . Proof. Write Z ⊗ I H for Φ ( Y ) ⊗ I H . Z ⊗ I H admits the representation Z ⊗ I H = diag ( π ( a ) , π ( a ) , ... ) , a i ∈ M. Thus, the operator Z ⊗ I H is as in Corollary 3 . 13, 1). Hence a = a . Fix now s ≥ Z ⊗ I H ) | G s = diag ( π s ( a s ) , π s +1 ( a s +1 ) , ... ) . By Lemma 3 . W s ( Z ⊗ I H ) | G s W ∗ s ∈ Alg Lat ρ s ( H ∞ ( E )), and has the matrixrepresentation W s ( Z ⊗ I H ) | G s W ∗ s = diag ( φ s ( a s ) ⊗ I K s , φ ( a s +1 ) ⊗ I K s , ... ) . The operator W s ( Z ⊗ I H ) | G s W ∗ s satisfies all conditions of Corollary 3 . 13, wherewe replace π by φ s ⊗ I K s . Hence a s = a s +1 .Since the choice of s is arbitrary, we obtain a = a = a = ... . Thus, Z ⊗ I H = Φ j ( Y ) ⊗ I H = φ ∞ ( a ) ⊗ I H . (cid:3) Remark 3.16. For the Fourier coefficient Φ j ( Y ) ⊗ I H with j ≥ 1, the abovetechnique is not working and we obtain only the following. Write Z j ⊗ I H for the j -th Fourier coefficient Φ j ( Y ) ⊗ I H , j ≥ 1, of some Y ⊗ I H ∈ Alg Lat ρ ( H ∞ ( E )).According to Theorem 3 . Z j ⊗ I H represented as a multiplication by sequence { ξ ( j )0 , ξ ( j )1 , ξ ( j )2 , ... } ⊂ E ⊗ j . Fix some s ≥ 0. Then, as it is easy to see from thedefinition of G s , the restriction ( Z j ⊗ I H ) | G s is represented by multiplication bysequence { ξ ( j ) s , ξ ( j ) s +1 , ... } ⊂ E ⊗ j . Let further, M s be the symmetric ρ s ( H ∞ ( E ))-coinvariant subspace of F ( E ) ⊗ π s K s , and write R s for the projection onto M s (with R = Q ). Then, by Lemma 3 . 14, and by Theorem 3 . 12, applied to therepresentation ρ s and the subspace M s , we obtain that R s W s ( Z j ⊗ I H ) W ∗ s | M s = R s W s ( T ξ ( j ) s ⊗ I H ) W ∗ s | M s . (19)In the following we shall drop the upper index for a general element ξ ∈ E ⊗ j .Now we may prove the reflexivity of ρ π ( H ∞ ( E )) where π : M → B ( H ) isa reducible representation of a factor. In the proof we shall use the followinglemma. Lemma 3.17. Let M be a factor and π : M → B ( H ) be a faithful normal rep-resentation of M on H . Let H ′ ⊆ H be a nontrivial π ( M ) -invariant subspace.Then1) If a ∈ M and π ( a ) | H ′ = 0 then a = 0 ;2) Let k ≥ , j ≥ and ξ ∈ E ⊗ j such that ξ ⊗ E ⊗ k ⊗ π H ′ = { } . Then ξ = 0 . a ∈ M and π ( a ) h ′ = 0 for every h ′ ∈ H ′ . Write P ′ for theprojection onto H ′ . Clearly, P ′ ∈ π ( M ) ′ . For every l ∈ H and h ′ ∈ H ′ onehave h l, π ( a ) h ′ i = 0. Hence, π ( a ) P ′ = 0. But the same is true if we replace P ′ by its central carrier C P ′ . Since M is factor, so is π ( M ) ′ and C P ′ = I H . Thus, π ( a ) = 0, and since π is faithful we get a = 0.2) We distinguish two cases k = 0 and k ≥ k = 0. If ξ ⊗ π H ′ = { } then for ζ ∈ E ⊗ j , h ′ ∈ H and l ∈ H , h ζ ⊗ l, ξ ⊗ h ′ i = 0 . Then h l, π ( h ζ, ξ i ) h ′ i = 0 , Since l ∈ H is arbitrary, we obtain π ( h ζ, ξ i ) h ′ = 0 . Write Q ′ for the projection in H onto H ′ . Clearly, Q ′ ∈ π ( M ) ′ . We get π ( h ζ, ξ i )) Q ′ = 0 . As in 1), since M is factor so is π ( M ) ′ and the central carrier C Q ′ = I H . Hence, π ( h ζ, ξ i ) = 0. But π is faithful and ζ ∈ E ⊗ j is arbitrary. Thus ξ = 0.Let now k ≥ 1. Then ξ ⊗ E ⊗ k ⊗ π H ′ = { } . For ζ ∈ E ⊗ j , θ , θ ∈ E ⊗ k , h ′ ∈ H and l ∈ H we have h ζ ⊗ θ ⊗ l, ξ ⊗ θ ⊗ h ′ i = 0 . Hence, h θ ⊗ l, ( φ k ( h ζ, ξ i ) θ ) ⊗ h ′ i = 0 . Since θ ∈ E ⊗ k and l ∈ H are arbitrary, we obtain( φ k ( h ζ, ξ i ) θ ) ⊗ h ′ = 0 . Write Q ′ for the projection in E ⊗ k ⊗ π H onto E ⊗ k ⊗ π H ′ . Clearly, Q ′ ∈ ( φ k ( M ) ⊗ I H ) ′ and ( φ k ( h ζ, ξ i ) ⊗ I H ) Q ′ = 0 . As in case when k = 0, since M is factor so is ( φ k ( M ) ⊗ I H ) ′ , and thecentral carrier C Q ′ = I E ⊗ k ⊗ π H . Hence, φ k ( h ζ, ξ i ) ⊗ I H = 0. But φ k is faithfuland ζ ∈ E ⊗ j is arbitrary. Thus ξ = 0. (cid:3) Theorem 3.18. Let M be any factor and let π : M → B ( H ) be a reduciblefaithful normal representation of M on a Hilbert space H . Then the algebra ρ π ( H ∞ ( E )) is reflexive. Y ⊗ I H ∈ Alg Lat ρ π ( H ∞ ( E )). We already have seen (even in amore general situation) that Φ ( Y ) ⊗ I H ∈ ρ π ( H ∞ ( E )). Thus we need to showthat, under our assumptions, Φ j ( Y ) ⊗ I H ∈ ρ π ( H ∞ ( E )), for every j ≥ M be an arbitrary factor. Since π is reducible, the Hilbert space H splits onto direct sum H = H ⊕ H of π ( M )-invariant subspaces H , H = (0),each of them is wandering with respect to the covariant representation ( V, σ ), σ ( · ) = φ ∞ ( · ) ⊗ I H .Set L k := L k ( H ) = E ⊗ k ⊗ π H ⊂ F ( E ) ⊗ π H , where L is the generalizedshift associated with ( V, σ ). Since ( V, σ ) is isometric, it is clear that E ⊗ k ⊗ π H ⊥ E ⊗ k ⊗ π H . Now set N k := H ⊕ L k . Since H and L k both are wandering and H ⊥ H , N k is a wandering subspace. N k generates the subspace in F ( E ) ⊗ π H M N k := ⊕ X s L s ( N k ) , which is unitarily isomorphic to the space F ( E ) ⊗ σ ′ N k = N k ⊕ E ⊗ σ ′ N k ⊕ ..., where σ ′ := σ | N k . It should be noted that in our assumption the representation σ ′ is faithful. This follows from Lemma 3 . 17, 1), since σ ′ | H = π | H and if π ( a ) N k = 0 for some a ∈ M , then π ( a ) | H = 0 for the nontrivial π ( M )-invariantsubspace H ⊂ H . By Lemma 3 . 17, 1), a = 0.Clearly, M N k is a ρ π ( H ∞ ( E ))-invariant subspace, and we denote by V k theunitary from M N k onto F ( E ) ⊗ σ ′ N k .Write Z j ⊗ I H for the operator Φ j ( Y ) ⊗ I H , Y ⊗ I H ∈ Alg Lat ρ π ( H ∞ ( E )).Then the restriction ( Z j ⊗ I H ) | M N k is in Alg Lat ρ π ( H ∞ ( E )) | M N k and theoperator V k ( Z j ⊗ I H ) V ∗ k is in Alg Lat ρ σ ′ ( H ∞ ( E )). Applying Corollary 3 . V k ( Z j ⊗ I H ) V ∗ k , we deduce that there is Z ′ j ∈ L ( F ( E )) such that Z ′ j ⊗ I N k = V k ( Z j ⊗ I H ) V ∗ k ∈ Alg Lat ρ σ ′ ( H ∞ ( E )) . By Theorem 3 . z i ∈ E ⊗ j , i = 0 , , , ... , such that(Φ j ( Z ′ j ) ⊗ I N k ) | E ⊗ k ⊗ σ ′ N k = ( T z k ⊗ I N k ) | E ⊗ k ⊗ σ ′ N k , k ≥ . Recall the formula Φ j ( Z ′ j ) ⊗ I N k = ( P s P s + j Z ′ j P s ) ⊗ I N k , where, as usual, P s is the projection P E ⊗ s , and write R s = P s ⊗ I N k .Take f ∈ H and θ ⊗ l ∈ L k arbitrary. Then(Φ j ( Z ′ j ) ⊗ I N k )( f + θ ⊗ l ) = z ⊗ ( f + θ ⊗ l ) . On the other hand(Φ j ( Z ′ j ) ⊗ I N k )( f + θ ⊗ l ) = R j ( V k ( Z j ⊗ I H ) V ∗ k ) R ( f + θ ⊗ l ) = ξ ⊗ f + ξ k ⊗ θ ⊗ l, { ξ k } k ≥ ⊂ E ⊗ j is the sequence which corresponds to Z j ⊗ I H by Theorem3 . 7. Thus, (Φ j ( Z ′ j ) ⊗ I N k )( f + θ ⊗ l ) = ( Z j ⊗ I H )( f + θ ⊗ l )or z ⊗ ( f + θ ⊗ l ) = ξ ⊗ f + ξ k ⊗ θ ⊗ l. (20)Set ζ := z − ξ . Hence L ζ | H = 0 . Using Lemma 3 . 17, 2), taking there k = 0, we obtain ζ = 0. Thus, ξ = z .From (20) we obtain now L ξ k | L k = L ξ | L k . Set ζ := ξ k − ξ . Thus L ζ | L k = 0. Again, using Lemma 3 . 17, 2) (with k ≥ ζ = 0, i.e. ξ k = ξ . Since k ≥ ξ = ξ = ... .This shows that Φ j ( Y ) ⊗ I H ∈ ρ π ( H ∞ ( E )) for any j ≥ 0, hence, Y ⊗ I H ∈ ρ π ( H ∞ ( E )), i.e. ρ π ( H ∞ ( E )) = Alg Lat ρ π ( H ∞ ( E )). (cid:3) Corollary 3.19. If M is a factor of type II or III , then for every faithfulnormal representation π : M → B ( H ) , the algebra ρ π ( H ∞ ( E )) is reflexive. Proof. If π is irreducible, then π ( M ) ′ = C I . Hence, π ( M ) = π ( M ) ′′ = B ( H ),i.e. π ( M ), and therefore M , is of type I . When this is not the case, π isreducible and the previous theorem applies. (cid:3) Remark 3.20. 1) In [2] Arias and Popescu proved the reflexivity of ρ ( H ∞ ( C n ))over the factor (of type I ) M = C , without the assumption of reducibility of π .2) In the previous chapter we saw that if M is a factor of type III then ρ π ( H ∞ ( E )) is even hyperreflexive. (cid:3) Here we consider the W ∗ -correspondence α M over a W ∗ -algebra M . We recallthat α ∈ End ( M ) and that p = α (1) is a projection in M . We set E = M ⊗ α M ,that is the self-dual completion of the algebraic tensor product M ⊗ M of algebraswith the relations ac ⊗ b = a ⊗ α ( c ) b and equipped with the inner productdefined by h a ⊗ b, c ⊗ d i = b ∗ α ( a ∗ c ) d . We identify M ⊗ α M with α ( M ) M andit is easy to see that the inner product in α ( M ) M turns to be h α ( a ) b, α ( c ) d i =( α ( a ) b ) ∗ α ( c ) d . The left action of M on E we define by a · ξ = α ( a ) ξ, where a ∈ M and ξ ∈ E . So, E has the structure of a W ∗ -correspondence over M and we write α M for it. In what follows we always assume α to be injective.30he map a α (1) a = pa gives us the identification of α M with α (1) M . Thus, E = α M = α (1) M = pM .Note that for every k ≥ α k (1) = α k − ( p ) α k − ( p ) ...α ( p ) p , and α k ( p ) ≥ α k +1 ( p ), for every k ≥ k ≥ 0, the map ξ ⊗ ... ⊗ ξ k α k − ( ξ ) ...α ( ξ k − ) ξ k , gives anidentification E ⊗ k = α k M , α k M = α k (1) M . The left action of M on α k M isgiven by the formula a · ξ k = α k ( a ) ξ k , a ∈ M and ξ k ∈ α k M The full Fockspace over α M is now F ( E ) = F ( α M ) = ⊕ X k ≥ α k M. For the left action φ ∞ of M on F ( α M ) we write α ∞ , which is now given by α ∞ = diag ( α , α, α , ... ). Thus, if ( ξ k ) ∈ F ( α M ) then φ ∞ ( a )( ξ k ) = α ∞ ( a )( ξ k ) =( α k ( a ) ξ k ).Let ξ ∈ E = α M and ξ k ∈ α k M , then T ξ ξ k = ξ ⊗ ξ k = α k ( ξ ) ξ k ∈ α k +1 M .Clearly, T α ( a ) ξb = α ∞ ( a ) T ξ α ∞ ( b ). Since every ξ ∈ E has a form ξ = α (1) a , a ∈ M , we get T ξ = T α (1) α ∞ ( a ). Thus, the operator T ξ is completely determinedby T α (1) = T p . Every ξ k ∈ α k M has the form α k (1) a k , a k ∈ M . Then T ξ ξ k = α k ( ξ ) ξ k = α k +1 (1) α k ( a ) a k . So, the set of generators of H ∞ ( α M ) consists ofleft multiplications α ∞ ( a ) by elements a of M , and the creation operator T α (1) .Let π : M → B ( H ) be a faithful normal representation of M on Hilbert space H . For every k ≥ 1, the space α k M ⊗ π H can be identified with π ( α k (1)) H = π ( α k − ( p )) H via ξ ⊗ ... ⊗ ξ k ⊗ h π ( α k − ( ξ ) ...α ( ξ k − ) ξ k ) h. For k = 0 this formula reduces to M ⊗ π H ∼ = π ( M ) H . So, the space F ( α M ) ⊗ π H can be identified with a subspace of l ( Z + , H ): F ( α M ) ⊗ π H ∼ = ⊕ X k ≥ π ( α k (1)) H. Let us consider the induced representation ρ ( H ∞ ( α M )) = π F ( α M ) ( H ∞ ( α M )).If Y ⊗ I H ∈ Alg Lat ( ρ ( H ∞ ( α M ))) then, as it follows from Theorem 3 . 7, the j -th Fourier coefficient is represented as multiplication by the matrix (we omitthe upper indices in L and the upper indices in ξ ∈ E ⊗ j ) ... ...... ... ... ... ... ... ... ... ... ... ...L ξ ... ... L ξ ... ... L ξ ... ...... ... ... ... ... ... , ξ s ∈ α j M . Hence there exist sequence { a s } s ≥ ⊂ M such that ξ s = α j (1) a s . Notice also that the adjoint operator ( T ξ ⊗ I H ) ∗ acts on α k M ⊗ π H by the formula T ∗ ξ ⊗ I H : ξ k ⊗ h = π ( α k (1) a k ) h π ( α k − ( a ∗ )) π ( α k (1) a k ) h,ξ = α (1) a , ξ k = α k (1) a .By Theorem 3 . 18, if π : M → B ( H ) is a reducible representation of a factor(in particular if M is a factor of type II or III ), then the algebra ρ π ( H ∞ ( α M ))is reflexive, for every endomorphism α ∈ End ( M ). In [9] E. Kakariadis, showedthat in the special case when α is a unitarily implemented automorphism of M ,the algebra ρ ( H ∞ ( α M )) is reflexive. We want first to show how Theorem 3 . v ∈ U ( M ) be some unitary in U ( M ), the unitary group of M . Let α ∈ Aut ( M ), and assume that α ( a ) = vav ∗ . Write u = π ( v ) ∈ U ( H ), where U ( H ) is the unitary group of the Hilbert space H . Hence, π ( α ( a )) = uπ ( a ) u ∗ .Fix some h ∈ H , some 0 < r < M := close { π ( b ) h + π ( vb ) rh + π ( v b ) r h + ... : b ∈ M } . It is easy to see that M is a ρ ( H ∞ ( α M ))-coinvariant subspace. Indeed, let x = π ( b ) h + π ( vb ) rh + π ( v b ) r h + ... ∈ M and let ξ = a ∈ α M , then( T ∗ ξ ⊗ I H ) x = π ( a ∗ ) π ( vb ) rh + π ( α ( a ∗ )) π ( v b ) r h + ... + π ( α s − ( a ∗ )) π ( v s ) r s h + ..., and since π ( α s − ( a ∗ )) π ( v s ) r s h = π ( v s − a ∗ ( v ∗ ) s − v s ) r s h = π ( v s − a ∗ vr ) r s − h ,we obtain that for c = a ∗ vr ( T ∗ ξ ⊗ I H ) x = π ( c ) h + π ( vc ) rh + π ( v c ) r h + ... ∈ M . If a ∈ M then ( α s ( a ∗ ) ⊗ I H ) π ( v s b ) r s h = π ( α s ( a ∗ ) v s b ) r s h = π ( v s a ∗ v ∗ s v s b ) r s h = π ( v s a ∗ b ) r s h . Hence,( α ∞ ( a ∗ ) ⊗ I H ) x = π ( a ∗ b ) h + π ( va ∗ b ) rh + π ( v a ∗ b ) r h + .... ∈ M . Thus M is indeed coinvariant.Take y := h + π ( v ) rh + π ( v ) r h + ... + π ( v s ) r s h + ... ∈ M . Then, using the notation of Theorem 3 . j ( Y ) ⊗ I H ) ∗ y = L ∗ ξ π ( v j ) r j h + L ∗ ξ π ( v j +1 ) r j +1 h + ... + L ∗ ξ s π ( v j + s ) r j + s h + ..., where ξ s ∈ E ⊗ j and L ξ s = ( T ξ s ⊗ I H ) | E ⊗ s ⊗ π H : E ⊗ s ⊗ π H → E ⊗ s + j ⊗ π H .Since (Φ j ( Y ) ⊗ I H ) ∗ y ∈ M , there exist some sequence { c i } i ⊂ M such that(Φ j ( Y ) ⊗ I H ) ∗ y = lim i [ h + π ( c i ) h + π ( vc i ) rh + π ( v c i ) r h + ... + π ( v s c i ) r s h + ... ] . i π ( c i ) h = L ∗ ξ π ( v j ) r j h = π ( a ∗ v j ) r j h, and lim i π ( v s c i ) r s h = L ∗ ξ s π ( v j + s ) r j + s h = π ( α s ( a ∗ s ) v j + s ) r j + s h, Then, lim i π ( c i ) h = π ( a ∗ v j ) r j h = π ( v j ) π ( α − j ( a ∗ )) r j h, andlim i π ( v s c i ) r s h = π ( α s ( a ∗ s ) v j + s ) r j + s h = π ( v j + s ) π ( α − ( j + s ) ( α s ( a ∗ s ))) r j + s h. But lim i π ( v s c i ) r s h = π ( v s ) r s lim i π ( c i ) h = π ( v j + s ) π ( α − j ( a ∗ )) r j + s h , and weget π ( v j + s ) π ( α − j ( a ∗ )) r j + s h = π ( v j + s ) π ( α − ( j + s ) ( α s ( a ∗ s )) r j + s h. Since π ( v k ) is unitary for every k , r is scalar and this equality holds for every h ∈ H , we obtain π ( α − j ( a ∗ )) = π ( α − ( j + s ) ( α s ( a ∗ s )) . Since π is faithful it follows that α − j ( a ∗ ) = α − ( j + s ) ( α s ( a ∗ s )) = α − j ( a ∗ s ) . Thus, a s = a for every s ≥ 0. We showed that if Y ⊗ I H ∈ Alg Lat ρ ( H ∞ ( α M ))then for every j ≥ 1, Φ j ( Y ) ⊗ I H is in ρ ( H ∞ ( α M )). Hence we proved Theorem 3.21. Let π : M → B ( H ) be a faithful normal representation of the W ∗ -algebra M . Then the algebra ρ π ( H ∞ ( α M )) is reflexive, whenever α is anautomorphism that is unitarily implemented. (cid:3) Corollary 3.22. If M is a factor and α ∈ Aut ( M ) , then ρ π ( H ∞ ( α M )) isreflexive for any normal representation π . Proof. The type II and type III cases follow from Corollary 3 . 19. If M is atype I factor, every α ∈ Aut ( M ) is unitarily implemented, thus the previoustheorem applies and we are done. (cid:3) The Hardy algebra compressed to M . In the previous chapter in formula (15) we defined the generalized symmet-ric ρ π ( H ∞ ( E ))-coinvariant subspace M . Remember that we write ρ for therepresentation ρ π . In this chapter we shall prove the reflexivity of the com-pression of ρ ( H ∞ ( E )) to the subspace M , i.e. we shall prove the reflexivity of Qρ ( H ∞ ( E )) | M , where we write Q = P M for the projection onto M . It shouldbe noted that even in the cases when we know that ρ ( H ∞ ( E )) is reflexive, thisresult does not follows immediately, since the algebras Alg Lat ( Qρ ( H ∞ ( E )) | M )and Q ( Alg Lat ρ ( H ∞ ( E ))) | M are not the same in general. Note also that thetheorem on the reflexivity of Qρ ( H ∞ ( E )) | M generalizes to our setting a resultof G. Popescu from [22, Theorem 4.5]. I thank Orr Shalit who pointed out thatthis theorem is also related to the fact that multiplier algebras on a reproducingkernel Hilbert space are always reflexive.Write Q k for the projection onto H ( k ) = W η η ( k ) ( H ). Clearly, Q k ≤ P k ⊗ I H .In particular, since M ⊗ π H ∼ = H = H (0), and E ⊗ π H = W η η ( H ) = H (1), wehave Q = P ⊗ I H and Q = P ⊗ I H .Set ˜ W t = X k ≥ e ikt Q k , and ˜ γ t = Ad ˜ W t , where Ad ˜ W t ( T ) = ˜ W t T ˜ W ∗ t for T ∈ B ( M ). Then { ˜ γ t } t ∈ R is an ultraweaklycontinuous action of R on B ( M ), which is the gauge automorphism group. Asin the previous chapter, the group { ˜ γ t } t ∈ R leaves invariant Qρ ( H ∞ ( E )) | M .The j -th Fourier coefficient of T ∈ B ( M ), associated with { ˜ γ t } R , is defined byΦ j ( T ) = 12 π Z π e − ijt ˜ γ t ( T ) dt. As for L ( F ( E )) this integral ultraweakly converges in B ( M ) and leaves invari-ant Qρ ( H ∞ ( E )) | M (since Qρ ( H ∞ ( E )) | M is { ˜ γ t } t ∈ R -invariant). Let Y be anelement of Alg Lat Qρ ( H ∞ ( E )) | M . Direct calculation gives the formulaΦ j ( Y ) = X k ≥ Q k + j Y Q k . Here we shall again use the upper index to indicate the degree of the generalelement of E ⊗ s , i.e. ξ ( s ) ∈ E ⊗ s . Lemma 4.1. If Y ∈ Alg Lat Qρ ( H ∞ ( E )) | M then for every j , the operator Φ j ( Y ) is in Alg Lat Qρ ( H ∞ ( E )) | M Proof. From Φ j ( Y ) = P k ≥ Q k + j Y Q k we conclude that Φ j ( Y ) = 0 for j < j ≥ 0. Note that ˜ W t has a closed range and by simple calculation we obtainthat ˜ W ∗ t = ˜ W − t and ˜ W t ( T ξ ⊗ I H ) | M = e it Q ( T ξ ⊗ I H ) ˜ W t | M . x = h + θ ⊗ h + θ (2)2 ⊗ h + ... + θ ( s ) s ⊗ h s ∈ M . Then Q ( T ξ ⊗ I H ) ˜ W t x = Q ( T ξ ⊗ I H )( h + e it θ ⊗ h + e it θ (2)2 ⊗ h + ... + e sit θ ( s ) s ⊗ h s ) = Q ( ξ ⊗ h )+ e it Q ( ξ ⊗ θ ⊗ h )+ e it Q ( ξ ⊗ θ (2)2 ⊗ h )+ ... + e sit Q s +1 ( ξ ⊗ θ ( s ) s ⊗ h s ) == e − it ˜ W t ( T ξ ⊗ I H )( h + θ ⊗ h + θ (2)2 ⊗ h + ... + θ ( s ) s ⊗ h s ) . It follows that if M ∈ Lat Qρ ( H ∞ ( E )) | M then also M t := ˜ W t ( M ) ∈ Lat Qρ ( H ∞ ( E )) | M .Let Y ∈ Alg Lat Qρ ( H ∞ ( E )) | M . Then˜ γ t ( Y )( M ) = ˜ W t Y ˜ W − t ( M ) = ˜ W t Y ( M − t ) ⊆ ˜ W t ( M − t ) = ˜ W t ˜ W − t ( M ) = M . Integrating, we get Φ j ( Y )( M ) ⊆ M , for all j . (cid:3) Proposition 4.2. Let Y ∈ Alg Lat Qρ ( H ∞ ( E )) | M and let us consider therestriction Φ j ( Y ) | H . Then1) For j = 0 there exists a ∈ M such that Φ ( Y ) | H = π ( a ) . 2) For j ≥ there exist ξ ( j )0 ∈ E ⊗ j such that Φ j ( Y ) | H = Q j L (0) ξ ( j )0 = Q j ( T ξ ( j )0 ⊗ I H ) | H . Proof. 1) Since Φ ( Y ) = P k Q k Y Q k , we obtain that Φ ( Y ) | H = Q Y Q . Picksome h ∈ H and set M h := Qρ ( H ∞ ( E )) h = Q [ π ( M ) h ⊕ E ⊗ π h ⊕ ... ] . Then M h is Qρ ( H ∞ ( E )) Q -invariant, h ∈ M h , and Φ ( Y ) h ∈ Q M h = π ( M ) h . There exist a net { a ι } ⊂ M such that Φ ( Y ) h = lim ι π ( a ι ) h . Let p ∈ π ( M ) ′ be any projection. Then there are nets { b ι } and { c ι } in M such thatΦ ( Y ) ph = lim ι π ( b ι ) ph and Φ ( Y ) p ⊥ h = lim ι π ( c ι ) p ⊥ h. Hence, Φ ( T ) h = lim ι π ( b ι ) ph + lim ι π ( c ι ) p ⊥ h, and applying p on both sides, we obtain p Φ ( Y ) h = lim ι π ( b ι ) ph = Φ ( Y ) ph. It follows that Φ ( Y ) commutes with p at h . Since h is arbitrary we obtain thatΦ ( Y ) p = p Φ ( Y ) on H . Since this holds for every projection from the vonNeumann algebra π ( M ) ′ , we obtain that Φ ( Y ) | H ∈ π ( M ) ′′ = π ( M ). Thus,Φ ( Y ) | H = π ( a ), for some a ∈ M . 35) For j ≥ j ( Y )( H ) = Q j Y Q ( H ) ⊂ E ⊗ j ⊗ π H . As for Φ ( Y ) wepick an arbitrary h ∈ H and form the Qρ ( H ∞ ( E )) Q -invariant subspace M h = Qρ ( H ∞ ( E )) h as above. Then Φ j ( Y ) h ∈ M h , and Φ j ( Y ) h ∈ Q j E ⊗ j ⊗ h ⊆ E ⊗ j ⊗ h . For any projection p ∈ π ( M ) ′ there are nets { θ ( j ) ι } , { ζ ( j ) ι } , { ϑ ( j ) ι } in E ⊗ j such that Φ j ( Y ) h = lim ι θ ( j ) ι ⊗ h , Φ j ( Y ) ph = lim ι ζ ( j ) ι ⊗ ph and Φ j ( Y ) p ⊥ h =lim ι ϑ ( j ) ι ⊗ p ⊥ h . HenceΦ j ( Y ) h = lim ι ζ ( j ) ι ⊗ ph + lim ι ϑ ( j ) ι ⊗ p ⊥ h, and by applying I j ⊗ p on both sides we obtain( I j ⊗ p )Φ j ( Y ) h = ( I j ⊗ p ) lim ι ζ ( j ) ι ⊗ ph = lim ι ζ ( j ) ⊗ ph = Φ j ( Y ) ph. Thus, ( I j ⊗ p )Φ j ( Y ) = Φ j ( Y ) p on H . Since p is an arbitrary projection in π ( M ) ′ we obtain that Φ j ( Y ) isintertwines the actions ι ( · ) and I j ⊗ ι ( · ) of π ( M ) ′ on H and on E ⊗ j ⊗ π H respectively, where ι is the identity action of π ( M ) ′ . By combining with theFourier transform U = U π we obtain the operator U Φ j ( Y ) : H → ( E π ) ⊗ j ⊗ ι H. Since U also intertwining the actions of π ( M ) ′ on E ⊗ j ⊗ π H and on ( E π ) ⊗ j ⊗ ι H ,as it pointed in the Chapter 2, formula (8). Hence there exist the unique ξ ( j )0 ∈ E ⊗ j such thatˆ ξ ( j ) ∗ ( η ⊗ ... ⊗ η k ⊗ h ) = L ∗ ξ ( j )0 (( I j − ⊗ η ) ...η k ( h )) , and in particular, ˆ ξ ( j ) ∗ ( η ⊗ j ⊗ h ) = L ∗ ξ ( j )0 (( I j − ⊗ η ) ...η ( h )) . Thus, from the formula 5, U ∗ ˆ ξ ( j ) ∗ = L ∗ ξ ( j )0 . Recall also that M = U ∗ ˜ M , and, in particular, H ( s ) = U ∗ s ˜ H ( s ), s ≥ j ( Y ) | H = Q j ( T ξ ( j )0 ⊗ I H ) | H . (cid:3) Lemma 4.3. The map X QXQ defines a homomorphism from the algebra Alg Lat ρ ( H ∞ ( E )) to the algebra QAlg Lat ρ ( H ∞ ( E )) | M . Proof. Enough to show that this map is multiplicative, that is QXY Q = QXQY Q , for every X, Y ∈ Alg Lat ρ ( H ∞ ( E )). But the coinvariant projec-tion Q can be represented as the difference I − Q ⊥ of two invariant projections.Thus, Q is semiinvariant and the map X QXQ is multiplicative.36 Let η ∈ D ( E π ). Then M η is Qρ ( H ∞ ( E )) Q -invariant. To see this, note, as itfollows from the preceding lemma, that for every ξ ∈ E and a ∈ M , Q ( T ξ ⊗ I H ) Q and Q ( φ ∞ ( a ) ⊗ I H ) Q are the generators of Qρ ( H ∞ ( E )) Q . Since Q M η = M η and M η is ρ ( H ∞ ( E ))-invariant we deduce that M η is Qρ ( H ∞ ( E )) Q -invariant.Thus, for every Y ∈ Alg Lat Qρ ( H ∞ ( E )) | M we have Y ∗ M η ⊆ M η , henceΦ j ( Y ) ∗ M η ⊆ M η for every j ≥ ( Y ) of a given Y ∈ Alg Lat Qρ ( H ∞ ( E )) | M .Take any h ∈ H , thenΦ ( Y ) ∗ ( h + η ( h ) + ... + η ( s ) ( h ) + ... ) = ( k + η ( k ) + ... + η ( s ) ( k ) + ... ) . We have Φ ( Y ) ∗ h = π ( a ∗ ) h = k andΦ ( Y ) ∗ η ( s ) ( h ) = η ( s ) ( k ) = η ( s ) ( π ( a ∗ ) h ) = ( φ ∞ ( a ∗ ) ⊗ I H ) η ( s ) ( h ) . Hence, Φ ( Y ) ∗ ( X s η ( s ) ( h )) = ( φ ∞ ( a ∗ ) ⊗ I H )( X s η ( s ) ( h )) , and we obtain that Φ ( Y ) ∗ M η = ( φ ∞ ( a ∗ ) ⊗ I H ) M η . Theorem 4.4. The algebra Qρ ( H ∞ ( E )) | M is reflexive. Proof. Every Y ∈ Alg Lat Qρ ( H ∞ ( E )) | M is a ultraweak limit of Cesaro sumsof its Fourier coefficients. Hence, enough to show that if Y ∈ Alg Lat Qρ ( H ∞ ( E )) | M then1) Φ ( Y ) | M = ( φ ∞ ( a ) ⊗ I H ) | M , and2) Φ j ( Y ) | M = Q ( T ξ ( j )0 ⊗ I H ) | M , for j ≥ j = 0. From the equality Φ ( Y ) ∗ M η = ( φ ∞ ( a ∗ ) ⊗ I H ) M η weconclude that Φ ( Y ) ∗ η ( s ) ( h ) = ( φ s ( a ∗ ) ⊗ I H ) η ( s ) ( h ) , for every η ∈ E π , h ∈ H and s ≥ 0. Hence,Φ ( Y ) ∗ M = ( φ ∞ ( a ∗ ) ⊗ I H ) M , and as a consequence Φ ( Y ) | M = ( φ ∞ ( a ) ⊗ I H ) | M . For 2) let j ≥ 1. We have Φ j ( Y ) ∗ M η ⊆ M η . ThenΦ j ( Y ) ∗ ( X s η ( s ) ( h )) = X s η ( s ) ( k ) . j ( Y ) ∗ ( η ( s ) ( h )) = 0 for 0 ≤ s ≤ j − k = Φ j ( Y ) ∗ ( η ( j ) ( h )) = ( Q j Y Q ) ∗ ( η ( j ) ( h )) = ( Q Y ∗ Q j )( η ( j ) ( h )) =( T ∗ ξ ( j )0 ⊗ I H ) η ( j ) ( h ). Then, for some k ∈ H , η ( s ) ( k ) = η ( s ) (( T ∗ ξ ( j )0 ⊗ I H )( η ( j ) ( h ))) = ( T ∗ ξ ( j )0 ⊗ I H ) η ( s + j ) ( h ) . 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