LIFTINGS OF COVARIANT REPRESENTATIONS OF W -CORRESPONDENCES

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1 Infinite Dimensional Analysis, Quantum Probability and Related Topics Vol. 13, No. 3 (2010) c World Scientific Publishing Company DOI: /S LIFTINGS OF COVARIANT REPRESENTATIONS OF W -CORRESPONDENCES SANTANU DEY Department of Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai , India dey@math.iitb.ac.in Received 28 April 2009 Revised 24 May 2010 Communicated by K. Sinha We generalize the notion of subisometric liftings of row contractions for liftings of completely contractive covariant representations of W -correspondences. A theory of characteristic functions for such liftings of covariant representations is presented. Keywords: W -correspondence; covariant representation; characteristic function; contractive lifting; subisometric lifting; coisometric; completely noncoisometric. AMS Subject Classification: 47A20, 47A15, 46L05, 46L10 1. Introduction Let the set of all adjointable operators on any Hilbert C -module G over some C -algebra be denoted by L(G). A Hilbert C -module G over a von Neumann algebra M can be equipped with a topology called the σ-topology (see Ref. 1) induced by the functionals f( ) = n=1 ω n( ξ n, ) whereξ n G,ω n the predual M of M and ω n ξ n <. A Hilbert C -module G over a von Neumann algebra M is called self-dual if for every continuous M-module map φ : G M there is an ξ φ Gso that φ(ξ) = ξ φ,ξ, ξ G. For self-dual Hilbert C -module G Paschke proved in Ref. 9 that L(G) is a von Neumann algebra. A W -correspondence E over a von Neumann algebra M is a self-dual Hilbert C -bimodule over M, where the corresponding left action ϕ:m L(E) is normal (i.e. σ-weak). We write ϕ(a)η = aη for all a M,η E. Definition 1.1. A pair (T,σ) is called a covariant representation of a W - correspondence of E over M on a Hilbert space H, if (i) T : E B(H) is a linear map that is continuous with respect to the σ-topology on E and the ultraweak topology on the set B(H) of all bounded operators on H. 511

2 512 S. Dey (ii) σ : M B(H) is a normal homomorphism. (iii) T (aξ) =σ(a)t (ξ),t(ξa) =T (ξ)σ(a) for all ξ E,a M. Moreover, if a covariant representation (T,σ)satisfies T (ξ) T (η) =σ( ξ,η ), ξ,η E it is called isometric. We remark that any row contraction T =(T 1,...,T d )onahilbertspaceh, i.e. T i B(H), T i Ti 1, can be encoded as the covariant representation (T,σ)of the W -correspondence C d over the von Neumann algebra C on H. In this picture σ maps every k C to the multiplication operator M k B(H) andif{e 1,...,e d } denote the standard basis of C d, then T (e i ):=T i for all i =1,...,d.Thiscovariant representation allows a coordinate-free treatment for the row contraction. Our prime objective is to use dilation theory of covariant representations to extend the theory of characteristic functions for liftings of row contractions of Ref. 3 to that of covariant representations of W -correspondences. This module approach is a generalization of the coordinate-free treatment mentioned above. Popescu developed an extensive theory ofdilationsofrowcontractions (see Ref. 10) partly motivated by the dilation theory of Sz. Nagy-Foias. In Ref. 11 he introduced for row contractions some operators called characteristic functions which completely classify a class of so-called completely noncoisometric row contractions. Characteristic functions for single contractions are related to the inner functions appearing in the context of shift-invariant subspaces of Hardy spaces in Beurling s theorem. Literature on characteristic functions for contractions and row contractions is rich (e.g., Refs. 8, 12, 3 and 14). Characteristic functions of covariant representations of W -correspondences have been studied by Muhly and Solel (see Refs. 6 and 7). For a W -correspondence E over a von Neumann algebra M and a normal representation σ : M B(H) the induced tensor product E σ H is the unique Hilbert space generated by elementary tensors ξ h (where ξ E and h H)that have as inner product: Define ξ 1 h 1,ξ 2 h 2 = h 1,σ( ξ 1,ξ 2 )h 2,ξ 1,ξ 2 E; h 1,h 2 H. E σ := {µ B(H, E σ H):µσ(a) =(ϕ(a) 1)µ a M}. This E σ is called the σ-dual of E. The concept of σ-dual was introduced in Ref. 5, and was introduced earlier and independently by Skeide in Ref. 16 under the name of commutant for von Neumann correspondences (i.e. when σ is just the defining representations of the von Neumann algebra M). Its open unit ball is denoted by D(E σ ). Now let us assume that (T,σ) is a covariant representation of E on H such that T as a map from E to B(H) is bounded. An operator T : E H Hcan be

3 Liftings of Covariant Representations of W -Correspondences 513 associated whose norm properties are related to those of T by the following lemma. On the algebraic tensor product of E with H (taken over M) T is defined as T (η h) :=T (η)h, η E, h H. (1.1) T is well-defined as T (ηa) =T (η)σ(a). The next lemma provides a dictionary to translate norm properties of T in terms of those of T and vice versa. Lemma 1.2. (Ref. 4) Let E be a W -correspondence over a von Neumann algebra M and let (T,σ) be a covariant representation of E on H. Then (i) T is completely contractive T 1 for all η 1,η 2,...,η n E (T (η i ) T (η j )) (σ( η i,η j )) in M n (B(H)). (ii) (T,σ) is isometric if and only if T is an isometry. We remark that T, when bounded is an element of E σ. Conversely, for each element µ E σ there is a representation (T,σ)ofE on H such that T = µ : E H H. Definition 1.3. Let E be a W -correspondence over a von Neumann algebra M. Let (C, σ C ) be a contractive covariant representation of E on Hilbert space H C. Then a contractive covariant representation (E,σ E )ofe on a Hilbert space H E H C is called a contractive lifting of (C, σ C )if (i) H C reduces σ E and for all a M σ E (a) HC = P HC σ E (a) HC = σ C (a). (ii) HC is invariant w.r.t. E(ξ) for all ξ E. (iii) P HC E(ξ) HC = C(ξ) for all ξ E. Set H A = HC,A(ξ) := E(ξ) H A and σ A (a) :=σ E (a) HA for all ξ E,a M. Observe that (A, σ A ) is also a covariant representation of E. The direct sum decomposition H E = H C H A induces a similar decomposition of E H E and thereby Ẽ (defined from E as in (1.1)) has the form ( ) ( ) C 0 B(E HC, H C ) B(E H A, H C ) Ẽ = B à B(E H, H) =, B(E H C, H A ) B(E H A, H A ) where C, B and à are defined from C, A and B respectively as in (1.1). Note that if (E,σ E ) is completely contractive then (C, σ C )and(a, σ A ) are also completely contractive. This follows easily by passing to Ẽ and using Lemma 1.2. After some preparation in the next section, the main results are discussed in Sec. 3. We first consider in Sec. 3 a class of liftings of contractive covariant representations of W -correspondences called subisometric liftings and provide a

4 514 S. Dey classification scheme for them in terms of characteristic functions. While studying subisometric liftings and those called coisometric liftings, we realize the hypothesis needed on a larger class of liftings of covariant representations which can be classified similarly using characteristic functions. We also provide a kind of functional model for these liftings which is useful for a finer understanding of the structure of the liftings. 2. Dilations Let E be a W -correspondence over a von Neumann algebra M. Definition 2.1. Let (T,σ) be a completely contractive covariant (c.c.c. for short) representation of E on H. An isometric dilation (V,π) of(t,σ) is an isometric covariant representation of E on H Hsuch that (V,π) is a lifting of (T,σ). A minimal isometric dilation (mid) of (T,σ) is an isometric dilation (V,π) onĥ for which Ĥ = span{v (ξ 1 ) V (ξ n )h : h H,ξ i E for i =1,...,n}. It is easy to check that minimal isometric dilation is unique up to unitary equivalence. The tensor product E E is defined as the unique W -correspondence generated by elementary tensors ξ η (where ξ,η E) that have as inner product: ξ 1 η 1,ξ 2 η 2 = η 1,ϕ( ξ 1,ξ 2 )η 2, ξ 1,ξ 2,η 1,η 2 E. Further one defines the full Fock module over M as F(E) = n=0 E n, where E 0 = M, and the tensor products and direct sum are those of W - correspondence. We will write F for F(E) to simplify notations. Let m 0 denote the element 1 0 F where 1 is the identity of E 0 = M. For a ξ E the creation operator L ξ on F(E) isgivenbyl ξ η = ξ η. We have an induced homomorphism ϕ n from M to L(E n )whichforeach a Mis given by ϕ n (a)(ξ 1 ξ 2 ξ n )=(ϕ(a)ξ 1 ) ξ 2 ξ n, Define an operator ϕ (a) := diag(a, ϕ(a),ϕ 2 (a),...), a M. ξ 1,...,ξ n E. Given a c.c.c. representation (T,σ)ofaW -correspondence E on a Hilbert space H, we consider the associated T : E H H. We set D,T := (1 T T ) 1 2 (in B(H)) and D T := (1 T 1 T ) 2 (in B(E H)). Let D,T := range D,T and D T = range D T. That every c.c.c. representation (T,σ)ofE has a mid (V,π)wasprovedinRef.6 by the following construction: The representation Hilbert space for this (V,π) is Ĥ = H F σ1 D T,

5 Liftings of Covariant Representations of W -Correspondences 515 where σ 1 is defined to be the restriction to D T of ϕ( ) 1 H. The representation π is given by π = σ σ F 1 ϕ with (σ1 F ϕ )(a) =ϕ (a) 1 DT for a M, andthenthelinearmapv : E B(Ĥ) is given in operator matrix form by: T (ξ) 0 0 D T (ξ ) 0 0 V (ξ) = (2.1) Moreover, if T T = 1, then (T,σ)issaidtobecoisometric. It is also known that the mid (V,π) iscoisometric if and only if (T,σ) is coisometric (see Ref. 5). Remark 2.2. Define L 1 DT : E B(F D T )by(l 1 DT )(ξ) =L ξ 1 DT. Then that (L 1 DT,ϕ ( ) 1 DT ) is an isometric covariant representation of E is immediate. Similarly (L 1 D,T,ϕ ( ) 1 D,T ) is also an isometric covariant representation of E. We recall another notion from Ref. 7 that is of importance to us. Suppose E is a W -correspondence over a von Neumann algebra M and σ : M B(H) isa normal representation, where H is a Hilbert space. For every µ D(E σ )wesetan operator µ (n) : H E n σ H by µ (n) := (1 E n 1 µ)(1 E n 2 µ) (1 E µ)µ (2.2) for n 1withµ (0) = 1 H. Now with each µ D(E σ ) we associate the operator called Poisson kernel K(µ) := (1 F (1 µ µ) 1 2 )µ (j) (2.3) j=0 which maps H to F σ H. In Proposition 10 of Ref. 7 it is shown that K(µ) isa contraction. We know that if (T,σ) is a c.c.c. representation of a W -correspondence E on a Hilbert space H, then µ := T D(E σ ). Let us define T n := (µ (n) ). Definition 2.3. Let (T,σ) be a c.c.c. representation of E on H. Then (T,σ)is called -stable if lim n T n ( T n ) = 0 in strong operator topology on B(H).

6 516 S. Dey 3. Liftings We consider two special cases of liftings before investigating the general case. Let E be a W -correspondence over a von Neumann algebra M. Let (C, σ C ) be a c.c.c. representation of E on H C and (E,σ E )onh E H C be a completely contractive lifting of (C, σ C ). Denote by (V E,π E )and(v C,π C ) the mids of (E,σ E ) and (C, σ C ) respectively. From the definition of lifting it is immediate that the space of the mid V C can be embedded as a subspace reducing V E. Definition 3.1. A lifting (E,σ E ) of a c.c.c. representation (C, σ C )onh E H C is called subisometric if the corresponding mids V E and V C are unitarily equivalent, i.e. there exists a unitary W : Ĥ E ĤC such that W HC = 1 HC,WV E (ξ) = V C (ξ)w for all ξ E and Wπ E (a) =π C (a)w for all a M. Remark 3.2. Alternatively, a subisometric lifting means the existence of a unitary W (same as above) such that Ṽ C (1 W )=W Ṽ E. Proposition 3.3. Let (C, σ C ) be a completely contractive covariant (c.c.c. for short) representation of W -correspondence E on H C. A completely contractive lifting (E,σ E ) on H E = H C H A of (C, σ C ) with ( ) C(ξ) 0 E(ξ) =, ξ E, B(ξ) A(ξ) is subisometric if and only if (A, σ A ) is -stable (see Definition 2.3) and B = D,A γ D C for an isometry γ : D,A D C. Proof. We assume that (E,σ E ) is a subisometric lifting of (C, σ C ). Let the mid V C of (C, σ C ) be realized on ĤC = H C (F D C ) as in (2.1). For the unitary W from Definition 3.1 let H A := W H A. This W will transfer (E,σ E ) to a unitarily equivalent representation (E,σ E )ofe on H E := H C H A, whose mid is V C. In operator matrix form we assume ( ) C(ξ) 0 E (ξ) =. B (ξ) A (ξ) Here H A is a coinvariant subspace of F D C for L ξ 1 C,ξ Eand hence L 1 C is an isometric lifting of (A,σ A ). Further, as (A,σ A )isacompressionof L 1 C, the representation (A,σ A )is -stable (and therefore also (A, σ A )). Since the Poisson kernel K(η),η D(E σa ) is now an isometry 7 from H A to F D,A, the representation L 1 D,A of E on F D,A is a mid of (A, σ A ). From the fact that mid is unique up to unitary equivalence, it is immediate that the map W HA

7 Liftings of Covariant Representations of W -Correspondences 517 which embeds H A in F D C is Poisson kernel type, namely H A h (1 F γd,a )µ (j) h, (3.1) j=0 where µ = Ã and γ : D,A D C is an isometry. Further as (E (ξ)) = (V C (ξ)) HE, we get (B (ξ)) = P HC (V C (ξ)) P HA. This together with (2.1) and (3.1) implies that B = D C γd,a. The converse follows easily using (2.1). Definition 3.4. The characteristic function of a subisometric lifting (E,σ E )of (C, σ C ) is defined as M C,E := W F DE : F D E F D C. The map Θ C,E : m 0 D C F D E given by is called the symbol of M C,E. Θ C,E := M C,E m0 D E This characteristic function is an isometry and has the following commutation property: M C,E (L ξ 1 E )=(L ξ 1 C )M C,E, ξ E. (3.2) For the special case of covariant representations of W -correspondence associated to row contractions, mentioned in the Introduction, the analogous commutation property is called multi-analytic property (see Refs. 3 and 11). Let (C, σ C ) be a c.c.c. representation of E on a Hilbert space H C and let (E,σ E ) and (E,σ E ) be two subisometric liftings of (C, σ C ). We call the characteristic functions M C,E and M C,E equivalent if there exist a unitary v : D E D E such that Θ C,E =Θ C,E v. (3.3) Theorem 3.5. Let (C, σ C ) be a c.c.c. representation of E on a Hilbert space H C. Two subisometric liftings (E,σ E ) and (E,σ E ) of (C, σ C ) are unitarily equivalent if and only if the corresponding characteristic functions M C,E and M C,E are equivalent. Proof. For the proof of the necessary part we assume that the liftings (E,σ E )and (E,σ E ) are c.c.c. representations on H E and H E, and U : H E H E is a unitary such that U HC = 1 HC and UE(ξ) =E (ξ)u, Uσ E (a) =σ E (a)u, ξ E,a M.

8 518 S. Dey The mids of unitarily equivalent row contractions are unitarily equivalent. Hence we extend U canonically to a unitary Û : Ĥ E ĤE defined between the spaces of mids (V E,π E )and(v E,π E )withû H E = U, and we get ÛV E (ξ) =V E (ξ)û, Ûπ E (a) =π E (a)û, ξ E,a M. As (E,σ E )and(e,σ E ) are subisometric, we also have unitaries W : Ĥ E ĤC and W : Ĥ E ĤC respectively with: Let us define From the following commuting diagram V C (ξ)w = WV E (ξ), W HC = 1 HC, V C (ξ)w = W V E (ξ), W HC = 1 HC. Ĥ E U C := W ÛW : Ĥ C ĤC. U C Ĥ C W W Û Ĥ E Ĥ C Ĥ E V E (ξ) W Û V C (ξ) Ĥ E V E (ξ) Ĥ C U ĤC C W V C (ξ) (3.4) we conclude that U C commutes with the V C (ξ) forξ E. Observe that U C HC = 1 HC because each of W, W and Û fixes H C pointwise. Therefore U C V C (ξ 1 )V C (ξ 2 ) V C (ξ n )h = V C (ξ 1 )V C (ξ 2 ) V C (ξ n )U C h = V C (ξ 1 )V C (ξ 2 ) V C (ξ n )h for ξ i E,h H C. Now by minimality of V C we deduce that U C = 1. Hence W =(U C ) W Û = W Û.Infact,Û maps m 0 D E ĤE onto m 0 D E ĤE. Let us define the unitary v := Û D E : D E D E. Then we see that Θ C,E =Θ C,E v. Conversely we show that if Θ C,E =Θ C,E v with a unitary v : D E D E,then the two subisometric liftings (E,σ E )and(e,σ E ) are unitarily equivalent. Let W

9 Liftings of Covariant Representations of W -Correspondences 519 and W be the corresponding unitaries from the subisometric lifting property. Then and so we define W H E = H C (F D C ) W (F D E ) = H C (F D C ) M C,E (F vd E ) = H C (F D C ) M C,E (F D E )=W H E, U := (W ) W HE : H E H E. As Wh = h = W h for h H C, we have Uh = h. In general for h H E and ξ E UE(ξ) h =(W ) WE (ξ)h =(W ) WP HE V E (ξ)h = P HE (W ) WV E (ξ)h = P HE (W ) V C (ξ)wh = P HE V E (ξ)(w ) Wh = E (ξ)uh, Identical computation yield Uσ E (a) =σ E (a)u, Hence E and E are unitarily equivalent. a M. An important property of coisometric liftings is established in the next proposition and this is parallel to Proposition 3.3. Proposition 3.6. Let E be a W -correspondence over a von Neumann algebra M. Let (C, σ C ) be a coisometric covariant representation of E on Hilbert space H C. If a covariant representation (E,σ E ) of E on a Hilbert space H E H C is a coisometric lifting of (C, σ C ) with ( ) C(ξ) 0 E(ξ) =, ξ E B(ξ) A(ξ) then there exists an isometry γ : D,A D C with B := γd,a. Proof. (E,σ E ) is coisometric if and only if C is coisometric B C =0 and ÃÃ + B B = 1. (3.5) If C is coisometric, then D C = 1 C C is the orthogonal projection onto the kernel of C. Using the first equation of (3.5) we get C B =( B C ) =0. Therefore range( B ) D C. From the second equation of (3.5) we get for h H A D,A h 2 = (1 ÃÃ )h, h = B B h, h = B h 2. So there exist an isometry γ : D,A range( B ) D C with γd,a h = B h for all h H A. Now we deal with the general case where (E,σ E ) is a contractive lifting of (C, σ C ). Because of the structure of liftings it is immediate that the space of mid

10 520 S. Dey (V C,π C ) is embedded in that of (V E,π E ). We introduce a c.c.c. representation (Y,π Y ) on the orthogonal complement K of the space of mid (V C,π C )toencode this. Hence we can get a unitary W such that W : H E (F D E ) H C (F D C ) K ˆV E (ξ)w = WV E (ξ), (π C π Y )(a)w = Wπ E (a), W HC = 1 HC with ˆV E (ξ) =V C (ξ) Y (ξ). We denote by the same W its restriction to the complement of H C too, i.e. With this we get W : H A (F D E ) (F D C ) K. (B(ξ)) = P HC (V E (ξ)) W HA = P HC [(V C (ξ)) (Y (ξ)) ]W HA =(D C (ξ )) P m0 D C W HA. Define H 1 A := {h H A : (Ãn ) h = h for all n N}. (3.6) From some easy observations similar to those appearing in Sec. 3 of Ref. 3 we can track down that W HA 1 K. (3.7) This eventually implies the existence of a contraction γ := P m0 D C W m0 D,A : D,A D C such that P m0 D C W HA = γd,a. We have shown that if (E,σ E ) is a c.c.c. lifting of (C, σ C )by(a, σ A )asabovethen B = D,A γ D C. (3.8) For the converse we start with two c.c.c. representations (C, σ C )and(a, σ A ), a contraction γ : D,A D C and B as in (3.8). Then for x H C,y H A x, C B y 2 = x, CD CγD,A y 2 = D C C x, γd,a y 2 D C C x 2 γd,a y 2 x, (1 C C )x y, (1 ÃÃ )y. As in Exercise 3.2 in Ref. 13 it follows that ( ) 1 C C C B 0 = 1 B C 1 ẼẼ. ÃÃ Thus Ẽ is a contraction. We summarize this in the following lemma: Lemma 3.7. Let (E,σ E ) be a lifting of (C, σ C ) by (A, σ A ). Then (E,σ E ) is completely contractive if and only if (C, σ C ) and (A, σ A ) are completely contractive and

11 Liftings of Covariant Representations of W -Correspondences 521 there exists a contraction γ : D,A D C such that B = D,A γ D C. A covariant representation (A, σ A ) of E is called completely noncoisometric (c.n.c.), if HA 1 = 0 (see (3.6)). Definition 3.8. A completely contractive lifting (E,σ E )of(c, σ C )by(a, σ A )is called reduced if (1) γ is resolving, i.e. for h H A (γd,a (A(ξ)) h =0forallξ E) (D,A (A(ξ)) h =0forallξ E), (2) (A, σ A ) is c.n.c. and Clearly -stable representations are c.n.c. and injective γ s are resolving. Thus subisometric liftings are reduced. Proposition 3.6 shows that coisometric liftings by c.n.c. covariant representations are also reduced. For a reduced lifting (E,σ E ) of (C, σ C ) the assumption that γ is resolving ensures that the c.n.c. property of (A, σ A ) will be transferred through the embedding W. From this and (3.7) by arguments analogous to those in Sec. 3 of Ref. 3 it turns out that if (E,σ E ) is a reduced lifting. (F D C ) W (F D E )=(F D C ) K, (3.9) Definition 3.9. The characteristic function of reduced lifting (E,σ E )of(c, σ C ) is defined as M C,E := P F DC W F DE. We remark that this M C,E also satisfy (3.2). Further this characteristic function coincides with the one defined in Definition 3.4 for a subisometric lifting. Set For x F D E we get C,E := (1 M C,E M C,E) 1 2 : F DE F D E. P K Wx 2 = (1 P F DC )Wx 2 = x 2 P F DC Wx 2 = x 2 M C,E x 2 = C,E x 2. This together with (3.9) give us a kind of functional model describing the dilation space of the reduced lifting E of C completely in terms of characteristic function and H C, namely W H A =[(F D C ) K] W (F D E ) =[(F D C ) C,E (F D E )] {M C,E x C,E x : x F D C }.

12 522 S. Dey Theorem Let E be a W -correspondence over a von Neumann algebra M. For any completely contractive covariant representation (C, σ C ) of E, the equivalence classes (under the equivalence relation of (3.3)) of characteristic functions are complete invariants for reduced liftings of (C, σ C ) up to unitary equivalence. Theorem 3.10 can be proved using the above functional model and doing necessary modification to the proof of Theorem 3.5. We set µ = Ã D(E σa )andd ξ h := (V E (ξ) E(ξ))h = D E (ξ h) forξ E and h H E. The expanded form of the symbol Θ C,E, computed using (2.1) and Definition 3.9 of the characteristic function, is the following: Case I. h H C. Θ C,E (d ξ h )=[D C(ξ h) γd,a B(ξ)h] (for the notation µ (j) see (2.2)). Case II. h H A. (1 γd,a )(1 µ (j) )(1 B(ξ)h) j=1 Θ C,E (d ξ h )= γµ d ξ h + (1 γd,a )(1 µ (j) )D A d ξ h. j=0 Let us briefly mention one potential good application of this theory to analytic crossed products of the type M Z + (see Sec. 6 of Ref. 6). Muhly and Solel showed in this last quoted work that one can associate a contraction t to every (σ-weakly continuous) representation of this crossed product. When t is c.n.c., its Sz. Nagy Foias characteristic function is equivalent to the characteristic function of the covariant representation associated to the representation of M Z +. This theory needs to be explored for liftings of covariant representations. References 1. M. Baillet, Y. Denizeau and J.-F. Havet, Indice dune esperance conditionelle, Compositio Math. 66 (1988) S. Dey and R. Gohm, Characteristic functions for ergodic tuples, Integral Eqns. Oper. Th. 58 (2007) S. Dey and R. Gohm, Characteristic functions of liftings, to appear in J. Oper. Th. 4. P. Muhly and B. Solel, Tensor algebras over C -correspondences: Representations, dilations, and C -envelopes, J. Funct. Anal. 158 (1998) P. Muhly and B. Solel, Hardy algebras, W -correspondences and interpolation theory, Math. Ann. 330 (2004) P. Muhly and B. Solel, Canonical models for representations of Hardy algebras, Integral Eqns. Oper. Th. 53 (2005) P. Muhly and B. Solel, The Poisson kernel for Hardy algebras, preprint (2007). 8. B. Sz.-Nagy and C. Foias, Harmonic Analysis of Operators on Hilbert Space (North- Holland, 1970). 9. W. Paschke, Inner product modules over B-algebras, Trans. Amer. Math. Soc. 182 (1973)

13 Liftings of Covariant Representations of W -Correspondences G. Popescu, Isometric dilations for infinite sequences of noncommuting operators, Trans. Amer. Math. Soc. 316 (1989) G. Popescu, Characteristic functions for infinite sequences of noncommuting operators, J. Oper. Th. 22 (1989) G. Popescu, Operator theory on noncommutative varieties II, Proc. Amer. Math. Soc. 135 (2007) V. I. Paulsen, Completely Bounded Maps and Operator Algebras (Cambridge Univ. Press, 2003). 14. R. Gohm, Noncommutative Markov chains and multi-analytic operators, to appear in J. Math. Anal. Appl. 15. M. Skeide, Generalised matrix C -algebras and representations of Hilbert modules, Math. Proc. R. Ir. Acad. 100A (2000) M. Skeide, Commutants of von Neumann modules, representations of B a (E) and other topics related to product systems of Hilbert modules, in Advances in Quantum Dynamics, South Hadley, MA, 2002, Contemp. Math., Vol. 335 (Amer. Math. Soc., 2003), pp