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OPERATOR ALGEBRAS AND SUBPRODUCT SYSTEMS ARISING FROM STOCHASTIC MATRICES ADAMDOR-ONANDDANIELMARKIEWICZ* Abstract. Westudysubproduct systems in thesense of Shalit and Solel arising from stochastic matrices on countable state spaces, and their associated operator algebras. We focus on the non-self-adjoint tensor algebra, and Viselter’s generalization of the Cuntz-Pimsner C*-algebra to the context of subproduct systems. Suppose that X and Y are Arveson-Stinespring subproduct systems associated to two stochastic matrices over a countable set Ω, and let T (X) and T (Y) + + be their tensor algebras. We show that every algebraic isomorphism from T (X) onto T (Y) is + + 4 automatically bounded. Furthermore, T (X) and T (Y) are isometrically isomorphic if and only + + 1 ifX andY areunitarilyisomorphicuptoa*-automorphismofℓ∞(Ω). WhenΩisfinite,weprove 0 thatT (X)andT (Y)arealgebraically isomorphicifandonlyifthereexistsasimilaritybetween 2 X and+Y up to a+*-automorphism of ℓ∞(Ω). Moreover, we provide an explicit description of the n Cuntz-Pimsner algebra O(X) in thecase where Ω is finite and thestochastic matrix is essential. a J 7 2 1. Introduction ] A In this paper we study the structure of tensor and Cuntz-Pimsner algebras (in the sense of O Viselter [Vis12]) associated to subproduct systems, and to what extent these algebras provide invariants for their subproduct systems. These algebras generalize the tensor and Cuntz-Pimsner . h operator algebras associated to C*-correspondences, which have been the focus of considerable t a interest by many researchers. tensor algebras of a C*-correspondence, in particular, have been m the subject of a deep study by Muhly and Solel [MS98, MS00, MS02], which has led into a far- [ reaching non-commutative generalization of function theory. We will focus on subproductsystems 1 associated to stochastic matrices, and in this context we prove several results which have a close v parallel in the work of Davidson, Ramsey and Shalit [DRS11] on the isomorphism problem of 2 tensor algebras of subproduct systems over C with finite dimensional (Hilbert space) fibers. 3 0 A subproduct system over a W*-algebra (and over the additive semigroup N) is a family M 7 Xn n N of W*-correspondences over endowed with an isometric comultiplication Xn+m . { } ∈ M → 1 Xn Xm which is an adjointable -bimodule map for every n,m. Subproduct systems were ⊗ N 0 first defined and studied for their own sake by Shalit and Solel [SS09], and in the special case 4 of = C they were also independently studied under the name of inclusion systems by Bhat 1 M : and Mukherjee [BM10]. Subproduct systems had appeared implicitly earlier in the work of many v researchers in the study of dilations of semigroups of completely positive maps (cp-semigroups for i X short) on von Neumann algebras and later C*-algebras (see for example [BS00, MS02, Mar03]). r Thestudyof cp-semigroupsisclosely related totheanalysis ofE -semigroups andproductsystems a 0 pioneered by Arveson and Powers (for a comprehensive introduction see [Arv03], and also [Ske03b] for product systems of Hilbert modules). Given a correspondence E over a C*-algebra , the Toeplitz C*-algebra (E) and the Cuntz- A T Pimsner C*-algebra (E) were introduced by Pimsner [Pim97], and modified by Katsura [Kat04] O in the case of non-injective left action of . As is well-known, in general the Cuntz-Pimsner A algebra does not provide a very strong invariant of the underlying correspondence. However, some information does remain. In the case of graph C*-algebras, for example, if a graph is row-finite, then its C*-algebra is simple if and only if the graph is cofinal and every cycle has an entry. And it Date: January 29, 2014. 2000 Mathematics Subject Classification. Primary: 47L30, 46L55, 46L57. Secondary: 46L08, 60J10. Key words and phrases. Non-self-adjoint operator algebras; tensor algebra; subproduct system; Cuntz-Pimsner algebra; cp-semigroup; stochastic matrix. ThefirstauthorwaspartiallysupportedbyGIF(German-IsraeliFoundation)researchgrantNo. 2297-2282.6/201, andthesecondauthorwaspartiallysupportedbygrant2008295fromtheU.S.-IsraelBinationalScienceFoundation. *corresponding author. 1 2 ADAMDOR-ONANDDANIELMARKIEWICZ is easy to find two graphs with d vertices and irreducible adjacency matrix whose C*-algebras are notisomorphic(see[Rae05]). Incontrast, insection 5weshowthatifX istheArveson-Stinespring subproduct system of a d d irreducible stochastic matrix, then (X) = C(T) M (C). More × O ∼ ⊗ d generally, we also provide an explicit description for the Cuntz-Pimsner algebra of a subproduct system associated to essential finite stochastic matrices. On the other hand, the non-self-adjoint tensor algebra (E) of a C*-correspondence E over + T A has often proven to be a strong invariant of the correspondence. Muhly and Solel [MS00] proved that if E and F are aperiodic C*-correspondences, then (E) is isometrically isomorphic to + T (F) if and only if E and F are isometrically isomorphic as -bimodules. Similarly, Katsoulis + T A andKribs[KK04]and Solel[Sol04]proved that ifG andG arecountable directed graphs, thenthe ′ tensoralgebras (G)and (G)areisomorphicasalgebras ifandonlyifGandG areisomorphic + + ′ ′ T T as directed graphs. See also Davidson and Katsoulis [DK11] for another important example of this phenomenon of increased acuity of the normed (non-self-adjoint) algebras as opposed to C*- algebras, perhaps first recognized in Arveson [Arv67] and Arveson and Josephson [AJ69]. The tensor algebras of subproduct systems were first considered by Solel and Shalit [SS09] in the special case of = C, and they analyzed the problem of graded isomorphism of their M tensor algebras. The general isomorphism problem for such subproduct systems was resolved by Davidson, Ramsey and Shalit [DRS11]. They proved that if X,Y are subproductsystems of finite- dimensional Hilbert space fibers, then (X) and (Y) are isometrically isomorphic if and only + + T T if X and Y are (unitarily) isomorphic. On the other hand, the recent work of Gurevich [Gur12] provides a useful contrast. Although in this paper we focus on subproduct systems over N, it is possible to consider more general semigroups. Gurevich studied subproduct systems over the semigroup N N, with finite dimensional × Hilbert space fibers. He proved in [Gur12] that subproduct systems of that type in a certain large class can be distinguished by their tensor algebras, however he also provided an example of two non-isomorphic subproduct systems over N N of finite dimensional Hilbert space fibers whose × tensor algebras are isometrically isomorphic. The following are our main results. Suppose that X and Y are Arveson-Stinespring subproduct systems over = ℓ (Ω), associated to two stochastic matrices over a countable set Ω, and let ∞ M (X) and (Y) be their tensor algebras. Then (X) and (Y) are isometrically isomorphic + + + + T T T T if and only if X and Y are unitarily isomorphic up to a *-automorphism of ℓ (Ω). Every algebraic ∞ isomorphism from (X) onto (Y) is automatically bounded. Furthermore, when Ω is finite, + + T T (X) and (Y) are algebraically isomorphic if and only if there exists a similarity between X + + T T and Y up to a *-automorphism of ℓ (Ω), in the appropriate sense. ∞ We now describe the structure of this paper. In section 2 we review some preliminary material. In section 3 we describe the subproduct system of a stochastic matrix, and in Theorem 3.8 we provide an effective isomorphism theorem for such objects. In section 4 we review some basic facts about the Cuntz-Pimsner algebra of a subproduct system, and in section 5 we characterize the Cuntz-Pimsner C*-algebra of the subproduct system of essential stochastic matrices. Finally, in section 6 we begin the study of the tensor algebra of a general subproduct system, culminating in the main results for the case of stochastic matrices in section 7. 2. Preliminaries Stochastic Matrices. We review the basic terminology and facts about stochastic matrices that are relevant to our study. Definition 2.1. Let Ω be a countable set. A stochastic matrix is a function P : Ω Ω R such + × → that for all i Ω we have P = 1. The set Ω is called the state set or space of the matrix ∈ j Ω ij P, and elements of Ω are calle∈d states of P. P Two stochastic matrices P and Q can be multiplied to obtain a new stochastic matrix PQ defined by (PQ) = P Q . Henceforth, we will denote by Pn the product of P with itself ik j Ω ij jk n times, and by P(n) :=∈(Pn) the (i,j)-th entry of Pn. ij P ij OPERATOR ALGEBRAS AND SUBPRODUCT SYSTEMS FROM STOCHASTIC MATRICES 3 Definition 2.2. Let P be a stochastic matrix on a state space Ω. Denote by Gr(P) the matrix representing the directed graph of P which is defined to be 1, P > 0 Gr(P) = ij ij 0, P = 0 ij (cid:26) Definition 2.3. Let P and Q be stochastic matrices on a state space Ω, let σ : Ω Ω be a → permutation with corresponding permutation matrix R = δ (σ(j)) . σ i We say that P is graph isomorphic to Q through σ and write P Q if R 1Gr(Q)R = Gr(P). (cid:2) ∼(cid:3)σ σ− σ We will also say that P is isomorphic to Q through σ and write P = Q if R 1QR = P. ∼σ σ− σ Thus we have that P Q if the directed graphs of P and Q are isomorphic, while P = Q if ∼σ ∼σ the weighted directed graphs of P and Q are isomorphic. Definition 2.4. Let P be a stochastic matrix over a state set Ω. A path in P is a path in the directed graph of P, that is a function γ : 0,...,ℓ Ω such that P > 0 for every γ(k)γ(k+1) { } → 0 k ℓ 1. The path γ is said to be a cycle if γ(0) = γ(ℓ). We will say that two states i,j ≤ ≤ − communicate if and only if there exists a path from i to j and vice-versa. It is clear that the communication relation is an equivalence relation. Note also that a path from i to j of length n exists if and only if Pn > 0. ij Definition 2.5. Let P be a stochastic matrix over a state set Ω, and let i Ω. ∈ (n) (1) The period of i is r(i) = gcd n P > 0 . If no such r(i) exists, or if r(i) = 1 we say { | ii } that i is aperiodic. (2) i Ω is said to be inessential in P if there is some j Ω and n N such that Pn > 0 but P∈m = 0 for all m N. P is said to be essential if it h∈as no iness∈ential states. ij ji ∈ (3) P is said to be irreducible if any pair i,j Ω communicates in P. ∈ Clearly, irreducible stochastic matrices are automatically essential. Further note that for an essential state i Ω, the number r(i) is always well-defined. ∈ Definition2.6. LetP beastochastic matrixoverastatesetΩ. Astate i Ωissaidtobe transient (n) ∈ if the series P converges, and otherwise recurrent. If i Ω is recurrent then it is null- n N ii ∈ (n)∈ recurrent if P 0, and positive-recurrent otherwise. We say that P is transient/recurrent Pii →n→∞ if all states are transient/recurrent in P, respectively. In an irreducible stochastic matrix, all states have the same period and classification in terms of recurrence type. We also note that recurrent states are essential (see [Chu60, Part I, Section 4, Theorem 4]). The next two theorems can be found in various forms in the literature, see [Chu60, Part I, Section 3], [Fel68, Chapter XV, Section 6]. We restate them in a form convenient for our purposes. Theorem 2.7. (Irreducible decomposition for essential matrices) Let P be an essential stochastic matrix over a state set Ω. Let (Ω ) be the partition of Ω into α α A ∈ equivalence classes of communicating states. With the appropriate enumeration of Ω, the matrix P decomposes into a block diagonal matrix whose diagonal blocks are irreducible stochastic matrices corresponding to the restriction of P to Ω Ω , for α A. α α × ∈ As a corollary, we observe that complete reducibility is equivalent to essentiality. Theorem 2.8. (Cyclic decomposition for periodic irreducible matrices) Let P be an irreducible stochastic matrix over a state set Ω with period r, and let ω Ω. For ∈ each ℓ = 0,...r 1, let Ω = j Ω P(n) > 0 = n ℓ mod r . Then the family (Ω )r 1 − ℓ { ∈ | ωj ⇒ ≡ } ℓ ℓ=−0 is a partition of Ω. Furthermore if j Ω then there exists N(j) such that for all n N(j) we ℓ ∈ ≥ (nr+ℓ) have P > 0. In fact, with an appropriate enumeration of Ω, there exist stochastic matrices ωj P ,...P such that P has the following cyclic block decomposition: 0 r 1 − 0 P0 0 ... ... .··.·. ...  0 0 Pr−2 ··· Pr−1 0 0 ···   Where the rows of P in this matrix decomposition are indexed by Ω for all 0 ℓ < r. ℓ ℓ ≤ 4 ADAMDOR-ONANDDANIELMARKIEWICZ Remark 2.9. We emphasize that the columns of P in the above matrix decomposition are indexed ℓ by Ω , where s(ℓ) =ℓ+1( mod r). Further, if P is finite d d, then P ,...,P are d d. s(ℓ) × 0 r−1 r × r We note that the cyclic decomposition is independent of our initial choice of ω in the sense that for P irreducible with period r and cyclic decomposition Ω ,...,Ω given by the previous 0 r 1 − theorem, if we were to pick a different ω Ω and a new partition Ω ,...Ω induced by it, a cyclic permutation of this partition would′y∈ield our original partition Ω′0,...,Ω′r−1. Furthermore, if 0 r 1 − (k) i Ω and j Ω are two states, let 0 ℓ < r be such that ℓ l l mod r. Now if P > 0 ∈ l1 ∈ l2 ≤ ≡ 2 − 1 ij then one must have k = mr+ℓ for some m N, and there exists some m N such that for all 0 ∈ ∈ (mr+ℓ) m m one has P > 0. (See [Chu60, Part I, Section 3, Theorems 3 & 4]) ≥ 0 ij RecallthatanirreduciblestochasticP ispositive-recurrentifandonlyifitpossessesastationary distribution, i.e. a vector π ℓ1(Ω) such that π 0 for all j Ω, π = 1 and π = π P ∈ j ≥ ∈ i i j i i ij for all j Ω. For the proof of the following theorem, see [Dur10, Theorem 6.7.2]. ∈ P P Theorem 2.10. (Convergence theorem for positive-recurrent irreducible matrices) LetP beanirreducible positive-recurrent stochastic matrix withstationary distribution π andperiod r 1. Let Ω ,...,Ω be a cyclic decomposition of Ω with respect to P as in Theorem 2.8. Given 0 r 1 ≥ − i Ω and j Ω , let 0 ℓ < r be such that ℓ (l l ) mod r. Then ∈ l1 ∈ l2 ≤ ≡ 2− 1 (mr+ℓ) lim P = π r. ij j m →∞ Definition 2.11. Let P be a stochastic matrix over Ω. i Ω is said to be amenable in P if ∈ limsup n P(n) = 1. P is said to be amenable if all states in Ω are amenable in P. n ii →∞ q The proof of the following fact is well-known and will be omitted. Lemma 2.12. Let P be a recurrent irreducible stochastic matrix, then P is amenable and for every n N we have that Pn is recurrent. ∈ Hilbert modules. We assume that the reader is familiar with the basic theory of Hilbert C*- modules, which can be found in [Pas73, Lan95, MT05]. We only give a quick summary of basic notions and terminology to clarify our conventions. Asusualinnerproductmodulesarerightmodules. Thedual module of aninnerproductmodule E over is theset of all bounded -modulemaps from E to , andit is denoted by E . AHilbert ′ A A A C*-module is called self-dual if the canonical embedding of E into E is surjective. ′ Definition 2.13. Let be a W*-algebra let E be a Hilbert C*-module over . The σ-topology M M on E is defined by the functionals f(·) = ∞n=1wn(hξn,·i) where ξn ∈ E and wn ∈ M∗ such that ∞n=1||wn||||ξn|| < ∞. P PHilbert C*-modules over a W*-algebra will be called Hilbert W*-modules. If E is a self-dual W*-module, then it is a dual Banach space (see [Pas73]), and the associated weak-* topology coincides with its σ-topology. If E is an inner productmoduleover a W*-algebra , then the inner productmodulestructure M of E can be naturally extended to E , which makes E into a self-dual Hilbert W*-module over ′ ′ . In this case we will refer to E as the self-dual extension of E. Furthermore, the canonical ′ M embedding of E into E maps onto a dense subset in the σ-topology of E , so that E is self dual ′ ′ if and only if it is σ-topology closed in E . ′ For E and F Hilbert W*-modules, let (E,F) denote the set of adjointable maps from E to L F. If E and F are self-dual W*-modules, then all bounded -module maps from E to F are M adjointable. It also turns out that bounded module maps between any two inner product modules over a W*-algebra behave well with respect to self dual completions, as the following proposition states. Proposition 2.14. Let E and F be inner-product modules over a W*-algebra , and let T : M E F be a bounded module map. Then T has a unique extension to a bounded module map T˜ :→E F , and T˜ = T . If E = F, then the map T T˜, restricted to the algebra of ′ ′ → || || || || → adjointable operators on E, is a faithful *-homomorphism from (E) to (E ). ′ L L OPERATOR ALGEBRAS AND SUBPRODUCT SYSTEMS FROM STOCHASTIC MATRICES 5 Definition 2.15. If E is a Hilbert C*-module over , and is another C*-algebra, then E is B A called a C*-correspondence from to if E is also a left -module such that the left action is A B A determined by a *-homomorphism φ : (E). In the case when = , E is called a C*- A → L A B correspondenceover . If and are W*-algebras then E iscalled a HilbertW*-correspondence A N M from to if in addition E is a self-dual module over and the left action φ of is normal. N M M N A key notion of C* and W*-correspondences is the internal tensor product. If E is a C*- correspondence from to with left action φ, and F is a C*-correspondence from to with A B B C left action ψ, then on the algebraic tensor product E F one defines a -valued pre-inner alg ⊗ C product satisfying x y ,x y = y ,ψ( x ,x )y on simple tensors. The usual quotient 1 1 2 2 1 1 2 2 h ⊗ ⊗ i h h i i and completion process yields the internal Hilbert C*-module tensor product of E and F, denoted by E F or E F, which is a C*-correspondence from to . In the case where , and are ψ ⊗ ⊗ A C A B C W*-algebras, taking the self dual completion yields the self-dual tensor product denoted also by E F or E F, which is a W*-correspondence from to . ψ ⊗ ⊗ A C The notion of self-dual direct sums of Hilbert C*-modules over W*-algebras was developed by Paschke. Definition 2.16. Let E be a family of self-dual W*-modules over . The ultraweak direct i i I { }∈ M sum of E is the subset X of the Cartesian product of E such that x is in X if the i i I i i I i { }∈ { }∈ { } sum x ,x converges ultraweakly. The inner product on X is defined to be x , y = i Ih i ii h{ i} { i}i x ∈,y , where the sum converges ultraweakly in . This direct sum is denoted by uwE or byi∈IPhEi whiien the context is that of self-dual modules.M ⊕i i i i P⊕ Base change. Suppose that E is a Hilbert C*-module (correspondence) over and ρ is a *- A automorphism of . Then one can define a new C*-module (correspondence) Eρ over . As a A A set, Eρ = E, but its operations are defined as follows: ξ a = ξ ρ(a) (also a ξ = ρ(a) ξ for ρ ρ · · · · correspondences) and ξ,η = ρ 1( ξ,η ) for all ξ,η E and a . In algebra, this operation ρ − h i h i ∈ ∈ A on modules is sometimes called a change of rings or base change. Definition2.17. LetE andF betwoC*-modulesover , andletρ : bea*-automorphism. A A → A We will say that a bounded linear map V : E F is a ρ-module (ρ-correspondence) morphism → if V : E Fρ is an -linear ( -correspondence) map, i.e. for all a and ξ E one → A A ∈ A ∈ has V(ξa) = V(ξ)ρ(a) (also V(aξ) = ρ(a)V(ξ) for correspondences). We will say that V is ρ- adjointable if V is adjointable as a map E Fρ, and we will denote by V( ,ρ) its adjoint, i.e. ∗ → V( ,ρ) :F E satisfies ξ,V(η) = ρ 1( ξ,V(η) ) = V( ,ρ)(ξ),η for all ξ F,η E. ∗ ρ − ∗ → h i h i h i ∈ ∈ Note that a ρ-adjointable V must be a bounded ρ-module morphism and V( ,ρ) is a ρ 1- ∗ − adjointable map with (V( ,ρ))( ,ρ−1) = V. Furthermore, if E, F and G are C*-modules over ∗ ∗ and we have V : E F a ρ-module/correspondence morphism and W : F G a τ- A → → module/correspondence morphism, then W V : E G is a (τ ρ)- module/correspondence ◦ → ◦ morphism respectively. Further, if V is ρ-adjointable and W is τ-adjointable, then W V is ◦ (τ ρ)-adjointable with (W V)( ,τ ρ) = V( ,ρ) W( ,τ). ∗ ◦ ∗ ∗ ◦ ◦ ◦ Furthermore, σ-topology continuity is automatic for ρ-adjointable maps. Indeed, suppose that V :E F is a ρ-adjointable map, and let η be a net in E converging in the σ-topology to η. α α → { } Let ξ F and w be such that w ξ < . Then we have that n n n n ∈ ∈M∗ || ||·|| || ∞ ∞ P ∞ w ( ξ ,V(η η) ) = (w ρ)( V( ,ρ)(ξ ),η η ) n n α n ∗ n α h − i ◦ h − i n=1 n=1 X X And since w ρ Vρ(ξ ) Vρ w ξ < , we have convergence of the net n n n n || ◦ ||·|| || ≤ || || || ||·|| || ∞ (Vη ) to Vη in the σ-topology in F. α We alsoPnote that the identity map ι :P(Eρ) E is a ρ-module isometric isomorphism. It ρ ′ ′ → follows that if E is self-dual, then the same holds for Eρ. It also leads to the following fact. Proposition 2.18. Let E and F be Hilbert W*-modules over a W*-algebra , and let ρ be a M *-automorphism of . Suppose that V : E F is a ρ-module morphism. Then V has a unique M → extension to a bounded ρ-module morphism V :E F , and V = V . ′ ′ → || || || || 6 ADAMDOR-ONANDDANIELMARKIEWICZ Proof. Using Proposition 2.14 we obtain a unique W*-module map V˜ : E (Fρ). By composing ′ ′ → with ι described in the paragraph preceding the propostion, we obtain that V = ι V˜ is a ρ ρ ◦ ρ-module morphism from E to F extending V. Since ι is a bijection, we also obtain uniqueness ′ ′ ρ of the extension. The norm condition holds because ι is isometric. (cid:3) ρ Definition 2.19. Let E and F be Hilbert W*-modules over and let ρ : be a *- M M → M automorphism. We say that a ρ-adjointable map W : E F is a ρ-coisometry if W( ,ρ) is ∗ → an isometry, and we will say that W is a ρ-unitary if it it is an isometric surjective ρ-module morphism. We observe that if U is a ρ-unitary then U 1 is a ρ 1-module morphism and U( ,ρ) = U 1. − − ∗ − Further note that a ρ-module map U is a ρ-unitary if and only if it is a ρ-adjointable module map satisfying UU( ,ρ) = Id and U( ,ρ)U = Id . This follows from the analogous and well-known ∗ F ∗ E theorem for (Id-)unitaries, and the preceding discussion. Subproduct systems. Definition 2.20. Let be a W* algebra, let X = Xn n N be a family of of Hilbert W*- M { } ∈ correspondences over and let U = Un,m : Xn Xm Xn+m n,m N be a family of maps. We will say that (X,U) isMa subproduct s{ystem over⊗ (an→d over th}e se∈migroup N) if the following M conditions are satisfied: (1) X = 0 (2) U iMs a coisometric mapping of W*-correspondences over for every n,m N n,m M ∈ (3) The maps U and U are given by the left and right actions of on X respectively, 0,n n,0 n and for every n,m N we have the following identity: M ∈ U (U I )= U (I U ) n+m,p n,m⊗ Xp n,m+p Xn ⊗ m,p In case the maps U are unitaries, we say that X is a product system. n,m When there is no ambiguity, we will suppress the reference to the family U of multiplication maps, and refer simply to a subproduct system X. Example 2.21. If E is a W*-correspondence over such that E = E (essential), the product system XE over N defined by XE = E n wherMe U is the nMatu·ral identification between n ⊗ n,m E n E m and E (n+m), is obviously a subproduct system. We call it the full product system. ⊗ ⊗ ⊗ ⊗ Definition 2.22. Let (X,UX) and (Y,UY) be subproduct systems over and respectively. M N We define a subproduct system X Y over , which we will call the direct sum of X and Y. ⊕ M⊕N (1) The n-th fiber W*-correspondence is given by (X Y) := X Y with left and right n n n ⊕ ⊕ multiplication of given by (m n) (ξ η) (m n) = mξm nηn, and inner ′ ′ ′ ′ M⊕N ⊕ · ⊕ · ⊕ ⊕ product given by ξ η,ξ η = ξ,ξ η,η . ′ ′ ′ ′ h ⊕ ⊕ i h i⊕h i (2) The subproduct maps UX Y are defined by n,m⊕ UX Y((ξ η ) (ξ η )) = UX (ξ η ) UY (ξ η ) n,m⊕ n n m m n,m n n n,m m m ⊕ ⊗ ⊕ ⊗ ⊕ ⊗ Definition 2.23. Let (X,UX) and (Y,UY) be two subproduct systems over a W*-algebra . A M family V = V of maps V :X Y is called a morphism of subproduct systems if n n n n { } → (1) ρ= V :X Y is a *-automorphism, 0 0 0 → (2) For all n = 0 the map V is a ρ-coisometric ρ-correspondence morphism. n (3) For all n,6m N the following identity hold: ∈ V UX = UY (V V ) n+m n,m n,m n m ◦ ◦ ⊗ When the family V is a family of ρ-unitaries, we say that X and Y are (unitarily) isomorphic n { } via ρ= V and write X = Y. 0 ∼ρ In [SS09] the notion of isomorphism of subproduct systems was defined so that the map V is 0 the identity on , yet the above variation will be more convenient for our purposes. M We now describe the general construction of Arveson-Stinespring subproduct systems. Let be a W*-algebra, and let θ be a completely positive contractive and normal map on . M M If ρ : B(H) is a normal representation, then we can define a new normal representation of M → OPERATOR ALGEBRAS AND SUBPRODUCT SYSTEMS FROM STOCHASTIC MATRICES 7 , which we will call a dilation of θ via ρ, as follows. We define H to be the Hausdorff θ M M⊗ completion of the algebraic tensor product H with respect to the sesquilinear form defined M⊗ by T h ,T h = h ,ρ(θ(T T ))h , for all T ,T ,h ,h H. h 1⊗ 1 2 ⊗ 2i h 1 1∗ 2 2i 1 2 ∈ M 1 2 ∈ Complete positivity of θ ensures that this sesquilinear form is positive semidefinite. We define a normal representation π of on H by π (S)(T h) = ST h. Moreover, the map θ θ θ M M⊗ ⊗ ⊗ W : H H given by W (h) = I h is a contraction which satisfies θ θ → M⊗ ⊗ ρ(θ(T)) = W π (T)W . θ∗ θ θ One easily checks that H is minimal in the sense that it is the smallest subspace of H θ θ M⊗ M⊗ containing W H and reducingπ . Itfollows that if(π,K,W) is atriple whereπ is arepresentation θ θ of onK suchthatρ(θ(T))= W π(T)W forallT andK isminimal,thenthereisaunitary ∗ M ∈ M U :K H suchthatU implementsaunitaryequivalencebetweenπandπ andUW = W U. θ θ θ → M⊗ We also define the associated intertwiner space (H, H) := X B(H, H) Xρ(T) = π (T)X T . θ θ θ LM M⊗ { ∈ M⊗ | ∀ ∈ M } Supposenow that is a von Neumann subalgebra of B(H) and ρ is the inclusion map. In this M case thetriple(π , H,W )willbecalled theminimalStinespringdilation of θ. In[MS02], the θ θ θ M⊗ intertwiner space (H, H) was shown to be a W*-correspondence over , with left and θ ′ LM M⊗ M right actions given by S X = (I S) X and X S = X S for S and X (H, H), ′ θ · ⊗ ◦ · ◦ ∈ M ∈ LM M⊗ and the -valued inner product is given by X,Y = X Y, for X,Y (H, H). ′ ∗ θ M h i ∈ LM M⊗ Now let φ be another completely positive contractive normal map on . Then we obtain a M representation π of on ( H) via dilation using π , and in an analogous way we θ,φ φ θ θ M M⊗ M⊗ have that the -correspondenceintertwiner space (H, ( H)), between the identity ′ φ θ M LM M⊗ M⊗ representation of and π . θ,φ M Definition 2.24. Let θ be a normal completely positive map on a W*-algebra . Let M θ(n)= (H, H) and θ(n,m) = (H, H), n,m N. θn θn θm L LM M⊗ L LM M⊗ M⊗ ∈ The Arveson-Stinespring subproduct system of θ is defined as follows: (1) The fibers are ( θ(n))n N with the aforementioned W*-correspondence structure over ′. (2) The subproductLmaps U∈θ : θ(n) θ(m) θ(n+m) are defined by Uθ = V ΨM n,m n,m n∗,m n,m L ⊗L → L where, (a) V : θ(n+m) θ(m,n) is defined by V (X) = Γ X where Γ : n,m L → L n,m n,m◦ n,m M⊗θn+m H ( H) is defined to be Γ (S h) = S I h. → M⊗θm M⊗θn n,m ⊗n+m ⊗m ⊗n (b) Ψ : θ(n) θ(m) θ(m,n) is defined by Ψ (X Y) = (I X)Y. n,m n,m L ⊗L → L ⊗ ⊗ We denote this subproduct system by ( θ,Uθ). L It was proven in [SS09] that this is indeed a subproduct system. This fact relies on the work in [MS02], where it was shown that V is an isometric correspondence morphism, and that Ψ is n,m n,m a correspondence isomorphism. A fundamental property of Arveson-Stinespring subproduct systems is that every subproduct system is (Id-)isomorphic to an Arveson-Stinespring subproduct system of a cp-semigroup (see [SS09, Corollary 2.10]). 3. Subproduct systems arising from stochastic matrices Let Ω be a countable set, and let ℓ (Ω) be the von Neumann algebra of bounded sequences ∞ indexed by Ω acting on the Hilbertspace ℓ2(Ω). Let us denote by e the canonical orthogonal i i Ω basis for ℓ2(Ω), and by p the collection of rank one pairw{ise}p∈erpendicular projections in j j Ω { } ∈ ℓ (Ω) defined by p (e ) = δ e . ∞ j i ij i In this section we will compute the Arveson-Stinespring subproductsystem of the cp-semigroup generated by a single unital positive normalmap on thevon-Neumann algebra ℓ (Ω). It is easy to ∞ see that such a map is determined uniquely by a stochastic matrix on Ω. This simple observation will be used repeatedly, hence we record it here for emphasis. We omit the straightfoward proof. 8 ADAMDOR-ONANDDANIELMARKIEWICZ Proposition 3.1. There is a 1-1 correspondence between unital positive normal maps θ :ℓ (Ω) ∞ → ℓ (Ω) and stochastic matrices P over Ω, where the relationship is given by ∞ e ,θ(p )e = P i j i ij h i The map just defined sends the composition of unital positive normal maps into the product of their respective stochastic matrices. Of course representations of stochastic matrices on Ω are dependent on its enumerations, hence permutations of Ω, or alternatively *-automorphisms of ℓ (Ω), willplay arole in thecontinuation. ∞ Recall that the *-automorphisms of ℓ (Ω) are in 1-1 correspondence with the permutations of Ω, ∞ because minimal projections must be sent to minimal projections via *-automorphisms. We can associate to every permutation σ : Ω Ω the automorphism ρ : ℓ (Ω) ℓ (Ω) given by σ ∞ ∞ → → ρ (f)= f σ 1. The inverse map is obtained as follows: if ρ is a *-automorphism, then for every σ − ◦ j Ωthereexistsauniqueσ (j) Ωsuchthatρ(p ) = p andσ isawell-definedpermutation. ∈ ρ ∈ j σρ(j) ρ Notation 3.2. We denote by the Schur(entrywise) multiplication of matrices A = [a ] and B = ij ∗ [b ] given by A B = [a b ], and let Diag be the map on matrices given by Diag([a ]) = [δ a ]. lk ij ij ij ij ij ∗ Notation 3.3. Let P and Q be non-negative matrices indexed by Ω. Define the set E(P) := (i,j) P > 0 which is the collection of edges in the weighted directed graph defined by P, and ij { | } the set E(P,Q) := (i,j,k) P Q > 0 . Also denote by √P and P♭ the matrices with (i,k)-th ij jk { | } entry given by (P ) 1, if (i,k) E(P) (√P) := P , and (P♭) := ik − ∈ ik ik ik (0, else p Theorem 3.4. Let P be a stochastic matrix over a state space Ω, and let θ be the unital positive normal map associated to P by the previous proposition. The Arveson-Stinespring subproduct system associated to θ is naturally (Id-)isomorphic to the following subproduct system, which will be denoted by Arv(θ) or Arv(P): (1) The n-th fiber is a W*-correspondence over ℓ (Ω), given by ∞ Arv(P) = [a ] a = 0 (i,j) / E(Pn) , sup a 2 < n ij ij ij { | ∀ ∈ | | ∞ } j Ω ∈ Xi∈Ω where left and right actions are given by multiplication as diagonal matrices. Given A,B ∈ Arv(P) , their W*-correspondence inner-product is given by n A,B = Diag(A B). ∗ h i (2) The subproduct maps are given by U (A B)= (√Pn+m)♭ (√Pn A) (√Pm B) n,m ⊗ ∗ ∗ · ∗ for n = 0 and m = 0, A Arv(P) and B (cid:2) Arv(P) . The maps U(cid:3) and U are given n m 0,n m,0 6 6 ∈ ∈ by left and right multiplication by elements of ℓ (Ω) respectively, considered as diagonal ∞ matrices. We call this presentation of Arv(P) the standard presentation of the Arveson-Stinespring subproduct system associated to the stochastic matrix P. Proof. For the computation of the n-th fibers, we fix an n N. We will follow the notation and ∈ construction described in Definition 2.24. For convenience we will write (n) instead of θ(n). L L Let us denote H = ℓ2(Ω), and consider the canonical inclusion of ℓ (Ω) into B(ℓ2(Ω)). Notice ∞ that the set p e constitutes an orthogonal set in ℓ (Ω) H since for i,j,k,ℓ N, j i (i,j) E(Pn) ∞ θn { ⊗ } ∈ ⊗ ∈ p e ,p e = e ,θn(p p )e = δ δ e ,θn(p )e = δ δ P(n). h k ⊗ ℓ j ⊗ ii h ℓ ∗k j ii kj iℓ h i j ii kj iℓ ij Furthermore, it is straightforward to check that p e is in fact an orthogonal basis j i (i,j) E(Pn) { ⊗ } ∈ for ℓ (Ω) H. ∞ θn ⊗ OPERATOR ALGEBRAS AND SUBPRODUCT SYSTEMS FROM STOCHASTIC MATRICES 9 Wenowshowthat (n)andArv(P) areisomorphicascorrespondencesover . LetX (n). n L M ∈ L Then there exist unique scalars (a ) such that a = 0 for (i,j) E(Pn) and for all k Ω, ijk i,j,k Ω ijk ∈ 6∈ ∈ X(e ) = a p e . k ijk j i ⊗ (i,j) E(Pn) X∈ Since X (n), it is a continuous linear map satisfying (T I)X = XT for all T ℓ (Ω) (see ∞ ∈ L ⊗ ∈ definition 2.24). On the other hand, if T = (c ) ℓ (Ω), then k k Ω ∞ ∈ ∈ XT(e ) = a c p e k ijk k j i ⊗ (i,j) E(Pn) X∈ (T I)X(e ) = a c p e k ijk j j i ⊗ ⊗ (i,j) E(Pn) X∈ Thus by uniqueness of representation, we must have for j = k that a = 0. Hence, if we define ijk 6 a := a , and denote A= [a ], we obtain, ij ijj ij X(e ) = a p e = p Ae j ij j i j j ⊗ ⊗ i : (i,j) E(Pn) X∈ where a = 0 for (i,j) / E(Pn), and Ae = (a ) ℓ2(Ω) for each j Ω. ij j ij i Ω ∈ ∈ ∈ ∈ The boundedness condition on X ensures that X X(e ) = p Ae j j j || || ≥ || || || ⊗ || Since X(ej) X(ej′) for all j = j′, we also have by Pythagoras that for every b ℓ2(Ω): ⊥ 6 ∈ X(b) 2 = b X(e ) 2 = b 2 X(e ) 2 b 2 sup p Ae 2 || || || j j || | j| || j || ≤ || ||2 || j ⊗ j|| j Ω Xj∈Ω Xj∈Ω ∈ Thus X = sup p Ae and, j Ω j j || || ∈ || ⊗ || p Ae 2 = p Ae ,p Ae = a 2 e ,θn(p )e = a 2P(n). || j ⊗ j|| h j ⊗ j j ⊗ ji | ij| h i j ii | ij| ij i Ω i Ω X∈ X∈ In this way each X (n) has a unique matrix A = [a ] in the set (n) of matrices indexed ij ∈ L E by Ω satisfying a = 0 for (i,j) / E(Pn) and sup a 2P(n) < . Conversely, it is easy to ij ∈ j i| ij| ij ∞ see that any matrix in (n) is obtained in this fashion, and this implements a bijection of (n) E P E (n) (n) with (n). In the remainder, we will denote by X = X = X = X the unique element L A [aij] A [aij] associated to A (P) determined by the identity n ∈ E X (e )= p Ae , k N. A k k k ⊗ ∈ A brief computation shows that the left and right actions are given by T X = (I T) X = A A · ⊗ ◦ X and X T = X , where T ℓ (Ω) is thought of as a diagonal matrix when multiplied T A A AT ∞ · · · ∈ with the matrix A. Furthermore, given A,B Arv(P) , notice that X ,X = X X is a an ∈ n h A Bi A∗ B element of ℓ (Ω) hence a diagonal matrix. A direct computation shows that for every k Ω ∞ ∈ X X e = X (p Be ) = p Ae , p Be e A∗ B k A∗ k k k k k k k ⊗ h ⊗ ⊗ i· Hence, ( X ,X ) = p Ae ,p Be = Ae ,θn(p )(Be ) A B jj j j j j j j j h i h ⊗ ⊗ i = a P(n)b = Diag((cid:10)(√Pn A) (√Pn(cid:11) B)) ij ij ij ∗ ∗ ∗ jj Xi∈Ω (cid:16) (cid:17) and it follows that X ,X = Diag((√Pn A) (√Pn B)). Establishing the fiberwise correspon- A B ∗ h i ∗ ∗ dence inner product for (n) We denote the family = (n) n N. Let us now focus on thEe subproduct maps. For thEat p{uErpos}e,∈fix n,m N. Let us observe that ∈ in analogy with the situation above, the set p p e is is an orthogonal basis k j i (i,j,k) E(Pn,Pm) { ⊗ ⊗ } ∈ 10 ADAMDOR-ONANDDANIELMARKIEWICZ for ℓ (Ω) ℓ (Ω) H, since for all i,j,k,i ,j ,k , ∞ θm ∞ θn ′ ′ ′ ⊗ ⊗ hpk ⊗pj ⊗ei,pk′ ⊗pj′ ⊗ei′i = δii′δjj′δkk′ hei,θn(pjθn(pk))eii= δii′δjj′δkk′ hei,θn(pj)eiiPj(km) (n) (m) = δii′δjj′δkk′ Pij Pjk . Furthermore, given Y (m,n), by a computation analogous to the one above involving the ∈ L intertwiner condition, there exist scalars c for (i,j,k) E(Pn,Pm) such that for each k Ω ijk ∈ ∈ Y(e ) = c p p e k ijk k j i ⊗ ⊗ (i,j,k) E(Pn,Pm) ∈X We may also define c = 0 for (i,j,k) / E(Pn,Pm). Furthermore, the norm of Y given by ijk ∈ (m,n) Y = sup Y(e ) . We denote such Y as Y = Y = Y where c = 0 for (i,j,k) / k k k∈Ωk k k [cijk] [cijk] ijk ∈ E(Pn,Pm). One can similarly compute Y and the inner product on (m,n) which would make it into a [∗cijk] L W* correspondence along with the usual left and right actions. As we shall see, these computations are unnecessary for the computation of our subproduct system. Define V : (n +m) (m,n) by the usual formula V (X) = Γ X, where Γ : n,m n,m n,m n,m L → L ◦ ℓ (Ω) H ℓ (Ω) ℓ (Ω) H is defined by Γ (a h) = a I h. ∞ θn+m ∞ θm ∞ θn n,m ⊗ → ⊗ ⊗ ⊗ ⊗ ⊗ ItisevidentthatV (X) = Γ X soinordertocomputeV ,alloneneedstodoiscompute n∗,m ∗n,m n∗,m ◦ Γ . So indeed, we compute Γ by computing the projection Q = Γ Γ onto the image of ∗n,m ∗n,m ∗n,m n,m Γ which is exactly ℓ (Ω) I H that has p I e as an orthogonal basis. n,m ∞ θm θn k i (i,k) E(Pn+m) ⊗ ⊗ { ⊗ ⊗ } ∈ In fact, as Hilbert spaces with the corresponding bases, ℓ∞(Ω) θm I θnH ∼= ℓ∞(Ω) θn+mH via ⊗ ⊗ ⊗ Γ . ∗n,m We run the aformentioned computation. Indeed, p I e ,p p (c ) k i k j ijk i Q(p p (c ) ) = h ⊗ ⊗ ⊗ ⊗ ip I e k ⊗ j ⊗ ijk i p I e 2 k ⊗ ⊗ i k i i : (i,k) E(Pn+m) || ⊗ ⊗ || X∈ And since, p I e ,p p (c ) = c P(n)P(m) and p I e 2 = p e 2 = P(n+m) h k⊗ ⊗ i k⊗ j⊗ ijk ii ijk ij jk || k⊗ ⊗ i|| || k⊗ i|| ik We obtain that, (n) (m) c P P j Ω ijk ij jk Q pk ⊗pj ⊗(cijk)i = P ∈ P(n+m) pk ⊗I ⊗ei j Ω i : (i,k) E(Pn+m) ik (cid:0)X∈ (cid:1) X∈ Now, since Γ = Γ Q, we have that ∗n,m ∗n,m (n) (m) c P P j Ω ijk ij jk Γ∗n,m pk ⊗pj ⊗(cijk)i = P ∈ P(n+m) pk ⊗ei (cid:0)Xj∈Ω (cid:1) i : (i,k)X∈E(Pn+m) ik (n) (m) (n) (m) We define the usual Ψ : (n) (m) (m,n) by Ψ (X X ) = (I X ) X and L ⊗L → L n,m A ⊗ B ⊗ A ◦ B obtain by a simple computation that for A= [a ] (n) and B = [b ] (m), ij lk ∈ E ∈ E (n) (m) (m,n) Ψ (X X )= Y n,m [aij]⊗ [blk] [aijbjk] So the multiplication maps are U : (n) (m) (n+m) given by U = V Ψ and n,m n,m n∗,m n,m L ⊗L → L we obtain: (n) (n) (m,n) U (X X )(e ) = (Γ Y )(e ) = n,m [aij]⊗ [blk] k ∗n,m◦ [aijbjk] k (n) (m) a P b P j Ω ij ij jk jk P ∈ P(n+m) pk ⊗ei i : (i,k)X∈E(Pn+m) ik In other words, (n) (m) (n+m) U (X X )= X n,m A ⊗ B (Pn+m)♭ (Pn A)(Pm B) ∗ ∗ · ∗ (cid:2) (cid:3)