Extensions and Dilations

1 Extensions and Dilations

In this chapter we are concerned with normal unital completely positive maps on von Neumann algebras which we call ‘stochastic maps’ for short. In the context of noncommutative stochastic processes, such maps play the role of transition operators and therefore they deserve a careful study on their own. To come as directly as possible to the heart of the matter and to new results, we have postponed a review of the basic structure theory of such maps to the Appendix (in particular A.1 and A.2) which can be used by the reader according to individual needs. Our first topic is an extension problem for stochastic maps which occurs naturally in connection with the GNS-construction. For concreteness, we start with some elementary computations in the easiest nontrivial case and then discuss the general case. Extension problems for completely positive maps are first discussed by W. Arveson in [Ar69], see also [ER00] for a recent survey. The additional point which we make consists in the inclusion of states. This is well motivated from the probabilistic point of view. Our main observation here is a duality between this extension problem and a dilation problem. The latter is described under the heading ‘weak tensor dilations’ of stochastic maps. While this is new as an explicit concept, it should be compared especially to some early dilation theories of completely positive maps, for example by D.E. Evans and J.T. Lewis [EL77], E.B. Davies [Da78], G.F. Vincent-Smith [Vi84]. It differs from them by insisting on a tensor product structure. We define an equivalence relation for weak tensor dilations which later turns out to be the correct one in the duality with the extension problem. To formulate the correspondence between extension and dilation, we have to consider commutants and the dual stochastic map on the commutants. Then under specified conditions every solution of the extension problem gives rise to an equivalence class of solutions of the dilation problem and conversely. Thus we see that these problems shed some light onto each other. The remainder of the chapter discusses further details of this correspondence, instructive special cases and further examples. Applications based on the fact that stochastic maps can be interpreted as transition operators in noncommutative probabil-

R. Gohm: LNM 1839, pp. 9–36, 2004. c Springer-Verlag Berlin Heidelberg 2004


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1 Extensions and Dilations

ity are postponed to the following chapters. The correspondence then proves to be a useful tool. Parts of the contents of Chapter 1 are also discussed in [Go1].

1.1 An Example with 2 × 2 - Matrices 1.1.1 A Stochastic Map
Consider a stochastic matrix

1−λ λ µ 1−µ

 with 0 < λ, µ < 1.

We think of it as an operator S : C2 → C2 ,




x1 x2




→

(1 − λ)x1 + λx2 µx1 + (1 − µ)x2

 .

S is a stochastic map, which means here that it is positive and unital. See A.1 for the general definition  µ and further discussion. There is an invariant λ , λ+µ probability measure φ = λ+µ in the sense that φ ◦ S = φ. We can apply the GNS-construction for the algebra A = C2 with respect 2 to the state φ, and we get the Hilbert space
√ H = C (with canonical scalar 1 √µ . Now the state φ is realized product) and the unit vector Ω = √λ+µ λ

 x1 0 x1 = Ω, Ω. Identifying A = C2 with as a vector state, i.e. φ x2 0 x2 the diagonal subalgebra of B(H) = M2 we have

 x1 0 (1 − λ)x1 + λx2 0 S: x= → 0 x2 0 µx1 + (1 − µ)x2 and Ω, xΩ = Ω, S(x)Ω. We shall now ask for stochastic maps Z : M2 → M2 (i.e.unital completely positive maps, see A.1.1) which extend S and satisfy Ω, xΩ = Ω, Z(x)Ω for all x ∈ M2 . 1.1.2 Direct Approach Let us try a direct approach. Anycompletely positive map Z : M2 → M2 can d be written in the form Z(x) = k=1 ak x a∗k with ak ∈ M2 . Introduce four vectors aij ∈ Cd , i, j = 1, 2, whose k-th entry is the ij-entry of ak (compare [K¨ u85b]). With the canonical scalar product and euclidean norm on Cd we get by direct computation: 



a11 2 x1 + a12 2 x2 a21 , a11 x1 + a22 , a12 x2 x1 0 = Z 0 x2 a11 , a21 x1 + a12 , a22 x2 a21 2 x1 + a22 2 x2 If Z is an extension of S we conclude that

1.1 An Example with 2 × 2 - Matrices

11

a11 2 = 1 − λ a12 2 = λ

a21 2 = µ

a22 2 = 1 − µ a11 , a21  = 0 a12 , a22  = 0 Now Ω, xΩ = Ω, Z(x)Ω for all x ∈ M2 means that Ω is a common eigen. . . , d(see A.5.1). Inserting vector for all a∗k , i.e. a∗k Ω = ωk Ω for k = 1,  ω1
√  .  d 1 √µ shows that the vector ΩP =  Ω = √λ+µ  ..  ∈ P := C is a unit λ ωd vector satisfying

λ µ a21 = a12 + a22 . ΩP = a11 + µ λ The operator Z is not changed if we apply a unitary in B(P) to the four vectors a11 , a12 , a21 , a22 (see A.2.3). Thus we can choose and fix the vectors ΩP , a11 , a21 in an arbitrary way except, of course, that they should satisfy the equations derived above. In particular, the two-dimensional plane spanned by them can be chosen arbitrarily. 1.1.3 Computations  11 11 for S. The general case only increases the amount of computation without introducing new ideas. We then have ΩP = a11 + a21 = a12 + a22 and we realize it by the right-angled and isosceles triangle given by       1 1 1 0   1 1 1  −1    ΩP =  0  , a11 =  0  , a21 =  0  . 2  2    .. .. .. . . .

For simplicity we show only the case λ = µ =

1 2,

i.e. the matrix

1 2




. Here .. means that for k ≥ 3 the entries are 0. Another right-angled and isosceles triangle is formed by ΩP , a12 , a22 . The most general form for a12 and a22 is given by

 1 1 1 1 a12 = , a22 = 2 γ 2 −γ with γ ∈ Cd−1 . Let γk be the k-th component of γ, in particular c := γ1 equals two times the second component of a12 . We have c ∈ C with | c |≤ 1 and 


1 11 1 1 0 γk−1 1 c a1 = , a2 = , ak = 2 11 2 −1 −c 2 0 −γk−1

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1 Extensions and Dilations

 for k = 3, . . . , d. It is easily checked that the operator Z = k ak · a∗k remains
 0 1− | c |2 the same if we replace the ak with k ≥ 3 by a3 = 12 and 0 − 1− | c |2 3 delete higher subscripts. Then the entries of Zc (x) := Z(x) = k=1 ak xa∗k with parameter c ∈ C, | c |≤ 1 can be computed and we find for x =
 x11 x12 : x21 x22 1 Zc (x) = 4




2 X + (1 + c)x12 + (1 + c)x21 (1 − c)x12 + (1 − c)x21 (1 − c)x12 + (1 − c)x21 2 X + (1 + c)x12 + (1 + c)x21

 ,

where X = x11 + x22 . In particular: The Zc for | c |≤ 1 are all different from each other. Their rank (see A.2.3) is easily checked to be 2 for | c |= 1 (where a3 = 0) and 3 otherwise. For general λ, µ we can proceed analogously. The result is the following. 1.1.4 Parametrization of the Set of Extensions
 1−λ λ Proposition: For the stochastic matrix with 0 < λ, µ < 1 µ 1−µ the set of completely positive extensions Z : M2 → M2 with Ω, xΩ = Ω, Z(x)Ω for all x ∈ M2 is parametrized by c ∈ C, | c |≤ 1. Explicitly: Zc (x) = a1 xa∗1 + a2 xa∗2 + a3 xa∗3 with 

  √  1√− λ λµ λ(1 − λ) λ(1 − µ) c  a1 = , a2 = , λµ 1 − µ − µ(1 − λ) − µ(1 − µ) c
  0 λ(1 − µ)(1− | c |2 ) , a3 = 0 − µ(1 − µ)(1− | c |2 ) 


Zc

x11 x12 x21 x22

 11

= x11 (1 − λ) + x22 λ √   + x12 λµ(1 − λ) + cλ (1 − λ)(1 − µ) √   λµ(1 − λ) + cλ (1 − λ)(1 − µ) + x21

 
x11 x12 = x11 µ + x22 (1 − µ) Zc x21 x22 22 √   + x12 λµ(1 − µ) + cµ (1 − λ)(1 − µ) √   + x21 λµ(1 − µ) + cµ (1 − λ)(1 − µ)

1.2 An Extension Problem


 x11 x12 = Zc x21 x22 12


 Zc

x11 x12 x21 x22

 = 21

13

   x12 (1 − λ)(1 − µ) − c λµ(1 − λ)(1 − µ)    + x21 λµ − c λµ(1 − λ)(1 − µ)    x12 λµ − c λµ(1 − λ)(1 − µ)    + x21 (1 − λ)(1 − µ) − c λµ(1 − λ)(1 − µ) .

Note that Zc1 +c2 = Zc1 + Zc2 , showing that the convex set of extensions is (affinely) isomorphic to {c ∈ C : | c |≤ 1}. For extremal points (| c |= 1) the rank of Zc (see A.2.3) is 2, otherwise it is 3.

1.2 An Extension Problem 1.2.1 The Set Z(S, φB ) of Extensions Suppose A ⊂ B(G) and B ⊂ B(H) are von Neumann algebras with cyclic vectors ΩG ∈ G and ΩH ∈ H. Restricting the corresponding vector states to A and B we get normal states φA and φB . Then consider a stochastic map S : (A, φA ) → (B, φB ), i.e. φB ◦ S = φA . This is the setting of A.1.3, and in A.1.2 the reader can find some motivation for it. We are interested in the following set Z(S, φB ) := {Z : B(G) → B(H) stochastic and Z|A = S and ΩG , xΩG  = ΩH , Z(x)ΩH  for all x ∈ B(G)}. (A, φA )  (B(G), ΩG )

S

/ (B, φB )

Z

 / (B(H), ΩH )

Note that in Proposition 1.1.4 we gave a characterization of Z(S, φ) for S given by a stochastic 2 × 2-matrix with an invariant state φ. 1.2.2 Z as a Convex Set Z is convex and closed in suitable topologies. For example we can use the topology of pointwise weak∗ -convergence. It is well known (and easy to check by a Banach-Alaoglu type of argument) that the set of stochastic maps is compact in this topology (see [Ar69], 1.2). Thus Z is the closed convex hull of its extremal points by the Krein-Milman theorem.

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1 Extensions and Dilations

1.2.3 Discussion Let us note some immediate observations. From S : A → B ⊂ B(H) we get a Stinespring representation S(a) = v ∗ a ⊗ 1I v, where a ∈ A, v : H → G ⊗ P is an isometry, P another Hilbert space, And we have corresponding d see A.2.4. ∗ a a a , where a ∈ A, ak ∈ B(G, H), Kraus decompositions S(a) = k k k=1 d = dimP, see A.2. The sum should be interpreted as a stop-limit if d = ∞. Then we have the ansatz Z(x) = v ∗ x ⊗ 1I v =

d 

ak x a∗k

k=1

for all x ∈ B(G). This is a stochastic map extending S. Concerning the states the following assertions are equivalent (see A.5.1): (1) ΩG , xΩG  = ΩH , Z(x)ΩH for all x ∈ B(G). (2) There is a unit vector ΩP ∈ P such that v ΩH = ΩG ⊗ ΩP . (3) There is a function ω : {1, . . . , d} → C, k → ωk such that a∗k ΩH = ωk ΩG for all k. If G = H and ΩG = ΩH =: Ω, then (3) means that Ω is a common eigenvector for all a∗k . Thus in an informal language we can restate our extension problem as follows: Can we find Stinespring representations or Kraus decompositions of S such that these additional properties are satisfied? And how many different Z can we construct in this way? Note that this extension problem is related to the results of W. Arveson in ([Ar69], 1.2) on extensions of completely positive maps. See also ([ER00], 4.1) for a recent account. But it is distinguished from these by including an additional condition about states which restricts the set of possible extensions. The additional condition is natural from a probabilistic point of view.

1.3 Weak Tensor Dilations 1.3.1 The Definition We want to relate the extension problem considered in the previous section to a dilation problem. We start by defining the latter. Definition: Let A, B, C be von Neumann algebras. A normal ∗ −homomorphism j : A → B⊗C (not necessarily unital) is called a weak tensor dilation (of first order) for a stochastic map S : A → B if there is a normal state ψ of C such that S = Pψ j. Here Pψ : B ⊗C → B denotes the conditional expectation determined by Pψ (b ⊗ c) = b ψ(c).

1.3 Weak Tensor Dilations

15

/B AE O EE EE Pψ E j EE " B⊗C S

Let us give some comments on this definition. We use von Neumann algebras and von Neumann tensor products. Of course the same definition is possible for C ∗ −algebras (dropping normality), but our main results concern von Neumann algebras. ‘Weak’ refers to the fact that the dilation is not assumed to be unital, similar to other weak dilation theories (see [BP94]). The dilation is called ‘tensor’ because the conditional expectation used is of tensor type. Such conditional expectations are also called slice maps. By ‘first order’ we mean that higher powers of S are not considered. We shall consider dilations which also work for higher powers in Chapter 2, and then it will turn out that the construction of dilations of first order is the decisive step in the construction of more sophisticated dilation theories. In Chapter 1 we always deal with dilations of first order, even if this is sometimes not written. When we say that the homomorphism j : A → B ⊗ C is not necessarily unital then this means that j(1I) does not necessarily coincide with the identity of B ⊗ C. But of course j(1I) is the identity of the von Neumann algebra j(A), i.e. in general a projection in B⊗C. We also did not assume the homomorphism j to be injective. But this can be achieved whenever needed by factoring all the objects involved by the kernel of j. Compare also 1.3.4. The definition of weak tensor dilations may be compared with early approaches to the dilation theory of completely positive maps, such as D.E. Evans and J.T. Lewis [EL77], E.B. Davies [Da78], G.F. Vincent-Smith [Vi84]. But it differs from these by explicitly demanding a tensor product structure. This feature becomes especially important when we consider classification, see Section 1.4. 1.3.2 Representations Applying the GNS-construction to (C, ψ) we get a Hilbert space K with a representation of C and a cyclic unit vector ΩK ∈ K representing the state ψ. Therefore we do not loose much if in the definition of weak tensor dilations above we replace (C, ψ) by (B(K), ΩK ). This is made precise by the equivalence relation for weak tensor dilations introduced in Section 1.4. If we are given a nondegenerate and faithful representation B ⊂ B(H) for the algebra B on a Hilbert space H, then it is natural in this context to identify H with H ⊗ ΩK ⊂ H ⊗ K. If the projection from H ⊗ K onto H is called pH , then we have Pψ (·) = pH · |H . With these conventions we have Lemma: j(1I) ≥ pH . Proof: pH j(1I) |H = Pψ j(1I) = S(1IA ) = 1IB = 1IH . 2

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1 Extensions and Dilations

1.3.3 Construction of Examples Let us consider some examples. The tensor dilations in [K¨ u85a] are in particular weak tensor dilations in our sense. They have additional properties: The homomorphism j is unital and the state ψ is faithful. More details are given in Section 2.1. Another class of examples is obtained as follows. If B is isomorphic to B(L) for some Hilbert space L, then we can construct a weak tensor dilation for S : A → B = B(L) starting with a Stinespring representation S(a) = v ∗ π(a)v, where v : L → Lˆ is an isometry into the Hilbert space Lˆ of the Stinespring representation and π is the representation of A on Lˆ (see A.2). In fact, let K be a Hilbert space with unit vector ΩK ∈ K and let u : Lˆ → L ⊗ K be an isometry such that vξ = u∗ (ξ ⊗ ΩK ) for all ξ ∈ L. Such isometries always exist if K is large enough: It is only required to extend the prescribed one-to-one u correspondence vL → L ⊗ ΩK in an isometric way. Using the von Neumann algebra tensor product we have B(L) ⊗ B(K) = B(L ⊗ K) (see [KR83], 11.2). Now j(a) := u π(a) u∗ is a weak tensor dilation of S: ξ1 ⊗ ΩK , j(a)ξ2 ⊗ ΩK  = ξ1 , v ∗ π(a)vξ2  = ξ1 , S(a)ξ2  for all ξ1 , ξ2 ∈ L. Lˆ hP ^>>PPPP >> PPPu∗ > PPP v >>> PPP / L⊗K L π(a)

S(a)

j(a)

 L⊗K 6 nnn v∗ nnn n n n u  nnnnn Lˆ  o @L

Conversely assume that j : A → B(L) ⊗ B(K) = B(L ⊗ K) with a unit vector ΩK ∈ K is a weak tensor dilation for S : A → B(L). If we define Lˆ := j(1I)L⊗K then the projection j(1I) : L ⊗ K → Lˆ is adjoint to the isometric embedding ˆ Thus u : Lˆ → L ⊗ K. By Lemma 1.3.2 with H = L we see that L ⊗ ΩK ⊂ L. if we define vξ = u∗ (ξ ⊗ ΩK )(= ξ ⊗ ΩK ) for all ξ ∈ L, then v : L → Lˆ is an isometry and S(a) = v ∗ j(a)v is a Stinespring representation for S. Given a weak tensor dilation it is therefore natural to identify L with L ⊗ ΩK and Lˆ with j(1I)L ⊗ K and consider u∗ : L ⊗ K → Lˆ as a coisometric extension of ˆ the isometry v : L → L. Note that j is unital if and only if u is unitary. If A = B = B(L), then (by A.2.4) there exists another Hilbert space P such that Lˆ = L ⊗ P. If further dim(Lˆ  vL) = dimL (dimP − 1), which is automatic if the spaces are finitedimensional, then it is possible (but not forced) to choose a Hilbert space K

1.3 Weak Tensor Dilations

17

with dimK = dimP and then u∗ as an arbitrary unitary from L ⊗ K to L ⊗ P extending v : L = L ⊗ ΩK → L ⊗ P. Then we get a unital dilation. If we look at the embedding L = L ⊗ ΩK ⊂ j(1I)(L ⊗ K) just as an abstract embedding of Hilbert spaces and ignore the tensor product structure, then weak tensor dilations with B = B(L) essentially reduce to Stinespring representations. We shall see however that some interesting problems arise if we do not ignore the tensor product structure. See Section 1.4, where we introduce an equivalence relation which is finer than just unitary equivalence of Stinespring representations. Using more refined techniques, it is possible to prove existence of weak tensor dilations also in the general case, i.e. for S : A → B with von Neumann algebras A and B. This can be done by a substantial improvement of the method above which involves consideration of Hilbert modules, see [GS]. We shall not use this result and do not include a proof here. 1.3.4 The Associated Isometry To develop further the theory of weak tensor dilations j for an operator S : A → B, we include states into our description. Assume that S : (A, φA ) → (B, φB ) in the setting of A.1.3, i.e. A ⊂ B(G) and B ⊂ B(H) and there are cyclic vectors ΩG ∈ G and ΩH ∈ H implementing the states φA and φB . As an important tool we define an associated isometry which provides a spatial implementation of the dynamics. The usefulness of such implementations for the study of quantum dynamical semigroups has already been noticed in early work of E.B. Davies, see for example [Da77]. To avoid confusion with the isometries in 1.3.3 we shall denote the associated isometry by v1 . Lemma: The operator v1 : G → H ⊗ K a ΩG → j(a) ΩH ⊗ ΩK

(for all a ∈ A)

defines an isometry which we call associated to j (with state φB ). A  G

j

v1

/ B⊗C  / H⊗K

Proof: We check the isometric property:

j(a) ΩH ⊗ ΩK 2 = ΩH ⊗ ΩK , j(a∗ a)ΩH ⊗ ΩK  = ΩH , S(a∗ a)ΩH  = ΩG , a∗ a ΩG  = a ΩG 2 .

2

As an application we remark that in this setting a weak tensor dilation j of S is always injective. In fact, if j(a) = 0 then for all a1 ∈ A also j(aa1 ) = 0, j(aa1 ) ΩH ⊗ ΩK = 0, aa1 ΩG = 0 and thus a = 0.

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1 Extensions and Dilations

1.3.5 The Minimal Version of an Associated Isometry As a first step in the classification of weak tensor dilations of first order we show that they always contain a minimal object. This is closely related to the associated isometry introduced above. For technical convenience we assume that the von Neumann algebra B has a separable predual and thus there exists a faithful normal state, see [Sa71], 2.1.9. Proposition: There is a smallest projection p ∈ B ⊗ B(K) with p ≤ j(1I) and such that j min (·) := p j(·) p is still a weak tensor dilation of first order of S by Pψ j min = S. If φB is faithful, then p is the projection onto the closure of (B  ⊗ 1I)v1 G. Here B  is the commutant of B on the GNS-space H. We call p the minimal projection and j min the minimal version of j. Proof: Choose a faithful normal state φB on B and apply Lemma 1.3.4 to get (H, ΩH ), φA , (G, ΩG ), v1 . Let B  be the commutant of B acting on H. We claim that p defined as the projection from H ⊗ K onto (B  ⊗ 1I) v1 G does the job. First we have p ∈ B ⊗ B(K) because its range (B  ⊗ 1I)v1 G is invariant for  B ⊗ 1I = (B ⊗ B(K)) . Then for a, a1 ∈ A, b ∈ B  j(a) b ⊗ 1I v1 a1 ΩG = b ⊗ 1I j(a) v1 a1 ΩG = b ⊗ 1I j(a)j(a1 ) ΩH ⊗ ΩK = b ⊗ 1I v1 aa1 ΩG ∈ (B  ⊗ 1I)v1 G, which shows that j(a) p = p j(a) p, i.e. jp (·) := p j(·) p is a homomorphism. For a = 1 the same computation shows that j(1I) b ⊗ 1I v1 a1 ΩG = b ⊗ 1I v1 a1 ΩG , i.e. p ≤ j(1I). Because v1 ΩG = ΩH ⊗ ΩK and ΩH is cyclic for B  it follows that p ≥ pH . Using this we get Pψ jp (·) = pH jp (·) |H = pH p j(·) p |H = pH j(·) |H = S(·). Now assume that q ∈ B ⊗ B(K) is any projection with q ≤ j(1I) such that jq (·) := q j(·) q is a weak tensor dilation of S by Pψ jq = S. We claim that q ≥ p, completing the proof of the proposition and showing in particular that there is a unique smallest projection, independent of the choice of φB . Using Lemma 1.3.1 for jq we find q = jq (1I) ≥ pH . Then for a ∈ A vq aΩG := jq (a) ΩH ⊗ ΩK = qj(a) ΩH ⊗ ΩK = qv1 aΩG . By Lemma 1.3.4 v1 and vq are isometries. This implies that the range of q includes v1 G and that vq = v1 . Because q ∈ B ⊗ B(K), its range is invariant for B  ⊗ 1I. Therefore its range includes (B  ⊗ 1I)v1 G, i.e. q ≥ p. 2 We add some remarks: Inspection of the proof above shows that the associated isometries for j and j min are the same. Conversely j min is already determined by its associated isometry v1 by j min (a) b ⊗ 1I v1 a1 ΩG = b ⊗ 1I v1 aa1 ΩG .

1.4 Equivalence of Weak Tensor Dilations

19

The minimal projection p and the minimal version j min do not depend on the state φB which has been introduced as a tool in the proof of Proposition 1.3.5. This follows from their defining properties. 1.3.6 Minimal Part of the Stinespring Representation If B = B(L) with a Hilbert space L, then we are in the setting discussed in 1.3.3. Then the minimal projection p for a weak tensor dilation j : A → B(L)⊗ B(K) = B(L ⊗ K) of S : A → B = B(L) is just a projection in B(L ⊗ K). In 1.3.3 we have constructed an associated Stinespring representation for S, and a moment’s thought convinces us that in this setting the minimal projection p is nothing but the projection onto the minimal part of this Stinespring representation. One only has to check that the two notions of minimality are the same.

1.4 Equivalence of Weak Tensor Dilations 1.4.1 An Equivalence Relation There is a natural equivalence relation for weak tensor dilations of first order of a stochastic map S : A → B. Let j1 , j2 be two such dilations. In Section 1.3 we have seen that there are various objects associated with a dilation j. If they are associated with j1 or j2 , then we use the corresponding subscript also for these objects. Proposition: For weak tensor dilations j1 , j2 of S the following assertions are equivalent: (a) There is a partial isometry w ∈ B(K1 , K2 ) such that v2 = (1I ⊗ w)v1 , where v1 , v2 are associated to j1 , j2 with a faithful state φB . (b) There is a partial isometry w ∈ B(K1 , K2 ) such that w ΩK 1 = ΩK 2 and j2min (·)(1I ⊗ w) = (1I ⊗ w)j1min (·) In this case 1I ⊗ w acts as a unitary from p1 (H ⊗ K1 ) to p2 (H ⊗ K2 ), where p1 , p2 are the minimal projections. We then say that j1 and j2 are equivalent: j1 ∼ j2 . j1min (a)

/ H ⊗ K1 ww w ww w w {w w S(a) / 1⊗w 1⊗w H cGG H GG ww w GG ww GG ww G  min  {ww (a) j2 / H ⊗ K2 H ⊗ K2 H ⊗ KcG1 GG GG GG GG

20

1 Extensions and Dilations

Remarks: We see from (b) that equivalence does not depend on the choice of the state φB used to define v1 and v2 . A dilation j is always equivalent to its minimal version j min . In fact, in this case the associated isometries are equal, see the proof of Proposition 1.3.5. Proof: Given (b) we get for a ∈ A v2 aΩG = j2min (a) ΩH ⊗ ΩK2 = j2min (a)(1I ⊗ w) ΩH ⊗ ΩK1 = (1I ⊗ w)j1min (a) ΩH ⊗ ΩK 1 = (1I ⊗ w)v1 aΩG , proving (a). Given (a) we find for any ξ ∈ G that

(1I ⊗ w)v1 ξ = v2 ξ = ξ = v1 ξ , i.e. 1I ⊗ w maps v1 ξ isometrically to v2 ξ. Therefore we get v1 ξ = (1I ⊗ w∗ )(1I ⊗ w)v1 ξ ∈ H ⊗w∗ wK1 . If {ηk } is an ONB of w∗ wK 1 and for some ξ ∈ G we write v1 ξ = ξk ⊗ ηk , then v2 ξ = (1I ⊗ w)v1 ξ = ξk ⊗ wηk and {wηk } are orthogonal unit vectors. Then for x ∈ B(H)

(1I ⊗ w)(x ⊗ 1I)v1 ξ = (x ⊗ 1I)(1I ⊗ w)v1 ξ = (x ⊗ 1I)v2 ξ

  =

xξk ⊗ wηk =

xξk ⊗ ηk = (x ⊗ 1I)v1 ξ . In particular (with b ∈ B  ) the operator 1I ⊗ w : p1 (H ⊗ K1 ) → p2 (H ⊗ K2 ) (b ⊗ 1I) v1 ξ → (b ⊗ 1I) v2 ξ is unitary. With a, a1 ∈ A we get by using (a) that j2min (a) (1I ⊗ w)v1 a1 ΩG = j2min (a) v2 a1 ΩG = v2 aa1 ΩG = (1I ⊗ w)v1 aa1 ΩG = (1I ⊗ w)j1min (a)v1 a1 ΩG , which means that j2min (·)(1I ⊗ w) and (1I ⊗ w)j1min (·) coincide on v1 G. Then they coincide also on (B  ⊗ 1I)v1 G = p1 (H ⊗ K1 ), because B  ⊗ 1I commutes with these terms. The unitarity shown above implies that 1I ⊗ w maps the orthogonal complement of p1 (H ⊗ K1 ) onto that of p2 (H ⊗ K2 ). Because of minimality both terms are equal to 0 there. Summing up: j2min (·)(1I ⊗ w) and (1I⊗w)j1min (·) coincide on the whole of H⊗K1 and are thus equal. This proves (b). 2 1.4.2 Equivalence and Unitary Equivalence For B = B(L) we have seen in 1.3.3 and 1.3.6 how weak tensor dilations correspond to Stinespring representations and also that minimality has the same meaning. But the equivalence considered here is stronger than just unitary equivalence of Stinespring representations: Here the intertwiner must have the form 1I ⊗ w. See Section 1.7 for examples where we have many non-equivalent weak tensor dilations.

1.5 Duality

21

1.5 Duality 1.5.1 Dual Stochastic Maps We want to establish a correspondence between the extension problem of Sections 1.1 and 1.2 on the one hand and the dilation problem of Sections 1.3 and 1.4 on the other hand. Consider S : (A, φA ) → (B, φB ) in the setting of A.1.3, i.e. A ⊂ B(G) and B ⊂ B(H) are von Neumann algebras with cyclic vectors ΩG ∈ G and ΩH ∈ H, implementing states φA on A and φB on B. Then these cyclic vectors also implement states φA and φB on the commutants A ⊂ B(G) and B  ⊂ B(H). The following duality is well known (see for example [AH78]). Lemma: There is a unique operator S  : (B  , φB ) → (A , φA ) such that ΩH , S(a) b ΩH  = ΩG , a S  (b ) ΩG  for all a ∈ A, b ∈ B  . S  is a stochastic map. We call it the dual map. Proof: For 0 ≤ a, b ≤ 1I ΩH , S(a)b ΩH 

≤

ΩH , S(a)ΩH  = ΩG , aΩG  = φA (a).

The commutant Radon-Nikodym theorem (see [KR83], 7.3.5) gives us a unique positive element in A which we call S  (b ), such that ΩH , S(a)b ΩH  = ΩG , aS  (b )ΩG . Inserting a = 1 shows that the operator S  defined in this way respects the states. For a1 , a2 ∈ A we have a1 ΩG , S  (1I)a2 ΩG  = ΩH , S(a∗1 a2 )ΩH  = ΩG , a∗1 a2 ΩG  = a1 ΩG , a2 ΩG , which proves that S  (1I) = 1I. Clearly S  is positive. With Sn := S ⊗ 1I : (A ⊗ Mn , φA ⊗ tr) → (B ⊗ Mn , φB ⊗ tr) one gets (S  )n = (Sn ) . Therefore n-positivity of S implies n-positivity of S  . Because S is completely positive also S  is completely positive. 2 1.5.2 From Dilation to Extension Now let j1 : A → B ⊗ B(K) be a weak tensor dilation of S, with associated isometry v1 : G → H ⊗ K. Define an operator Z  : B(H) → B(G), Proposition:

x → v1∗ x ⊗ 1I v1 .

Z  ∈ Z(S  , φA ).

Proof: Z  is given in a Stinespring representation, i.e. it is stochastic. The associated isometry v1 : aΩG → j1 (a) ΩH ⊗ ΩK satisfies v1 ΩG = ΩH ⊗ ΩK

22

1 Extensions and Dilations

which implies ΩH , xΩH  = ΩG , Z  (x)ΩG for all x ∈ B(H). For all a1 , a2 ∈ A and b ∈ B  we get a1 ΩG , v1∗ b ⊗ 1I v1 a2 ΩG  = j1 (a1 )ΩH ⊗ ΩK , b ⊗ 1I j1 (a2 ) ΩH ⊗ ΩK  = ΩH ⊗ ΩK , j1 (a∗1 a2 )(b ΩH ) ⊗ ΩK  = ΩH , Pψ j1 (a∗1 a2 ) b ΩH  = ΩH , S(a∗1 a2 )b ΩH  = ΩG , S  (b )a∗1 a2 ΩG  = a1 ΩG , S  (b )a2 ΩG , i.e. v1∗ b ⊗ 1I v1 = S  (b ).

2

1.5.3 From Extension to Dilation Let us look for a kind of converse for the preceding result. Proposition: Given Z : B(G) → B(H) with Z ∈ Z(S, φB ). Then there exists a weak tensor dilation j1 : B  → A ⊗ B(P) of S  with associated isometry v1 : H → G ⊗ P such that Z(x) = (v1 )∗ x ⊗ 1I v1 for all x ∈ B(G). Note that j1 and v1 are not commutants of other objects, but connection to the dual map S  .



denotes the

Proof: By the Stinespring representation theorem (see A.2) applied to the stochastic map Z we find an isometry v1 : H → G ⊗ P such that Z(x) = (v1 )∗ x ⊗ 1I v1 . Because S = Z|A we also have S(a) = (v1 )∗ a ⊗ 1I v1 for a ∈ A. Let p be the projection from G ⊗ P onto the minimal part of the Stinespring representation of S, i.e. onto the closure of (A ⊗ 1I) v1 H. We want to use a result of W. Arveson (see [Ar69], 1.3, see also [Ta79], IV.3.6) on the lifting of commutants. The following version of it is adapted to our needs: If H1 , H2 are Hilbert spaces, v : H1 → H2 an isometry, E ⊂ B(H2 ) a selfadjoint algebra with EvH1 = H2 , then there is an isomorphism j from {v ∗ Ev} onto E  ∩ {vv ∗ } satisfying j(·)v = v· (with · representing elements of {v ∗ Ev} ). Note that the last condition determines j uniquely: For ξ ∈ H1 and

∈ E we have j(·) vξ = j(·)vξ = v · ξ In fact, if we define j in this way, then it is possible to check the properties stated above. This is done in [Ar69]. We apply this for H1 = H, H2 = p (G ⊗ P) and v = v1 . Define E := p A ⊗ 1I p = A ⊗ 1I p . Then because p (G ⊗ P) ⊃ v1 H we get {(v1 )∗ Ev1 } = {(v1 )∗ A ⊗ 1I v1 } = S(A) ⊃ B  and now Arveson’s result yields a homomorphism j1 : B  → {A ⊗ 1I p }c , where we have introduced the notation c to denote the commutant on H2 . For any projection e in a von Neumann algebra M which is represented on a Hilbert space it is always true that on the range of e the commutant of eMe

1.5 Duality

23

equals M e (see [Ta79], II.3.10). Here this means that {p A ⊗ B(P) p }c = (A ⊗ 1I)p and thus {(A ⊗ 1I)p }c = (p A ⊗ B(P) p )cc = p A ⊗ B(P) p ⊂ A ⊗ B(P). We conclude that j1 : B  → A ⊗ B(P) and j1 (1I) ≤ p . Applying j1 (·)v1 = v1 · to ΩH gives for b ∈ B  v1 b ΩH = j1 (b )v1 ΩH Because ΩG , xΩG  = ΩH , Z(x)ΩH  for all x ∈ B(G) it follows by A.5.1 that there is a unit vector ΩP ∈ P such that v1 ΩH = ΩG ⊗ ΩP . Thus v1 b ΩH = j1 (b ) ΩG ⊗ ΩP , which expresses the fact that v1 is the isometry associated to j1 . It remains to prove that if ψ is the vector state given by ΩP then Pψ j1 = S  . For a1 , a2 ∈ A and b ∈ B  we get a1 ΩG , Pψ j1 (b )a2 ΩG  = (a1 ⊗ 1I) ΩG ⊗ ΩP , (a2 ⊗ 1I)j1 (b ) ΩG ⊗ ΩP  = a1 ⊗ 1I v1 ΩH , a2 ⊗ 1I v1 b ΩH  = ΩH , (v1 )∗ a∗1 a2 ⊗ 1I v1 b ΩH  = ΩH , S(a∗1 a2 )b ΩH  = ΩG , a∗1 a2 S  (b )ΩG  2 = a1 ΩG , S  (b )a2 ΩG , i.e. Pψ j1 = S  .

1.5.4 One-to-One Correspondence The correspondence between dilation and extension worked out so far takes an especially nice form if we assume that the cyclic vectors ΩG and ΩH are not only cyclic but also separating, i.e. if we consider standard representations. In this case we write T : (A, φA ) → (B, φB ),

T  : (B  , φB ) → (A , φA )

instead of S, S  . It is easy to check that (T  ) = T , and there is a duality in all statements about algebras and commutants. The distinguished vector states are cyclic for algebras and commutants and we can apply 1.5.2 and 1.5.3 with S = T and with S = T  . Theorem: There is a one-to-one correspondence between (1)weak tensor dilations j1 : A → B ⊗ B(K) of T modulo equivalence (see 1.4.1) (2)stochastic maps Z  : B(H) → B(G) such that Z  ∈ Z(T  , φA ). The following objects are also in one-to-one correspondence to those above and show explicitly how the correspondence (1) ↔ (2) works: (3a) isometries v1 : G → H ⊗ K such that v1∗ b ⊗ 1I v1 = T  (b ) for all b ∈ B  and v1 ΩG = ΩH ⊗ ΩK , with a Hilbert space K and a unit vector ΩK ∈ K, modulo the equivalence relation given by v2 = (1I ⊗ w)v1 with a partial isometry w ∈ B(K1 , K2 ).

24

1 Extensions and Dilations

(3b) {a1 , . . . , ad } ⊂ B(H, G) with d ∈ [1, . . . , ∞] (∞ included, in this case the sums below are stop-limits) d such that T  (b ) = k=1 ak b a∗k and a∗k ΩG = ωk ΩH for all k with a function ω : {1, . . . , d} → C, k → ωk modulo the equivalence relation given by: {a1 , . . . , ad1 } ∼ {a†1 , . . . , a†d2 } if and only if there is a complex d2 × d1 -matrix w representing a partial isometry  ∗  † ∗  (a1 ) a1  ..   ..  with  .  = w  .  . (a†d2 )∗

a∗d1

Proof: Given a stochastic map Z  as in (2), write down a Stinespring representation and a Kraus decomposition Z  (x) = v1∗ x ⊗ 1I v1 =

d 

ak x a∗k .

k=1

Now ΩH , xΩH  = ΩG , Z  (x)ΩG  corresponds to v1 ΩG = ΩH ⊗ ΩK in (3a) and to a∗k ΩG = ωk ΩH for all k in (3b), see A.5.1. The uniqueness assertion for Stinespring representations of Z  correspond to the equivalence relation in (3a) and the uniqueness assertion for Kraus decompositions of Z  correspond to the equivalence relation in (3b), see A.2. This shows the one-to-one correspondences (3a) ↔ (3b) ↔ (2). Given (1), i.e. a weak tensor dilation j1 , the associated isometry v1 satisfies the properties in (3a). Indeed, the nontrivial part of this has been done in 1.5.2. Equivalence of weak tensor dilations is the same as the equivalence in (3a) by 1.4.1. Thus to see that (1) ↔ (3a) is one-to-one it only remains to check that each of the isometries in (3a) can be realized as an associated isometry of some j1 . But this has been proved in 1.5.3. In fact, we only have to read 1.5.3 for Z  instead of Z. 2 We add some remarks. In (1) one may replace equivalence classes by minimal versions. The weak tensor dilation j1 constructed with the methods of 1.5.3 is already minimal. In fact (with  removed) it has been noted there that j1 (1I) ≤ p. But in our standard representation we also know from 1.3.5 that the projection p onto (B  ⊗ 1I) v1 G is the minimal projection for j1 . Thus j1 (1I) = p and j1 is minimal. 1.5.5 Discussion Let us summarize what has been achieved in this section. There is a correspondence between the extension problem defined in Section 1.2 for T  and the dilation problem defined in Section 1.3 for T . We have characterized the equivalence relation on dilations which makes this correspondence one-to-one. Equivalence tells us essentially that only the minimal representation space

1.6 The Automorphic Case

25

p(H ⊗ K) for the Stinespring representation of T  is relevant: It is invariant for the dilation of T and allows to define its minimal version. There is an analogous theorem with the role of T and T  interchanged, dealing with extensions Z of T based on weak tensor dilations j1 of T  . In Chapter 2 we shall interpret T as a transition operator, and we shall speak of Z as an extended transition operator and of Z  as a dual extended transition operator. Note however that Z and Z  are not dual in the sense of 1.5.1. Remember that the equivalence classes of weak tensor dilations of first order do not depend on the choice of states, see 1.4.1. From the one-to-onecorrespondence above we infer the remarkable result that also the set of extensions is essentially independent of the choice of states. More precisely, Z(T, φ1 ) and Z(T, φ2 ) may contain different operators, but there is a canonical bijection between these sets. Note also that these sets are always non-empty by the existence result in [GS], mentioned in 1.3.3.

1.6 The Automorphic Case 1.6.1 Conditional Expectations Recall that a conditional expectation P on a von Neumann algebra can be defined as a normal projection of norm one onto a subalgebra. Further properties familiar to a probabilist already follow from that, in particular P is stochastic and P (b1 ab2 ) = b1 P (a)b2 for all b1 , b2 in the range of P . For this topic compare [Sa71, Ta72] and [KR83], 8.5.85-86. When we speak of conditional expectations we shall always assume that the distinguished state is invariant, as usual in probability theory. Note that in noncommutative probability theory conditional expectations do not always exist, see [Ta72, AC82]. 1.6.2 Adjoints Again we start with an already well-known setting: a weak tensor dilation of first order j1 : A → B ⊗ C of a stochastic map S : (A, φA ) → (B, φB ), with the relevant states represented by unit vectors ΩG , ΩH , ΩK on the GNS-Hilbert spaces. In particular (φB ⊗ ψ) ◦ j1 = φA . By Proposition 1.5.2 there exists a stochastic map Z  ∈ Z(S  , φA ) defined by Z  : B(H) → B(G), Z  (x) = v1∗ x ⊗ 1I v1 , where v1 is the associated isometry of j1 . Proposition: Assume that there exists a conditional expectation P1 from B⊗C onto j1 (A). Then Z  (B) ⊂ A and Z  |B = S + , where S + is a φB −adjoint of S (i.e. S + : B → A is a stochastic map satisfying S + (b)ΩG , aΩG  = bΩH , S(a)ΩH  for a ∈ A, b ∈ B, see [AC82, K¨ u88a]). Proof: We associate to each b ∈ B the element a1 ∈ A such that P1 (b ⊗ 1I) = j1 (a1 ). Then with a2 , a ∈ A

26

1 Extensions and Dilations

Z  (b) a2 ΩG , a ΩG  = v1∗ b ⊗ 1I v1 a2 ΩG , a ΩG  = b ⊗ 1I j1 (a2 ) ΩH ⊗ ΩK , j1 (a) ΩH ⊗ ΩK  = P1 (b ⊗ 1I) j1 (a2 ) ΩH ⊗ ΩK , j1 (a) ΩH ⊗ ΩK  = j1 (a1 a2 ) ΩH ⊗ ΩK , j1 (a) ΩH ⊗ ΩK  = a1 a2 ΩG , a ΩG , i.e. Z  (b) = a1 ∈ A. With a2 = 1I it follows that Z  (b) ΩG , a ΩG  = b ⊗ 1I ΩH ⊗ ΩK , j1 (a) ΩH ⊗ ΩK  2 = b ΩH , S(a) ΩH . 1.6.3 Automorphic Tensor Dilations Consider the special case A = B. Then we can define a weak tensor dilation (of first order) j1 of S : A → A to be automorphic if there is an automorphism α1 of A ⊗ C such that j1 (a) = α1 (a ⊗ 1I) for all a ∈ A. Compare 1.3.1. Note that an automorphic j1 is unital. /A A JJ O JJ j JJ 1 JJ Pψ JJ  $ α1 / A⊗C A⊗C S

If in addition we have a state φ on A and if φ as well as ψ on C are faithful normal states and if further α1 : (A ⊗ C, φ ⊗ ψ) → (A ⊗ C, φ ⊗ ψ), then we call it an automorphic tensor dilation of first order (omitting ‘weak’, compare Section 2.1). Note that φ is invariant for S. / (A, φ) (A, φ) Q O QQQ QQQj1 QQQ Pψ QQQ  ( α1 / (A ⊗ C, φ ⊗ ψ) (A ⊗ C, φ ⊗ ψ) S

For automorphic tensor dilations the conditional expectation P1 considered in 1.6.2 always exists by an application of a theorem of Takesaki [Ta72] (see [K¨ u85a]). We also have all results of 1.5.4 to our disposal. For the rest of this section we shall examine which simplifications and improvements of our theory are possible for automorphic tensor dilations. We can then use the fact that the representation of A ⊗ C on H ⊗ K is standard (see [Sa71, Ha75]). The problems of existence and classification of automorphic tensor dilations have been extensively investigated by B. K¨ ummerer, see [K¨ u85a, K¨ u85b, K¨ u88a, K¨ u88b] and Section 2.1.

1.6 The Automorphic Case

27

1.6.4 Duality for Automorphic Tensor Dilations Assume that we are given an automorphic tensor dilation of S. By the GNSconstruction, A is represented on a Hilbert space H with cyclic and separating vector ΩH corresponding to φ. In the same way C is represented on K with ΩK corresponding to ψ. Now define a unitary u1 ∈ B(H ⊗ K) associated to α1 : u1 a ⊗ c ΩH ⊗ ΩK := α1 (a ⊗ c) ΩH ⊗ ΩK . In particular u1 ΩH ⊗ ΩK = ΩH ⊗ ΩK . With H  H ⊗ ΩK ⊂ H ⊗ K we have u1 |H = v1 , where v1 is the isometry associated to j1 . It is easy to check that the automorphism Ad (u1 ) of B(H ⊗ K) leaves A ⊗ C invariant and Ad (u1 )|A⊗C = α1 . It also leaves A ⊗ C  invariant, defining an automorphism α1 := Ad (u1 )|A ⊗C  of A ⊗ C  . Theorem: For an automorphic tensor dilation j1 of T with given automorphism α1 we always have a (naturally associated) extended transition operator Z as well as a dual extended transition operator Z  . Duality is completed by an associated automorphic tensor dilation j1 for T  , and both j1 and j1 can be extended to J1 and J1 which are automorphic weak tensor dilations of Z and Z  . On the level of the automorphisms duality is nothing but inversion. Proof: The following definitions are indeed quite natural, and checking the statements about them is straightforward. They will also make clear the role of inversion. J1 : B(H) → B(H ⊗ K), Z : B(H) → B(H),

x → Ad (u1 )(x ⊗ 1I) x → pH J1 (x)|H

J1 is a weak tensor dilation of Z J1 |A = j1 Z(x) = pH u1 x ⊗ 1I j1 : A → A ⊗ C  ,

u∗1 |H

Z ∈ Z(T, φ) =

(v1 )∗

x ⊗ 1I v1

with v1 := u∗1 |H

a → Ad (u∗1 )(a ⊗ 1I) = (α1 )−1 (a ⊗ 1I)

j1 is a weak tensor dilation of T  : A → A ,

a → pH (α1 )−1 (a ⊗ 1I)|H

v1 is the associated isometry for j1 And the dual version: J1 : B(H) → B(H ⊗ K), Z  : B(H) → B(H),

x → Ad (u∗1 )(x ⊗ 1I) x → pH J1 (x)|H

J1 is a weak tensor dilation of Z 

28

1 Extensions and Dilations

J1 |A = j1

Z  ∈ Z(T  , φ )

Z  (x) = pH u∗1 x ⊗ 1I u1 |H = v1∗ x ⊗ 1I v1 j1 : A → A ⊗ C,

with v1 := u1 |H

a → Ad (u1 )(a ⊗ 1I) = α1 (a ⊗ 1I)

j1 is a weak tensor dilation of T v1 is the associated isometry for j1 2 Corollary: Z(a ) = pH α1 (a ⊗ 1I) pH = (T  )+ (a ) Z  (a) = pH (α1 )−1 (a ⊗ 1I) pH = T + (a) Proof: See 1.6.2. 2

1.7 Examples 1.7.1 Example 1.1 Revisited In Section 1.1 we considered extensions of the stochastic matrix
 1−λ λ with 0 < λ, µ < 1. µ 1−µ   Using the notation of the previous sections we
√have  A = B = A = B = 1 √µ . The vector Ω is cyclic C2 , G = H = C2 , Ω := ΩG = ΩH = √λ+µ λ and separating and the restriction φ of the vector state to the algebra A is normal and faithful. We can apply our theory for the stochastic map

 x1 (1 − λ)x1 + λx2 2 2  + → . T =T =T :C →C , x2 µx1 + (1 − µ)x2

In fact, in Proposition 1.1.4 we have characterized the set Z(T  , φ ). It consists of a family of stochastic maps Zc : M2 → M2 parametrized by c ∈ C, | c |≤ 1. Theorem 1.5.4 gives a one-to-one correspondence to equivalence classes of weak dilations jc : C2 → C2 ⊗ B(K), where K is a Hilbert space. These dilations may be computed using Arveson’s construction applied in the proof of Proposition 1.5.3 or they may be guessed from the considerations of Section 1.1. We give the result and then verify the required properties. Recall the vectors aij ∈ C3 = K, i, j = 1, 2 from Section 1.1: √     λµ 1 − λ   , a11 =  λ(1 − λ)  , λ(1 − µ) c a12 =   2 0 λ(1 − µ)(1− | c | )

1.7 Examples



a21

√   λµ =  − µ(1 − λ)  , 0



29



1 − µ , a22 =  − µ(1 − µ) c − µ(1 − µ)(1− | c |2 )

with c ∈ C, | c |≤ 1. Denote by pij the one-dimensional projection from K to C aij (complex conjugation in all components). Now we define jc : C2 → C2 ⊗ M3

 1 0 (1) b → ⊗ j (b) + ⊗ j (2) (b), 0 1 with homomorphisms j (1) , j (2) : C2 → M3 determined by

 1 1 (1) (2) j = p11 , j = p12 , 0 0

 0 0 j (1) = p21 , j (2) = p22 . 1 1 From a11 , a21  = a12 , a22  = 0 it follows that j (1) and j (2) are embeddings of C2 into M3 . Note that only the second of these depends on the parameter c. We  choose  the unit vector in K needed to specify the weak dilation as 1 ΩK =  0 . Preparing the following computations we note that 0 p11 ΩK = a11 p22 ΩK = a22

µ p12 ΩK = a12 λ λ p21 ΩK = a21 . µ If ψ is the vector state given by ΩK then

  1  0 Pψ jc (b) = Pψ ⊗ j (2) (b) ⊗ j (1) (b) + 1 0

 0 1 ΩK , j (2) (b)ΩK . = ΩK , j (1) (b)ΩK  + 1 0 Inserting the definition of j (1) and j (2) , it is now readily checked that Pψ jc = T , i.e. jc is a weak tensor dilation of T . Further, if       1 0 0

1 = ΩK =  0  , 2 =  1  , 3 =  0  0 0 1

30

1 Extensions and Dilations

are the canonical basis vectors of C3 , then by the definition of the vectors aij in Section 1.1 we have 3  aij = (a∗k )ji k , and Zc (x) =

3 k=1

k=1

ak x a∗k is a Kraus decomposition of Zc .

To compute the isometry vc associated to jc , assume that b = Then vc b ΩH = jc (b) ΩH ⊗ ΩK

=

=

=

=

b1 b2



∈ C2 .

   

 1 1 λ 1 0 ⊗ j (1) (b)  0  + ⊗ j (2) (b)  0  0 µ+λ 1 0 0  


1 µ 1 ⊗ (b1 p11 + b2 p21 )  0  µ+λ 0 0  
 1 λ 0 + ⊗ (b1 p12 + b2 p22 )  0  µ+λ 1 0



 √ 1  √ 1 1 ⊗ a11 + b2 λ ⊗ a21 b1 µ 0 0 µ+λ

 √  √ 0 0 + b1 µ ⊗ a12 + b2 λ ⊗ a22 1 1



 3 √ 1  ∗ √ 1 1 ∗ + (ak )12 b2 λ (ak )11 b1 µ 0 0 µ+λ k=1

 √ √ 0 0  ⊗ k +(a∗k )21 b1 µ + (a∗k )22 b2 λ 1 1

=




µ µ+λ

3 

a∗k (b ΩH ) ⊗ k .

k=1

This implies that vc (ξ) =

3 

a∗k (ξ) ⊗ k

for all ξ ∈ H,

k=1

which means that indeed vc is the isometry featuring in the Stinespring representation of Zc which corresponds to the Kraus decomposition Zc (x) = 3 ∗ k=1 ak x ak . But this means that with the jc defined above we have found a representative of the equivalence class of weak dilations which corresponds to Zc as described in Theorem 1.5.4.

1.7 Examples

31

1.7.2 Further Discussion Let us add some remarks, restricting to the case µ = λ = 12 for simplicity. In this case we have     1 1 1  1 , c a11 = 1 , a12 = 2 2  2 0 1− | c |     1 1 1 1 . a22 =   −c a21 =  −1  , 2 2 0 − 1− | c |2 For which c do j (1) (C2 ) and j (2) (C2 ) commute? One-dimensional projections, such as the pij involved in their definition, commute if and only if the corresponding vectors are either multiples of each other or orthogonal. Thus the only solutions to our question are c = 1 and c = −1. In these two cases we can easily write down equivalent versions of these dilations using only commutaC p21  C2 . Let us replace p11 , p21 tive algebras and restrict M3 to
Cp11
⊕  1 0 by the canonical basis vectors , of C2 . 0 1 For c = 1 we have p11 = p12 and p21 = p22 and thus j1 : C2 → C2 ⊗ C2




  1 b1 1 0 1  b= → b1 ⊗ + ⊗ b2 0 0 1 0



  1 0 0 0  . + b2 ⊗ + ⊗ 0 1 1 1 For c = −1 we have p11 = p22 and p21 = p12 and thus j−1 : C2 → C2 ⊗ C2




  1 b1 1 0 0  b= → b1 ⊗ + ⊗ b2 0 0 1 1



  1 0 0 1  . + b2 ⊗ + ⊗ 0 1 1 0 These are automorphic dilations in the sense of 1.6.3. In fact, we can define automorphisms α1 extending j1 and α−1 extending j−1 by the following selfexplaining diagrams. Compare also Section 4.2.



 1 1 1 0 ⊗ ← ⊗ 0 0 0 1 α1 :
 ↓
 ↑

 0 1 0 0 ⊗ → ⊗ 1 0 1 1

32

1 Extensions and Dilations

α−1





 1 1 1 0 ⊗ ← ⊗ 0 0 0 1 :
↓
 ↑

 0 0 0 1 ⊗ → ⊗ 1 1 1 0

The corresponding operators Zc are given as follows
 x11 x12 (with x = ∈ M2 ): x21 x22 
1 x11 + x12 + x21 + x22 0  Z1 (x) = = ΩH , xΩH  1I 0 x11 + x12 + x21 + x22 2 
1 x11 + x22 x12 + x21  Z−1 (x) = 2 x12 + x21 x11 + x22 If c = ±1 then j (1) (C2 ) and j (2) (C2 ) do not commute. As shown in Proposition 1.4.1, for an equivalent version of the dilation we have on the minimal representation space a common unitary transform w for these embeddings and their relative position is thus unchanged. This does not exclude the possibility to realize equivalent non-minimal versions of these dilations by commuting algebras. 1.7.3 A Class of Maps on M2 We give another class of examples. Consider the following family of stochastic maps, studied in [K¨ u85a, Ep91]: Tρ : (M2 , tr) → (M2 , tr)

 a11 a12 a11 ρ a12 a= → , a21 a22 ρ a21 a22 ρ ∈ C with | ρ |≤ 1. Let ρ be fixed from now on. This can be put into our setting by choosing G = H = C2 ⊗ C2 and A = B = M2 = A = B  , with A = B acting as M2 ⊗ 1I and A = B  acting as 1I ⊗ M2 . Then a vector representing tr is given by



 1  1 1 0 0  √ Ω := ΩG = ΩH = ⊗ + ⊗ . 0 0 1 1 2 This gives the GNS-representation. Tρ : A → B and Tρ : B  → A are given by the same formula, i.e. we also have
  
  b11 b12 b11 ρ b12 Tρ : b = →  . b21 b22 ρ b21 b22

1.7 Examples

33

All automorphic tensor dilations (see 1.6.3 and Section 2.1) have the form
a=

j1 : M2 → M2 ⊗ C = M2 (C) 
 

a11 a12 1I 0 1I 0 a11 1I a12 1I · , → · a21 a22 0u 0 u∗ a21 1I a22 1I

where u is a unitary in some von Neumann algebra C, and with a faithful normal state ψ on C satisfying ψ(u) = ρ (see [K¨ u85a] and [Ep91], 3.3). Now j1 (A) ⊂ M2 ⊗ span{1I, u, u∗ }, which suggests to define an equivalent version of j1 as follows. Let µψ be the probability measure on the unit circle C satisfying C z n dµψ = ψ(un ) for all n ∈ Z, i.e. µψ is the spectral measure for u with respect to ψ. Now define K := L2 (C, µψ ) and ΩK = 1I (constant function on C). Identifying u with multiplication by the variable z on L2 (C, µψ ) we have j1 : M2 → L∞ (C, µψ , M2 ) ⊂ M2 ⊗ B(K) (functions with values in M2 ). Let us compute the associated isometry v1 : H → H ⊗ K C ⊗ C2 → C2 ⊗ C2 ⊗ L2 (C, µψ ) = L2 (C, µψ , C2 ⊗ C2 )
 a11 a12 2 2 (functions with values in C ⊗ C ). For a = ∈ A we have a21 a22 



1  a11 1 a12 0  ∈ C2 ⊗ C2 , ⊗ + ⊗ a ΩH = √ a21 a22 0 1 2 2

1  j1 (a) ΩH ⊗ΩK = √ 2


i.e. for ξ=




ξ11 ξ21




 1 z a12 0  a11 ∈ L2 (C, µψ , C2 ⊗C2 ), ⊗ + ⊗ z a21 a22 0 1






 1 ξ12 0 ⊗ + ⊗ ∈ C2 ⊗ C2 ξ22 0 1

we get





 ξ11 1 z ξ12 0 ⊗ + ⊗ ∈ L2 (C, µψ , C2 ⊗ C2 ) z ξ21 ξ22 0 1



 1 1 1 0  ξ11 ⊗ ⊗ 1I + ξ12 ⊗ ⊗z 0 0 0 1



 0 1 0 0 + ξ21 ⊗ ⊗ z + ξ22 ⊗ ⊗ 1I ∈ C2 ⊗ C2 ⊗ L2 (C, µψ ). 1 0 1 1

v1 ξ =

We see that two dilations of this type are equivalent (i.e. v2 = (1I ⊗ w)v1 according to Proposition 1.4.1) if and only if there is a partial isometry w : L2 (C, µ1 ) → L2 (C, µ2 ) mapping 1I → 1I, z → z, z → z. This is the case if

34

1 Extensions and Dilations

  and only if (additionally to the first Fourier coefficient ρ = C zdµ1 = C zdµ2 characterizing Tρ ) the second Fourier coefficients of µ1 and µ2 are the same:   z 2 dµ1 = z 2 dµ2 =: ρ2 . C

C

It is well-known that given ρ one is left with the freedom to choose ρ2 ∈ {c ∈ C : | c − ρ2 |≤ 1− | ρ |2 }. This follows from the parametrization of positive definite sequences by Schur parameters (see [FF90], XV). We conclude that for | ρ |< 1 there is a continuum of different solutions. Let us compute the corresponding dual extended transition operators Z  : M2 ⊗ M2 → M2 ⊗ M2 . If

    b11 b12 b11 b12  b= , b = ∈ M2 , b21 b22 b21 b22





 1 ξ12 0 ξ= ⊗ + ⊗ , ξ22 0 1 



1 η12 0 η11 ⊗ + ⊗ , η= η21 η22 0 1 ξ11 ξ21



then we have for all x = b ⊗ b ∈ M2 ⊗ M2 ξ, Z  (x)η = ξ, v1∗ x ⊗ 1I v1 η = v1 ξ, b ⊗ b ⊗ 1I v1 η







 ξ 1 z ξ12 0 η11 1 z η12 0 =  11 ⊗ + ⊗ ,b ⊗b +b ⊗b  z ξ21 ξ22 z η21 η22 0 1 0 1


 
 ξ11 ξ11 η11 z η12 =  ,b  b11 +  ,b  b12 z ξ21 z η21 z ξ21 η22


 
 z ξ12 z ξ12 η11 z η12 + ,b  b21 +  ,b  b22 ξ22 z η21 ξ22 η22 = b11 b11 ξ 11 η11 + b12 b11 ξ 11 η21 ρ + b21 b11 ξ 21 η11 ρ + b22 b11 ξ 21 η21 + b11 b12 ξ 11 η12 ρ + b12 b12 ξ 11 η22 + b21 b12 ξ 21 η12 ρ2 + b22 b12 ξ 21 η22 ρ + b11 b21 ξ 12 η11 ρ + b12 b21 ξ 12 η21 ρ2 + b21 b21 ξ 22 η11 + b22 b21 ξ 22 η21 ρ + b11 b22 ξ 12 η12 + b12 b22 ξ 12 η22 ρ + b21 b22 ξ 22 η12 ρ + b22 b22 ξ 22 η22 .

1.7 Examples

We conclude that
  10  Z ⊗ b 00
   01 Z ⊗ b 00
   00 Z ⊗ b 10
   00 Z ⊗ b 01


=
=


10 00 01 00




⊗




⊗






b11 ρ b12 ρ b21 b22




=

ρ b11 b12 ρ2 b21 ρ b22



10 00



35

⊗ Tρ (b ),

 ρ b11 ρ2 b12 b21 ρ b22

 
 00 b11 ρ b12 00 = ⊗ = ⊗ Tρ (b ). 01 ρ b21 b22 01 =

Note also that  Z b⊗ 

 Z b ⊗

00 10





10 00 00 01

⊗

 

 


= Tρ (b) ⊗
= Tρ (b) ⊗

10 00 00 01

 ,  ,

which is an instance of Proposition 1.6.2. Note that for | ρ |< 1 the stochastic map Z  is different from the tensor product Tρ ⊗ Tρ . In fact, the vector state given by Ω is not invariant for Tρ ⊗ Tρ . Because the setting is symmetric in algebra and commutant one gets analogous formulas for the extended transition operator Z. In particular the set Z of extensions (see Section 1.2) is parametrized by ρ2 varying in a disc. 1.7.4 Maps on Mn An automorphism α of the matrix algebra Mm can always be written as α(x) = Ad U (x) = U xU ∗ , where U = (Uij ) ∈ Mm is unitary. The GNSm m Hilbert space  with respect to the tracial state is C ⊗ C with cyclicm vector − 12 Ω=m k δk ⊗ δk , where δk are the canonical basis vectors in C . Here Mm acts as Mm ⊗ 1I. We have a unitary operator u : Cm ⊗ Cm → Cm ⊗ Cm associated to α(x) given by u (x Ω) = α(x) Ω. ¯ where (U ¯ )ij = Uij (complex conjugation). Lemma: u = U ⊗ U,  ¯sj Ers and thus Proof: For a matrix unit Eij we have U Eij U ∗ = r,s Uri U  1 ¯sj Ers δk ⊗ δk α(Eij ) Ω = m− 2 Uri U r,s,k

=m

− 12



¯kj δr ⊗ δk Uri U

r,k 1

¯ δj = U ⊗ U ¯ Eij Ω. = m− 2 U δi ⊗ U

2

Now assume that Mm = Mn ⊗Md and α1 = Ad U with U ∈ M n ⊗Md . Then it d is more convenient to represent U as a block matrix, i.e. U = i,j=1 Uij ⊗Eij

36

1 Extensions and Dilations

with Uij ∈ Mn . We consider j1 = α1 |Mn ⊗1I as a dilation of T : Mn → Mn , b → Ptr (j1 (b)). Then a Kraus decomposition of T can be computed as follows:  ∗ Uij b Ukl ⊗ Eij Elk ) T (b) = Ptr (U b ⊗ 1I U ∗ ) = Ptr ( = Ptr (

i,j,k,l



∗ Uij b Ukj ⊗ Eik ) = d−1



i,j,k

∗ Ukj b Ukj .

j,k

We want to compute Z  : Mn ⊗ Mn → Mn ⊗ Mn , where Mn ⊗ 1I is the original ¯ algebra and 1I ⊗ Mn is the commutant. Using the lemma we have u1 = U ⊗ U n n n n d d and the associated isometry v1 : C ⊗ C → C ⊗ C ⊗ C ⊗ C satisfies  1 δk ⊗ δk v1 ξ ⊗ ξ  = u1 ξ ⊗ ξ  ⊗ d− 2 1

= d− 2



k

¯  ⊗ δk ) U (ξ ⊗ δk ) ⊗ U(ξ

k

− 12

=d

  

   ¯jk ξ  ⊗ δj . U Uik ξ ⊗ δi ⊗

i

k

j

Now from ξ ⊗ ξ  , Z  (b ⊗ b )η ⊗ η   = v1 ξ ⊗ ξ  , b ⊗ b v1 η ⊗ η    ¯jk η   ¯jk ξ  , b Uik η ⊗ b U = d−1 Uik ξ ⊗ U k,k i,j

−1

=d



∗ ¯jk η ⊗ η   ¯ ∗ b U ξ ⊗ ξ  , Uik b Uik ⊗ U jk

k,k i,j

we conclude that Z  (x)

=

d 

aij x a∗ij

i,j=1

with

aij = d−1

d 

∗ ∗ ¯jk Uik ⊗U

for all i, j.

k=1

Similarly (with U and U ∗ interchanged) we get Z(x)

=

d 

bij x b∗ij

i,j=1

with

−1

bij = d

d 

¯kj Uki ⊗ U

for all i, j.

k=1

It is a nice exercise to check directly that Z extends T .