Hilbert Modules and Stochastic Dilation of a Quantum Dynamical Semigroup on a von Neumann Algebra

Commun. Math. Phys. 205, 377 – 403 (1999) Communications inMathematical

Physics© Springer-Verlag 1999

Hilbert Modules and Stochastic Dilation of a QuantumDynamical Semigroup on a von Neumann Algebra

Debashish Goswami1,?, Kalyan B. Sinha2

1 Indian Statistical Institute, Delhi Centre, 7, SJSS Marg, New Delhi - 110016, India.E-mail: [email protected]

2 Indian Statistical Institute, Delhi Centre and Jawaharlal Nehru Centre for Advanced Scientific Research,Bangalore, India

Received: 15 June 1998/ Accepted: 4 March 1999

Abstract: A general theory for constructing a weak Markov dilation of a uniformlycontinuous quantum dynamical semigroupTt on a von Neumann algebraA with respectto the Fock filtration is developed with the aid of a coordinate-free quantum stochasticcalculus. Starting with the structure of the generator ofTt , existence of canonical structuremaps (in the sense of Evans and Hudson) is deduced and a quantum stochastic dilationof Tt is obtained through solving a canonical flow equation for maps on the right FockmoduleA⊗0(L2(R+, k0)), wherek0 is some Hilbert space arising from a representationof A′. This gives rise to a∗-homomorphismjt of A. Moreover, it is shown that everysuch flow is implemented by a partial isometry-valued process. This leads to a naturalconstruction of a weak Markov process (in the sense of [B-P]) with respect to Fockfiltration.

1. Introduction

Given a uniformly continuous quantum dynamical semigroupTt on a von NeumannalgebraA, a general theory for constructing a (weak) Markov dilation ofTt with respectto the Fock-filtration is developed. While doing this, we introduce in a natural way (inSect. 2) a coordinate-free stochastic calculus and quantum Ito formula which combinesthe initial space and the Fock space. The Sect. 3 is devoted to the solution of a class ofquantum stochastic differential equations, both of the Hudson–Parthasarathy as well asthe Evans–Hudson types. Here we find that the language of Hilbert rightA-modules isvery useful to describe a quantum stochastic flow equation which is now a differentialequation for maps on the moduleA⊗0(L2(R+, k0)), wherek0 is a certain Hilbert spaceassociated with a representation ofA′. The proof of the∗-homomorphism property ofthe solutionjt (x) of the flow equation (see [Mo-S]) becomes particularly transparent inthis language needing no extra assumptions as in [Mo-S]. We also prove that every such

? Research partially supported by the National Board of Higher Mathematics, India.

378 D. Goswami, K. B. Sinha

flow can be implemented by a partial isometry-valued process, and that{jt , jt (1)Et }t≥0is an example of a weak Markov process (see [B-P]), whereEt denotes the conditionalexpectation in Fock space . We would like to add that there is some overlap in the studyof Evans–Hudson flows in Sect. 3 with the work of Lindsay and Wills ([L-W]).

Let us consider a unital von Neumann algebraA in B(h), whereh is a (not neces-sarily separable) Hilbert space. Let(Tt )t≥0 be a uniformly continuous quantum dynam-ical semigroup (that is, a contractive, normal, completely positive semigroup) with thebounded generatorL : A → A. It is known due to Christensen and Evans ([C-E]) thatthere exist a Hilbert spaceK, bounded operatorR : h → K, normal∗-representationπ : A → B(K) and` ∈ A such that,

L(x) = R∗π(x)R + `∗x + x`. (1.1)

We say that(Tt )t≥0 is conservative ifL(1) = 0, which is equivalent to saying thatTt (1) = 1 for t ≥ 0. It is simple to note that in case whenπ(1) = 1 andL(1) = 0,

L(x) = R∗π(x)R − 1

2R∗Rx − 1

2xR∗R + i[H, x], (1.2)

whereH = i(l+ 12R

∗R), a self-adjoint element ofA, andR∗π(x)R ∈ A for all x ∈ A.

By replacingK by K ≡ π(1)K; andR by R ≡ π(1)R, it is clear that

R∗π(x)R + `∗x + x` = R∗π(1)π(x)π(1)R + `∗x + x` = L(x).

Since the range ofπ(x) is contained inK for all x ∈ A, π may be thought of as a∗-representation fromA to B(K). In view of this, we may assume thatπ(1) = 1. Thatwe can also putL(1) = 0 follows by a slight modification of the reasoning in Theorem2.13 of [B-P].

It has been observed elsewhere ([S, A-L]) that the symmetric or bosonic Fock spaceoften acts as a model for a heat-bath or reservoir. While the evolution of the state of thecombined system, consisting of the observed physical object and the reservoir to whichit is coupled, is given by a quantum stochastic differential equation (or equivalently inthe dual picture, by a quantum stochastic flow equation of the observables), that of theobserved subsystem is given by some kind of averaging or expectation with respect tothe Fock variables. Thus though the total evolution is not given by a group, the evolutionof the observed subsystem is given by a quantum dynamical semigroup. However, inmost cases of physical interest, the semigroup is expected to be only strongly continuousand not uniformly continuous as has been assumed here. In this study, we are interestedonly in the structural aspects of the theory and the more realistic cases of a stronglycontinuous dynamical semigroup can often be discussed as a suitable limit of a sequenceof uniformly continuous ones and will be treated elsewhere.

2. A Coordinate-Free Quantum Stochastic Calculus

Thus we shall assume thatπ(1) = 1K, L(1) = 0 for the rest of the article. Our presentaim is to develop a coordinate-free theory of quantum stochastic calculus, which will beneeded for constructing a dilation of(Tt )t≥0.

Hilbert Modules and Stochastic Dilation of a Quantum Dynamical Semigroup 379

2.1. Basic processes.LetH1,H2 be two Hilbert spaces andAbe a (possibly unbounded)linear operator fromH1 to H1 ⊗ H2 with domainD. For eachf ∈ H2, we define alinear operator〈f,A〉 with domainD and taking value inH1 such that,

〈〈f,A〉u, v〉 = 〈Au, v ⊗ f 〉 (2.1)

for u ∈ D, v ∈ H1. This definition makes sense because we have,|〈Au, v ⊗ f 〉| ≤‖Au‖ ‖f ‖ ‖v‖, and thusH1 3 v → 〈Au, v ⊗ f 〉 is a bounded linear functional.Moreover,‖〈f,A〉u‖ ≤ ‖Au‖ ‖f ‖, for all u ∈ D, f ∈ H2. Similarly, for each fixedu ∈ D, v ∈ H1, H2 3 f → 〈Au, v ⊗ f 〉 is a bounded linear functional, and hencethere exists a unique element ofH2, to be denoted byAv,u, satisfying

〈Av,u, f 〉 = 〈Au, v ⊗ f 〉 = 〈〈f,A〉u, v〉. (2.2)

We shall denote by〈A, f 〉 the adjoint of〈f,A〉, whenever it exists. Clearly, ifA isbounded, then so is〈f,A〉 and‖〈f,A〉‖ ≤ ‖A‖ ‖f ‖. Similarly, for anyT ∈ B(H1⊗H2)

andf ∈ H2, one can defineTf ∈ B(H1,H1 ⊗ H2) by settingTf u = T (u ⊗ f ). Forany Hilbert spaceH, we denote by0(H) and0f (H) the symmetric Fock space andthe full Fock space ofH. For a systematic discussion of such spaces, the reader maybe referred to [Par], from which we shall borrow all the standard notations and results.Now, we define a mapS : 0f (H2) → 0(H2) by setting,

S(g1 ⊗ g2 ⊗ · · · ⊗ gn) = 1

(n− 1)!∑σ∈Sn

gσ(1) ⊗ · · · ⊗ gσ(n), (2.3)

and linearly extending it toH⊗n

2 , whereSn is the group of permutations ofn objects.Clearly,‖S|H⊗n

2‖ ≤ n. We denote byS the operator 1H1 ⊗ S.

Let us now define the creation operatora†(A) abstractly which will act on the linearspan of vectors of the formvg⊗n

andve(g) (whereg⊗ndenotesg ⊗ · · · ⊗ g︸︷︷︸

n times

), n ≥ 0,

with v ∈ D, g ∈ H2. It is to be noted that we shall often omit the tensor product symbol⊗ between two or more vectors when there is no confusion. We define,

a†(A)(vg⊗n

) = 1√n+ 1

S((Av)⊗ g⊗n

). (2.4)

It is easy to observe that∑n≥0

1

n! ‖a†(A)(vg⊗n

)‖2 < ∞, which allows us to define

a†(A)(ve(g)) as the direct sum⊕n≥0

1

(n!) 12

a†(A)(vg⊗n

). We have the following simple

but useful observation, the proof of which is straightforward and hence omitted.

Lemma 2.1.1. For v ∈ D, u ∈ H1, g, h ∈ H2,

〈a†(A)(ve(g)), ue(h)〉 = 〈Au,v, h〉〈e(g), e(h)〉 = d

dε〈e(g + εAu,v), e(h)〉|ε=0.

(2.5)


In the same way, one can define annihilation and number operators inH1 ⊗ 0(H2)

for A ∈ B(H1,H1 ⊗ H2) andT ∈ B(H1 ⊗ H2) as:

a(A)ue(h) = 〈A, h〉 ue(h),3(T )ue(h) = a†(Th)ue(h).

One can also verify that in this casea†(A) is the adjoint ofa(A) on H1 ⊗ E(H2),whereE(H2) is the linear span of exponential vectorse(g), g ∈ H2. Next, to definethe basic processes, we need some more notations. Letk0 be a Hilbert space,k =L2(R+, k0), kt = L2([0, t])⊗ k0, k

t = L2((t,∞))⊗ k0, 0t = 0(kt ), 0t = 0(kt ), 0

= 0(k). We assume thatR ∈ B(h, h⊗ k0) and defineR1t : h⊗ 0t → h⊗ 0t ⊗ kt fort ≥ 0 and a bounded interval1 in (t,∞) by,

R1t (uψ) = P((1h ⊗ χ1)(Ru)⊗ ψ),

whereχ1 : k0 → kt is the operator which takesα to χ1(·)α for α ∈ k0, andP is thecanonical unitary isomorphism fromh⊗k⊗0 toh⊗0⊗k. We define the creation fielda

†R(1) on either of the domains consisting of the finite linear combinations of vectors

of the formut ⊗ f t⊗nor of ut ⊗ e(f t ) for ut ∈ h⊗ 0t , f

t ∈ 0t , n ≥ 0, as:

a†R(1) = a†(R1t ), (2.6)

wherea†(R1t ) carries the meaning discussed before Lemma 2.1.1, withD = E(H2),H1 = h⊗ 0t , H2 = kt . Similarly the two fieldsaR(1) and3T (1) can be defined as:

aR(1)(ute(ft )) = ((

∫1

〈R, f (s)〉ds)ut )e(f t ), (2.7)

and forT ∈ B(h⊗ k0),

3T (1)(ute(ft )) = a†(T 1f t )(ut e(f

t )). (2.8)

In the above,T 1f t

: h⊗ 0t → h⊗ 0t ⊗ kt is defined as,

T 1f t (uαt ) = P(1 ⊗ χ1)(T (uft )⊗ αt ), (2.9)

andT ∈ B(h ⊗ L2((t,∞), k0)) is given by,T (uϕ)(s) = T (uϕ(s)), s > t , andχ1 isthe multiplication byχ1(·) onL2((t,∞), k0). Clearly,‖T ‖ ≤ ‖T ‖, which makesT 1

f t

bounded. We note here that objects similar toaR(.), a†R(.) and3T (.) were used in [H-

P2], however in a coordinatized form. In what follows, we shall assume that(Ht )t≥0 and(H ′

t )t≥0 are two operator-valued Fock-adapted processes (in the sense of [Par]), havingall vectors of the formve(ft )ψt in their domains, wherev ∈ h, ft ∈ kt , ψt ∈ 0t . Wealso assume that there exist constantsc(t, f ) andc′(t, f ) such that fort ≥ 0,

sup0≤s≤t

‖Hs(ue(f ))‖ ≤ c(t, f )‖u‖, sup0≤s≤t

‖H ′s(ue(f ))‖ ≤ c′(t, f )‖u‖. (2.10)

We shall often denote an operatorB and its trivial extensionB⊗I to some bigger

space by the same notation, unless there is any confusion in doing so. We also denote theunitary isomorphism fromh⊗k0 ⊗0(k) ontoh⊗0(k)⊗k0 and that fromh⊗k⊗0(k)


ontoh ⊗ 0(k) ⊗ k by the same letterP . Clearly,HtP acts on any vector of the formw ⊗ e(g), wherew ∈ h⊗algk0, g ∈ k and sup

0≤s≤t‖HsP (we(g))‖ ≤ c(t, g)‖w‖. This

allows one to extendHtP on the whole of the domain containing vectors of the formwe(g), w ∈ h⊗k0, g ∈ k. We denote this extension again byHtP . Similarly we defineH ′t P . WhenP is taken to be the isomorphism fromh⊗ k ⊗ 0(k) ontoh⊗ 0(k)⊗ k,

we defineHtP andH ′t P in an exactly parallel manner.

Next we prove a few preliminary results which will be needed for establishing thequantum Ito formula in the next subsection.

Lemma 2.1.2. Let1,1′ ⊆ (t,∞) be intervals of finite length,R, S ∈B(h, h⊗ k0); u, v ∈ h; g, f ∈ k. Then we have,

〈Hta†R(1)(ve(g)), H

′t a

†S(1

′)(ue(f ))〉= e〈gt ,f t 〉{〈HtR1t (ve(gt )), H ′

t S1′t (ue(ft ))〉

+ 〈〈f t ,HtR1t 〉ve(gt ), 〈gt ,H ′t S1′t 〉ue(ft )〉}

=∫

1∩1′〈(HtPR)(ve(g)), (H ′

t P S)(ue(f ))〉ds

+∫1

∫1′

〈〈f (s),HtPR〉(ve(g)), 〈g(s′), H ′t P S〉(ue(f ))〉ds ds′. (2.11)

Proof. For the present proof, we make the convention of writingdf (ε)dε

|ε=ε0 for the

limit limn→∞n(f (ε0 + 1

n)− f (ε0)) whenever it exists, andR1 will denote(1 ⊗ χ1)R ∈

B(h, h⊗ k) for R ∈ B(h, h⊗ k0). Let us now choose and fix orthonormal bases{eν}ν∈Jand{kα}α∈I of h ⊗ 0t and0t respectively(t ≥ 0). We also choose subsetsJ0 andI0,which are at most countable, ofJ andI respectively as follows. LetJ0 be such that〈HtPR1(ve(g)), eν ⊗ kα〉 = 0 = 〈eν ⊗ kα, H

′t P S1′(ue(f ))〉 for all α ∈ I whenever

ν 6∈ J0. Fixing thisJ0, we chooseI0 to be the union ofIν,n, ν ∈ J0, n = 1,2, . . . ,∞,such that

〈e(gt + 1

n(HtPR1)eν,ve(gt )), kα〉 = 0 = 〈kα, e(f t + 1

n(H ′

t P S1′)eν ,ue(ft ))〉for all α 6= Iν,n whenn < ∞, and

〈e(gt ), kα〉 = 0 = 〈kα, e(f t )〉 for α 6= Iν,∞.

We have now,

〈Hta†R(1)(ve(g)), H ′

t a†S(1′)(ue(f ))〉

=∑ν∈J0α∈I0

〈Hta†R(1)(ve(g)), eν ⊗ kα〉〈eν ⊗ kα, H

′t a

†S(1′)(ue(f ))〉

=∑ν∈J0α∈I0

(d

dε〈e(gt + ε(HtPR1)eν ,ve(gt )), kα〉|ε=0)× (

d

dη〈e(f t + η(H ′

t P S1′ )eν ,ue(ft )), kα〉|η=0)

=∑ν∈J0

∂2

∂ε∂η(∑α∈I0

〈e(gt + ε(HtPR1)eν ,ve(gt )), kα〉〈kα, e(f t + η(H ′t P S1′ )eν ,ue(ft ))〉|ε=0=η)


=∑ν∈J0

∂2

∂ε∂η(〈e(gt + ε(HtPR1)eν ,ve(gt )), e(f

t + η(H ′t P S1′ )(eν ,ue(ft ))〉|ε=0=η)

=∑ν∈J0

e〈gt ,f t 〉(〈(HtPR1)eν ,ve(gt ), (H ′t P S1′ )eν ,ue(ft )〉

+ 〈(HtPR1)eν ,ve(gt ), f t 〉〈gt , (H ′t P S1′ )eν ,ue(ft )〉).

Before proceeding further, let us justify the intermediate step in the above calcu-lations, which involves an interchange of summation and limit, by appealing to thedominated convergence theorem. Indeed, for any fixedα ∈ I0, ψ, ψ ′ ∈ kt , if we write

k(n)α for the projection ofkα on kt

⊗n(n ≥ 0), then〈e(gt + εψ), kα〉 can be expressed

as∑i≥0 c

(α)i εi , wherec(α)i = ∑

n≥i 1√n!(ni

)〈gt⊗(i) ⊗ ψ⊗(n−i), k(n)α 〉, wheregt⊗(i) ≡

gt ⊗ · · · ⊗ gt︸︷︷︸i−times

, andψ⊗(n−i) ≡ ψ ⊗ · · · ⊗ ψ︸︷︷︸(n−i)−times

. It can be easily verified that the above is an

absolutely summable power series inε, converging uniformly forε ∈ [0,M], say, forany fixedM > 0. Similar analysis can be done for〈kα, e(f t + ηψ ′)〉. By Mean ValueTheorem and some straightforward estimate, we have that forε, η, ε′, η′ in [0,M],

1

|(ε − ε′)(η − η′)|∑α∈I0

|(〈e(gt + εψ), kα〉 − 〈e(gt + ε′ψ), kα〉)

×(〈kα, e(f t + ηψ ′)〉 − 〈kα, e(f t + η′ψ ′)〉)|

≤∑

n≥0,m≥0,0≤i≤n,0≤j≤m

i.j.Mi+j−2

√n!m!

(n

i

)(m

j

)

×∑α∈I0

|〈gt⊗(i) ⊗ ψ⊗(n−i), k(n)α 〉〈k(n)α , f t⊗(i) ⊗ ψ ′⊗(m−j) 〉|

≤∑n,m,i,j

ijMi+j−2

√n!m!

(n

i

)(m

j

)‖gt⊗(i) ⊗ ψ⊗(n−i)‖ ‖f t⊗(j) ⊗ ψ ′⊗(m−j) ‖

[since{k(n)α }α∈I0 are mutually orthogonal for any fixedn,

with ‖k(n)α ‖ ≤ 1 ∀α]≤∑n≥0m≥0

mn‖gt‖ ‖f t‖ (M‖gt‖ + ‖ψ‖)n−1(M‖f t‖ + ‖ψ ′‖)m−1

√m!n! < ∞.

This allows us to apply dominated convergence theorem.Let us now choose a countable subsetI ′

0 of I so that 0= 〈(HtPR1)eν,ve(gt ), kα〉 =〈kα, (H ′

t P S1′)eν ,ue(ft )〉 for α not in I ′0, for all ν ∈ J0.


Clearly, we have∑ν∈J0

〈(HtPR1)eν,ve(gt ), (H ′t P S1′)eν ,ue(ft )〉

=∑

ν∈J0,α∈I ′0

〈(HtPR1)(ve(gt )), eν ⊗ kα〉〈eν ⊗ kα, (H′t P S1′)(ue(ft ))〉

= 〈(HtPR1)(ve(gt )), (H ′t P S1′)(ue(ft ))〉.

We choose sequencesω(n), ω′(n) of vectors which can be written as finite sums of theform,ω(n) = ∑

v(n)i ⊗β(n)i , ω′(n) = ∑

i u(n)i ⊗α(n)i , whereu(n)i , v

(n)i ∈ h, β(n)i , α

(n)i ∈

k0, andω(n) → Rv, ω′(n) → Su asn → ∞. Then we have,

‖HtP (1 ⊗ χ1) (ω(n) ⊗ e(gt ))−HtPR1(ve(gt ))‖

≤ c(t, g)‖ω(n) − (Rv)‖ |1| → 0 asn → ∞,

where|1| denotes the Lebesgue measure of1. Similarly,‖H ′

t P (1⊗χ1′) (ω′(n) ⊗e(ft ))− (H ′t P S1′)(ue(ft ))‖ → 0 asn → ∞. Hence we obtain

〈(HtPR1)(ve(gt )), (H ′t P S1′(ue(ft ))〉

= limn→∞〈HtP (1 ⊗ χ1)(ω

(n)e(gt )),H′t P (1 ⊗ χ1′)(ω′(n)e(ft ))〉

= limn→∞

∫〈Ht(

∑i

v(n)i ⊗ e(gt )⊗ β

(n)i ), H ′

t (∑i

u(n)i ⊗ e(ft )⊗ α

(n)i )〉χ1∩1′(s)ds

= limn→∞ |1 ∩1′|〈(HtP )(ω(n)e(gt )), (H ′

t P )(ω′(n)e(ft ))〉

= |1 ∩1′|〈(HtPR)(ve(gt )), (H ′t P S)(ue(ft ))〉

=∫

1∩1′〈HtPR(ve(gt )),H ′

t P S(ue(ft ))〉ds.

Moreover, ∑ν∈J0

〈(HtPR1)eν,ve(gt ), f t 〉〈gt , (H ′t P S1′)eν ,ue(ft )〉

=∑ν∈J0

〈〈f t ,HtPR1〉(ve(gt )), eν〉〈eν, 〈gt ,H ′t P S1′ 〉 (ue(ft ))〉

= 〈〈f t ,HtPR1〉(ve(gt )), 〈gt , (H ′t P S1′)〉(ue(ft ))〉,

where the last step follows by Parseval’s identity, noting the fact that forν 6∈ J0,〈〈f t ,HtPR1〉(ve(gt )), eν〉 = 0 because for suchν, 〈(HtPR1)(ve(gt )), eν ⊗ kα〉 = 0for all α ∈ I ; and similarly〈eν, 〈gt , (H ′

t P S1′)〉(ue(ft ))〉 = 0 ∀ν 6∈ J0.We complete the proof by observing that

〈f t ,HtR1t 〉 =∫1

〈f (s),HtPR〉ds, and

〈gt ,H ′t S1′t 〉 =

∫1′

〈g(s′),H ′t P S〉ds′.


To see this, it is enough to note that forω ∈ h, ht ∈ kt , we have,

〈〈f t ,HtR1t 〉(ve(gt )), ωe(ht )〉 =∫1

〈(HtPR)(ve(gt )), ωe(ht )⊗ f t (s)〉ds,

which can be justified by consideringω(n) as before and applying dominated convergencetheorem.Since1 ⊆ (t,∞) and hencef t (s) = f (s) for s ∈ 1, the above expression can nowbe written as∫

1

〈(HtPR)(ve(gt )), we(ht )f (s)〉ds =∫1

〈〈f (s),HtPR〉(ve(gt )), we(ht )〉ds.

This completes the proof.utRemark 2.1.3.If Ht andH ′

t are bounded, then (2.11) of Lemma 2.1.2 holds withu, v

replaced by arbitrary vectors inh⊗ 0t andf, g by the same inkt .

Lemma 2.1.4. LetT , T ′ ∈ B(h⊗ k0). Then we have,

〈(HtT 1gt )(ve(g)), (H ′t T

1′f t )(ue(f ))〉

=∫

1∩1′〈HtPT P ∗(ve(g)g(s)),H ′

t P T′P ∗(ue(f )f (s))〉ds,

and

〈gt ,H ′t T

1′f t 〉 =

∫1′

〈g(s),H ′t T

′f (s)〉ds,

whereT ′f (s) ∈ B(h⊗ 0(k), h⊗ 0(k)⊗ k0) is defined in (2.9).

The proof is omitted since it is very similar to that of Lemma 2.1.2.

Lemma 2.1.5. For η ∈ k0, 〈η,HtPR〉ve(g) = Ht((〈η,R〉v)e(g)), wherev ∈ h, g ∈k.

Proof. It is easy to see that by virtue of (2.10), for every fixedg, f ∈ k, η ∈ k0, t ≥ 0,〈Ht(ve(g)), ue(f )〉 = 〈v,Mtu〉 defines an operatorMt ∈ B(h). Let Mt = Mt ⊗ 1k0.Then we have, forw = v ⊗ α, w′ = u⊗ β;α, β ∈ k0, u, v ∈ h,

〈HtP (we(g)), P (w′e(f ))〉 = 〈w, Mtw′〉.

By the density ofh⊗algk0 in h ⊗ k0, we have that〈HtP (we(g)), P (w′e(f ))〉 =〈w, Mtw

′〉 for all w,w′ ∈ h⊗ k0. Thus

〈〈η,HtPR〉ve(g) ue(f )〉 = 〈HtP ((Rv)e(g)), ue(f )η〉= 〈HtP ((Rv)e(g)), P (uηe(f ))〉 = 〈Rv, Mt (uη)〉 = 〈Rv, (Mtu)⊗ η〉

= 〈〈η,R〉v, Mtu〉 = 〈Ht((〈η,R〉v)e(g)), ue(f )〉.This completes the proof, since the vectors of the formue(f ) are total inh⊗ 0(k). ut


2.2. Stochastic integrals and Left Quantum Ito formulae.Following [H-P1] and [Par],we call an adapted process(Ht )t≥0 satisfying sup

0≤s≤t‖Hsve(g)‖ ≤ c(t, g)‖v‖ (for all

v ∈ h, f ∈ k), to besimple if Ht is of the form,

Ht =m∑i=0

Htiχ[ti ,ti+1)(t),

wherem is an integer(≥ 1), and 0≡ t0 < t1 < · · · < tm < tm+1 ≡ ∞. If M denotesone of the four basic processesaR, a

†R and3T andtI , and if (Ht ) is simple, then we

define the left and right integralst∫

0HsM(ds) and

t∫0M(ds)Hs respectively in the natural

manner:

t∫0

HsM(ds) =m∑i=0

HtiM([ti , ti+1) ∩ [0, t]),

t∫0

M(ds)Hs =m∑i=0

M([ti , ti+1) ∩ [0, t])Hti .

We callHt to beregular if t 7→ Ht(ue(f )) is continuous for all fixedu ∈ h andf ∈ k.Also note that ifHt is regular, then so is the extensionHtP . The next proposition givesthe quantum Ito formulae for simple integrands.

Proposition 2.2.1. Let u, v ∈ h; f, g ∈ L2(R+, k0);R, S,R′, S′ ∈ B(h, h ⊗ k0) andlet T , T ′ ∈ B(h ⊗ k0). Furthermore, assume thatE,F,G,H andE′, F ′,G′, H ′ areadapted simple processes satisfying the bound given at the beginning of this subsection,and that

Xt =∫ t

o

(Es3T (ds)+ FsaR(ds)+Gsa

†S(ds)+Hsds

),

X′t =

∫ t

0

(E′s3T ′(ds)+ F ′

saR′(ds)+G′sa

†S′(ds)+H ′ds

).

Then we have,

(i) (first fundamental formula)

〈Xtve(g), ue(f )〉 (2.12)

=∫ t

0ds⟨{⟨f (s), EsPTg(s)⟩+Fs 〈R, g(s)〉+Gs 〈f (s), S〉+Hs}(ve(g)), ue(f )

⟩.

(ii) (second fundamental formula or Quantum Ito formula). For this part suppose thatf, g ∈ k ∩ L∞(R+, k0). Then⟨

Xtve(g),X′t ue(f )

⟩=∫ t

0ds[⟨Xsve(g), {

⟨g(s), E′

sPT′f (s)

⟩+ F ′

s

⟨R′, f (s)

⟩+G′s

⟨g(s), S′⟩+H ′

s }(ue(f ))⟩]

+∫ t

0ds[⟨{⟨f (s), EsPTg(s)⟩+ Fs 〈R, g(s)〉 +Gs 〈f (s), S〉 +Hs }(ve(g)),X′

sue(f )⟩]+


+∫ t

0ds[⟨EsPTg(s)(ve(g)), E

′sPT

′f (s)(ue(f ))

⟩+ ⟨EsPTg(s)(ve(g)),G

′sPS

′(ue(f ))⟩

+⟨GsPS(ve(g)), E

′sPT

′f (s)(ue(f ))

⟩+ ⟨GsPS(ve(g)),G

′sPS

′(ue(f ))⟩]. (2.13)

Since the proof is very similar in spirit to the proof in [H-P1, Par], it is omitted. However,a comment with regard to the notation used above is in order. For example, for almost alls ∈ R+, the expressionEsPTg(s)(ve(g)) is to be understood as(Es ⊗ Ik0)P (Tg(s)v ⊗e(g)) = (Es⊗Ik0)P (T (v⊗g(s))⊗e(g)) ∈ h⊗0⊗k0.Thus the operatorEsPTg(s) mapsh⊗0 into h⊗0⊗ k0 and therefore by the discussion in Subsect. 2.1,

⟨f (s), EsPTg(s)

⟩mapsh⊗ 0 into h⊗ 0.

For a simple integrandHt , one can easily derive the following estimate by Gronwall’sLemma as in [Par].

Lemma 2.2.2. Letv, g,Xt be as in Proposition 2.2.1 (ii). Then one has

||Xtve(g)||2 ≤ et∫ t

0ds[||{EsPTg(s) +GsPS}(ve(g))||2

+||{⟨g(s), EsPTg(s)⟩+ Fs 〈R, g(s)〉 + 〈g(s),GsPS〉 +Hs}(ve(g))||2]. (2.14)

The extension of the definition ofXt to the case when the coefficients(E, F,G,H) areregular is now fairly standard and we have the following result:

Proposition 2.2.3. The integralXt with regular coefficients(E, F,G,H) exists as aregular process and the first and second fundamental formulae as well as the estimate(2.14) remain valid in such a case.

Corollary 2.2.4. (i) Assume that in the above propositionE,F,G,H satisfyC =sup0≤t≤t0(||Es || + ||Fs || + ||Gs || + ||Hs ||) < ∞. Suppose furthermore thatR, S, Tare functions oft such thatt 7→ R(t)u, S(t)u, T (t)ψ are strongly continuous forubelonging to a dense subspaceD ⊆ Dom(R(t))∩Dom(S(t)) ⊆ h andψ ≡ u⊗f (t) ∈Dom(T (t)) ⊆ h ⊗ k0 for all t ∈ [0, t0] with f ∈ C, the set of all bounded continuousfunctions inL2(R+, k0). Then the integral:

X(t) =∫ t

0(Es3T (ds)+ FsaR(ds)+Gsa

†S(ds)+Hsds)

defines an adapted regular process satisfying the estimate (2.14) with the constant co-efficientsT ,R, S replaced byT (s), R(s) andS(s) respectively.

(ii) In the first part of the corollary, if we replaceT (t), R(t), S(t) by adapted processesdenoted by the same symbols respectively but withD replaced byD ⊗alg E(Ct ), whereCt = C ∩ kt , then the conclusions as in (i) remain valid.

Proof. (i) Clearly we can choose sequencesT (n)(t), R(n)(t), S(n)(t) of simple coeffi-cients such thatT (n)(t)ψ,R(n)(t)u andS(n)(t)u converge toT (t)ψ,R(t)u andS(t)urespectively foru andψ as mentioned in the statement of the corollary. With these,we can define the integralX(n)(t) on u ⊗ e(f ) in a natural way using Proposition2.2.3. The hypotheses of continuity of the coefficients will allow one to pass to the limitin the integral as well by using the estimate (2.14). For example,|| ∫ t0 Es(3T (n) (ds) −3T (m)(ds))ue(f )||2 ≤ Cet ||e(f )||2 ∫ t0(1+||f (s)||2)||(T (n)(s)−T (m)(s))(uf (s))||2ds→ 0 asm, n → ∞. The estimate for||X(t)ue(f )|| will also follow by continuity.


(ii) This part follows easily from (i) with obvious adaptations. For instance, in the esti-mate above we shall have instead|| ∫ t0 Es[3T (n)(ds)−3T (m)(ds)]ue(f )||2 ≤Cet

∫ t0 ds(1 + ||f (s)||2)||(T (n)(s)− T (m)(s))(u⊗ e(fs)⊗ f (s))||2||e(f s)||2. ut

Remark 2.2.5.Instead of the left integral, one could as well have dealt with the rightintegral

∫ t0 M(ds)Hs and obtained formulae similar to those in Propositions 2.2.1 and

2.2.3.

Remark 2.2.6.(i) The Ito formulae derived in Proposition 2.2.3 can be put in a conve-nient symbolic form. Letπ0(x) denotex ⊗ 10(k) andπ0(x) denotex ⊗ 1k0. Then theIto formulae are:aR(dt)π0(x)a

†S(dt) = R∗π0(x)Sdt, 3T (dt)π0(x)3T ′(dt) = 3Tπ0(x)T ′(dt),

3T (dt)π0(x)a†S(dt) = a

†T π0(x)S

(dt), aS(dt)π0(x)3T (dt) = aT ∗π0(x)S(dt), and theproducts of all other types are 0.

(ii) The coordinate-free approach of quantum stochastic calculus developed here in-cludes the old coordinatized version as presented in [Par ]. Let us consider for example,for f ∈ L2(R+, k0), the operatorRf defined byRf u = u ⊗ f , for u ∈ h. It is easyto see that the creation and annihilation operatorsa†(Rf ), a(Rf ) coincide with thecreation and annihilation operatorsa†(f ) anda(f ) (respectively) defined in [Par ] as-sociated withf . Indeed, it is easy to see that(Rf )u,v = 〈u, v〉f for u, v ∈ h. Thus, forg, l ∈ k, 〈a†(Rf )ve(g), ue(l)〉 = d

dε(〈e(g + ε〈u, v〉f ), e(l)〉)|ε=0 = e〈g,l〉〈v, u〉〈f, l〉

= 〈v, u〉 ddε

〈e(g + εf ), e(l)〉|ε=0 = 〈v, u〉〈a†(f )e(g), e(l)〉. It is also clear that〈Rf , g〉= 〈f, g〉 and hencea(Rf )(ve(g)) = 〈f, g〉ve(g) = v(a(f )e(g)). Finally, the numberoperator3(T ) in the sense of [Par ] forT ∈ B(k) can be identified with31h⊗T .

3. Quantum Stochastic Differential Equations

3.1. Equations of Hudson–Parthasarethy (H-P) type.We consider the quantum stochas-tic differential equations (q.s.d.e.) of the form,

dXt = Xt(aR(dt)+ a†S(dt)+3T (dt)+ Adt), (3.1)

dYt = (aR(dt)+ a†S(dt)+3T (dt)+ Adt)Yt , (3.2)

with prescribed initial valuesX0⊗1 andY0⊗1 respectively, withX0, Y0 ∈ B(h), whereR, S ∈ B(h, h⊗ k0), T ∈ B(h⊗ k0), A ∈ B(h).

Proposition 3.1.1. The q.s.d.e.’s (3.1) and (3.2) admit unique solutions as regular pro-cesses.

Proof. The standard proofs of existence and uniqueness of solutions along the lines ofthat given in [Par] (Sect. 26 for the left equation and Sect. 27 for the right equation) workhere also. For the iteration process in the case of the right equation to make sense, onehas to take into account Corollary 2.2.4(ii) while interpreting the right integrals involved.The estimates in the same corollary also prove the regularity of the solutions as well asthe estimates : sup0≤s≤t {||Xsue(f )|| + ||Ysue(f )||} ≤ c(t, f )||u||, for u ∈ h, f ∈ Cand some constantc(t, f ). ut


We now consider a pair of special q.s.d.e.’s:

dUt = Ut(a†R(dt)+3T−1(dt)− aT ∗R(dt)+ (iH − 1

2R∗R)dt), U0 = I, (3.3)

dWt = (aR(dt)+3T ∗−I (dt)− a†T ∗R(dt)− (iH + 1

2R∗R)dt)Wt , W0 = I ; (3.4)

whereT is a contraction inB(h⊗ k0),R ∈ B(h, h⊗ k0) andH is a selfadjoint elementof B(h). Then we have:

Proposition 3.1.2 (see [Mo] also).

(i) The solutions of both Eq. (3.3) and (3.4) exist as regular contraction-valued pro-cesses andWt = U∗

t .(ii) If furthermoreT is a co-isometry, thenWt is an isometry, or equivalentlyUt is a

co-isometry.(iii) If T is unitary, thenUt is a unitary process.

Proof. (i) We have already seen the existence and uniqueness of the solutionsUt andWt

in the previous proposition. A simple calculation using the second fundamental formulain Proposition 2.2.3 and the right equation (3.4) give foru, v ∈ h andf, g ∈ C:

〈Wtve(g),Wtue(f )〉−〈ve(g), ue(f )〉=∫ t

0

⟨Wsve(g),

⟨g(s), (T T ∗−I )f (s)

⟩Wsue(f )

⟩,

(3.5)

which implies thatWt is a contraction for allt . SinceWt ∈ B(h ⊗ 0), an applicationof the first fundamental formula in Proposition 2.2.3 shows thatUt admits a boundedextension (which we denote also byUt ) to the whole ofh⊗ 0 and thatU∗

t = Wt .

(ii) The relation (3.5) shows clearly thatWt is an isometry if and only ifT is a co-isometry.

(iii) We note the following simple facts:

(a) For fixedg, f ∈ L2(R+, k0)∩L∞(R+, k0) andt ≥ 0, there exists a unique operatorMf,gt ∈ B(h) such that〈v,Mf,g

t u〉 = 〈Ut(ve(g)), Ut (ue(f ))〉.(b) SettingMf,g

t = Mf,gt ⊗ 1k0, we have for allw,w′ ∈ h⊗ k0,

〈UtP (we(g)), UtP (w′e(f ))〉 = 〈w, Mf,gt w′〉.

It is an easy computation using the Quantum Ito formulae (Proposition 2.2.3) to verifythat,

〈v,Mf,gt u〉 − 〈ve(g), ue(f )〉 =

t∫0

ds[−〈v,Mf,gs 〈T ∗R, f (s)〉u〉

− 〈u, 〈g(s), T ∗R〉Mf,gs u〉 + 〈v,Mf,g

s 〈g(s), R〉u〉 + 〈vg(s), Mf,gs ((T − 1)(uf (s)))〉

+ 〈v,Mf,gs (iH − 1

2R∗R)u〉 + 〈v, 〈R, f (s)〉Mf,g

s u〉

+ 〈vg(s), (T ∗ − 1)Mf,gs (uf (s))〉 + 〈v, (−iH− 1

2R∗R)Mf,g

s u〉


+ 〈Rv, Mf,gs (Ru)〉 + 〈Rv, Mf,g

s (T − 1)(uf (s))〉+ 〈vg(s), (T ∗ − 1)Mf,g

s (Ru)〉 + 〈vg(s), (T ∗ − 1)Mf,gs (T − 1)(uf (s))〉].

Let us consider mapsYi, i = 1, . . . ,5 from [0,∞) × B(h) to B(h) given by:Y1(s, A) = −A〈T ∗R, f (s)〉 − 〈g(s), T ∗R〉A + A〈g(s), R〉 − 1

2(AR∗R + R∗RA) +

i[A,H ]+〈R, f (s)〉A,Y2(s, A) = R∗AR, whereA = A⊗1k0,Y3(s, A) = 〈g(s), {(T ∗−1)A+A(T −1)+(T ∗−1)A(T −1)}f (s)〉,Y4(s, A) = 〈(T ∗−I )A∗R, f (s)〉,Y5(s, A) =〈(T ∗ − I )AR, g(s)〉∗. Then it follows that

〈v,Mf,gt u〉 − 〈ve(g), ue(f )〉 =

∫ t

s

〈v,5∑i=1

Yi(s,Mf,gs )u〉ds,

i.e.,

dMf,gt

dt=

5∑i=1

Yi(t,Mf,gt ).

We also have thatMf,g0 ≡ 〈e(g), e(f )〉I is a solution since the isometry property of

T implies thatYi(t, I ) = 0 ∀ i. Moreover,Yi ’s are linear and bounded, hence by theuniqueness of the solution of the Banach space valued initial value problem, we concludethatMf,g

t = Mf,g0 for all t , or equivalently thatUt is an isometry. ut

3.2. Fock modules.For any Hilbert spaceH, we denote byA ⊗alg H the subspace ofB(h, h⊗ H) consisting of finite sums of the form

∑xi ⊗ αi , wherexi ∈ A, αi ∈ H.

We also equipA ⊗alg H with anA-valued inner product〈·, ·〉 onA ⊗alg H by defining〈x ⊗ α, y ⊗ β〉 = x∗y〈α, β〉 and extending this linearly. One denotes byA ⊗c∗ Hthe completion ofA ⊗alg H under the operator norm inherited fromB(h, h ⊗ H). It

is easy to see that‖X‖ = ‖X∗X‖ = ‖〈X,X〉‖ 12 . This is a usual HilbertC∗ module

with A being the underlying algebra. For a comprehensive study of such structures,the reader may be referred to [Lan]. However, instead of the norm topology, we needto topologizeA ⊗alg H by the inherited strong operator topology fromB(h, h ⊗ H)and the closure ofA ⊗alg H under this topology will be denoted byA ⊗ H. Note thatA ⊗ H 3 Xn → X ∈ A ⊗ H if and only if Xnu → Xu ∀u ∈ h. It is clear that〈·, ·〉 extends naturally to bothA ⊗c∗ H andA ⊗ H; and they also have a natural rightA-module action, namely,(Xa)u := X(au) for a ∈ A, X ∈ A ⊗c∗ H or A ⊗ H.For other applications of Hilbert modules in quantum stochastic processes, the reader isreferred to [A-L] and [Sk].

Lemma 3.2.1. Any elementX of A ⊗ H can be written as,X = ∑α∈J xα ⊗ γα, where

{γα}α∈J is an orthonormal basis ofH and xα ∈ A. The above sum over a possiblyuncountable index setJ makes sense in the usual way: it is strongly convergent and∀u ∈h, there exists an at most countable subsetJu of J such thatXu = ∑

α∈Ju(xαu)⊗ γα.Moreover, once{γα} is fixed,xα ’s are uniquely determined byX.

Proof. Setxα = 〈γα,X〉 as per notation of Sect. 2.1. Clearly, ifX ∈ A ⊗alg H, xα ∈ Afor all α. Since any element ofA ⊗ H is a strong limit of elements fromA ⊗alg H; andsinceA is strongly closed, if follows thatxα ∈ A for an arbitraryX ∈ A⊗H. Now, for a


fixedu ∈ h, letJu be the (at most countable) set of indices such that∀α ∈ Ju, ∃ vα ∈ hwith 〈Xu, vα ⊗ γα〉 6= 0. Then for anyv ∈ h andγ ∈ H, we have withcγα = 〈γα, γ 〉,

〈Xu, v ⊗ γ 〉 =∑α∈Ju

cγα 〈Xu, v ⊗ γα〉 =∑α∈Ju

cγα 〈〈γα,X〉u, v〉

=∑α∈Ju

〈xαu, v〉〈γα, γ 〉 = 〈∑α∈Ju

(xα ⊗ γα)u, v ⊗ γ 〉;

that is,X = ∑α∈J xα ⊗ γα in the sense described in the statment of the lemma. Given

{γα}, the choice ofxα ’s is unique, because for any fixedα0, 〈γα0, X〉 = xα0, whichfollows from the previous computation if we takeγ to beγα0. utCorollary 3.2.2. Let X, Y ∈ A ⊗ H be given byX = ∑

α∈J xα ⊗ γα and Y =∑α∈J yα ⊗ γα as in the lemma above. For any finite subsetI of J , if we denote byXI

andYI the elements∑α∈I xα⊗γα and

∑α∈I yα⊗γα respectively, thenlimI 〈XI , YI 〉 =

〈X, Y 〉, where the limit is taken over the directed family of finite subsets ofJ with usualpartial ordering by inclusion.

Proof. The proof is an easy adaptation of Lemma 27.7 in [Par].utWe give below a convenient necessary and sufficient criterion for verifying whether

an element ofB(h, h⊗ H) belongs toA ⊗ H.

Lemma 3.2.3. LetX ∈ B(h, h⊗H). ThenX belongs toA⊗H if and only if〈γ,X〉 ∈ Afor all γ in some dense subsetE of H.

Proof. ThatX ∈ A ⊗ H implies〈γ,X〉 ∈ A ∀ γ ∈ H has already been observed in theproof of the previous lemma. For the converse, first we claim that〈γ,X〉 ∈ A for allγ inE(whereE is dense inH) will imply 〈γ,X〉 ∈ A for all γ ∈ H. Indeed, for anyγ ∈ H thereexists a netγα ∈ E such thatγα → γ , and hence‖〈γ,X〉−〈γα,X〉‖ ≤ ‖γα−γ ‖‖X‖ →0. Now let us fix an orthonormal basis{γα}α∈J of H and writeX =

∑α∈J

〈γα,X〉 ⊗ γα

by Lemma 3.2.1. Clearly, the netXI indexed by finite subsetsI of J (partially orderedby inclusion) converges strongly toX. SinceXI ∈ A ⊗alg H for any such finite subsetI (as〈γα,X〉 ∈ A ∀ α), the proof follows by noting thatA is strongly closed.ut

In caseH = 0(k), we call the moduleA ⊗ 0(k) as the right FockA-module over0(k), for short theFock module, and denote it byA ⊗ 0.

3.3. Solution of Evans–Hudson type q.s.d.e.In the previous subsections, we have con-sidered q.s.d.e.’s on the Hilbert spaceh⊗ 0. Now we shall study an associated class ofq.s.d.e.’s, but on the Fock moduleA ⊗ 0. This is closely related to the Evans–Hudsontype of q.s.d.e.’s ([Ev, Par]).

For this part of the theory, we assume that we are given thestructure maps, thatis, the triple of normal maps(L, δ, σ ), whereL ∈ B(A), δ ∈ B(A,A ⊗ k0) andσ ∈B(A,A ⊗ B(k0)) satisfying:

(S1) σ(x) = π(x)− x ⊗ Ik0 ≡ 6∗(x ⊗ Ik0)6− x ⊗ Ik0, where6 is a partial isometryin h⊗ k0 such thatπ is a∗-representation onA.

(S2) δ(x) = Rx − π(x)R, whereR ∈ B(h, h ⊗ k0) so thatδ is aπ -derivation, i.e.δ(xy) = δ(x)y + π(x)δ(y).


(S3) L(x) = R∗π(x)R+ lx + xl∗, wherel ∈ A with the conditionL(1) = 0 so thatLsatisfies the second order cocycle relation withδ as coboundary, i.e.

L(x∗y)− x∗L(y)− L(x)∗y = δ(x)∗δ(y) ∀x, y ∈ A.Given the generatorL of a q.d.s., that one can choosek0 and6 such that the hypotheses(S1)–(S3) are satisfied will be established in the next section.To describe Evans–Hudson flow in this language, it is convenient to introduce a map2

encompassing the triple(L, δ, σ ) as follows:

2(x) =(L(x) δ†(x)

δ(x) σ (x)

), (3.6)

wherex ∈ A, δ†(x) = δ(x∗)∗ : h ⊗ k0 → h, so that2(x) can be looked upon as abounded linear normal map fromh⊗ k0 ≡ h⊗ (C ⊕ k0) into itself. It is also clear from(S1)-(S3) that2mapsA into A⊗B(k0). The next lemma sums up important propertiesof 2.

Lemma 3.3.1. Let2 be as above. Then one has:

(i) 2(x) = 9(x)+K(x ⊗ 1k0)+ (x ⊗ 1

k0)K∗, (3.6)

where9 : A → A ⊗ B(k0) is a completely positive map andK ∈ B(h⊗ k0).

(ii) 2 is conditionally completely positive and satisfies the structure relation:

2(xy) = 2(x)(y ⊗ 1k0)+ (x ⊗ 1

k0)2(y)+2(x)Q2(y), (3.8)

whereQ =(

0 00 1h⊗k0

).

(iii) There existsD ∈ B(h⊗ k0) such that||2(x)ζ || ≤ ||(x ⊗ 1k0)Dζ || ∀ζ ∈ h⊗ k0.

Proof. Define the following maps with respect to the direct sum decompositionh⊗ k0 =h⊕ (h⊗ k0):

R =(

0 0R −I

), π(x) =

(x 00 π(x)

),K =

(l 0R −1

21h⊗k0

), 6 =

(1h 00 6

).

Then it is easy to see that (i) is verified with9(x) = R∗6∗(x ⊗ 1k0)6R = R∗π(x)R.

Clearly,9 is completely positive. That2 is conditionally completely positive and sat-isfies the structure relation (3.8) is also an easy consequence of (i) and (S1)-(S3). Theestimate in (iii) follows from the structure of9 given above with the choice ofD as

D = ||6R|| 6R + ||K|| 1h⊗k0

+K∗. utWe now introduce the basic processes. Fixt ≥ 0, a bounded interval1 ⊆ (t,∞),

elementsx1, x2, . . . , xn ∈ A and vectorsf1, f2, . . . , fn ∈ k; u ∈ h. We define thefollowing: (

aδ(1)(

n∑i=1

xi ⊗ e(fi))

)u =

n∑i=1

aδ(x∗i )(1)(ue(fi)),(

a†δ (1)(

n∑i=1

xi ⊗ e(fi))

)u =

n∑i=1

a†δ(xi )

(1)(ue(fi)),

392 D. Goswami, K. B. Sinha(3σ (1)(

n∑i=1

xi ⊗ e(fi))

)u =

n∑i=1

3σ(xi)(1)(ue(fi)),(IL(1)(

n∑i=1

xi ⊗ e(fi))

)u =

n∑i=1

|1|(L(xi)u)⊗ e(fi)),

where|1| denotes the length of1.

Lemma 3.3.2. The above processes are well defined onA⊗algE(k) and they take valuesin A ⊗ 0(k).

Proof. First note thate(f1), . . . , e(fn) are linearly independent wheneverf1, . . . , fnare distinct from which it is easy to see that

∑ni=1 xi⊗e(fi) = 0 impliesxi = 0∀i, when-

everfi ’s are distinct. This will establish that the processes are well defined. The secondpart of the lemma will follow from Lemma 3.2.3 with the choice of the dense setE to beE(k)andH =0(k)and by some simple computation, noting the fact thatL, δ, σ are struc-ture maps. For example,〈e(g), a†

δ (1)(x ⊗ e(f ))〉 = 〈e(g), e(f )〉 ∫1

〈g(s), δ(x)〉ds ∈A, which shows that the range ofa†

1 is in A ⊗ 0(k). Similarly, one verifies that〈e(g),3σ (1)(x ⊗ e(f ))〉 = 〈e(g), e(f )〉 ∫

1

〈g(s), σ (x)f (s)〉ds ∈ A, sinceσ(x) ∈A ⊗ B(k0). ut

Next, we want to consider the solution of an equation of the Evans–Hudson typewhich in our notation can be written as:

Jt = idA⊗0 +∫ t

0Js ◦ (a†

δ + aδ +3σ + IL)(dt), 0 ≤ t ≤ t0, (3.9)

where the solution is looked for as a map fromA ⊗ 0 into itself. For this, we first needan abstract lemma which allows us to interpret the above integral on the right-hand sideand to get an appropriate bound for such integrals.

Lemma 3.3.3 (The Lifting Lemma). LetH be a Hilbert space andV be a vector space.Letβ : A ⊗alg V → A ⊗ H be a linear map satisfying the estimate

||β(x ⊗ η)u|| ≤ cη||(x ⊗ 1H′′)ru|| (3.10)

for some Hilbert spaceH′′ and r ∈ B(h, h ⊗ H′′) (both independent ofη) and forsome constantcη depending onη. Then, for any Hilbert spaceH′, we can define amapβ : (A ⊗ H′) ⊗alg V → A ⊗ (H ⊗ H′) by β(x ⊗ f ⊗ η) = β(x ⊗ η) ⊗ f forx ∈ A, η ∈ V, f ∈ H′. Moreover,β admits the estimate

||β(X ⊗ η)u|| ≤ cη||(X ⊗ 1H′′)ru||, (3.11)

whereX ∈ A ⊗ H′.

Proof. LetX ∈ A⊗H′ be given by the strongly convergent sumX = ∑xα⊗eα, where

xα ∈ A and{eα} is an orthonormal basis ofH′. It is easy to verify that||β(∑ xα ⊗ eα ⊗η)u||2 = ∑ ||β(xα ⊗ η)u||2 ≤ c2

η

∑α ||(xα ⊗ 1H′′)ru||2 = c2

η||(X ⊗ 1H′′)ru||2, and

thusβ is well defined and admits the required estimate.ut


Corollary 3.3.4. If we takeV = C and identifyA⊗algV withA, thenβ : A → A⊗H isa bounded normal map andβ is also a bounded normal map fromA⊗H′ toA⊗H⊗H′.The proof of this corollary is a simple consequence of the estimates.

We now want to define∫ t

0 Y (s)◦(a†δ+aδ+3σ+IL)(ds), whereY (s) : A⊗algE(k) →

A ⊗ 0(k) is an adapted strongly continuous process satisfying the estimate

supo≤t≤t0

||Y (t)(x ⊗ e(f ))u|| ≤ ||(x ⊗ 1H′′)ru||, (3.12)

for x ∈ A, f ∈ C and whereH′′ is a Hilbert space andr ∈ B(h, h ⊗ H′′). In this theintegrals corresponding toaδ andIL belong to one class while the other two belong toanother. In fact, we define

∫ t0 Y (s) ◦ (aδ + IL)(ds)(x ⊗ e(f )) by setting it to be equal

to∫ t

0 Y (s)((L(x) + 〈δ(x∗), f (s)〉)⊗ e(f ))ds. For the integral involving the other two

processes, we need to considerY (s) : A ⊗ k0 ⊗ E(ks) → A ⊗ 0s ⊗ k0 as is given bythe previous lemma and fixx ∈ A andg ∈ C (see Corollary 2.2.4). Define two mapsS(s) : h⊗alg E(Cs) → h⊗ 0s ⊗ k0 andT (s) : h⊗alg E(Cs)⊗ k0 → h⊗ 0s ⊗ k0 by

S(s)(ue(fs)) = Y (s)(δ(x)⊗ e(fs))u,

andT (s)(ue(gs)⊗ f (s)) = Y (s)(σ (x)f (s) ⊗ e(gs))u.

By virtue of the hypotheses onY (s), the lifting lemma and the fact thats 7→ e(gs) isstrongly continuous, the familiesS andT satisfy the hypotheses of Corollary 2.2.4 (ii).Therefore we can define the integral

∫ t0 Y (s) ◦ (3σ (ds)+ a†

δ (ds)(x⊗ e(f ))u by setting

it to be equal to(∫ t

0 3T (ds)+ a†S(ds)

)ue(f ). Thus we have:

Proposition 3.3.5. The integralZ(t) ≡ ∫ t0 Y (s) ◦ (a†

δ + aδ + 3σ + IL)(ds), whereY (s) satisfies (3.12) is well defined onA ⊗alg E(C) as a regular process. Moreover, theintegral satisfies an estimate:

||{Z(t)(x ⊗ e(f )}u||2 ≤ 2et∫ t

0exp(||f s ||2){||Y (s)(2(x)

f (s)⊗ e(fs))u||2 +

||〈f (s), Y (s)(2(x)f (s) ⊗ e(fs))u||2}ds, (3.13)

where2was as defined earlier,Y (s) = Y (s)⊕Y (s) : A⊗ k0⊗algE(Cs) → A⊗0s⊗ k0,f (s) = 1 ⊕ f (s) andf (s) is identified with0 ⊕ f (s) in k0.

Proof. We have already seen that the integral is well defined. The estimate (3.13) followsfrom the estimate (2.14) in Corollary 2.2.4 by settingE = F = G = H = I andrecalling the definition of2. ut

Now we are ready to prove the main result of this section .

Theorem 3.3.6. (i) There exists a unique solutionJt of equation (3.9), which is anadapted regular process mappingA ⊗ E(C) into A ⊗ 0. Furthermore, one has anestimate

sup0≤t≤t0

||Jt (x ⊗ e(g))u|| ≤ C′(g)||(x ⊗ 10f (k)

)Et0u||,

whereg ∈ C, k = L2([0, t0], k0), Et ∈ B(h, h⊗0f (k)) and0f (k) is the full Fockspace overk.


(ii) Settingjt (x)(ue(g)) = Jt (x ⊗ e(g))u, we have(a) 〈jt (x)ue(g), jt (y)ve(f )〉 = 〈ue(g), jt (x∗y)ve(f )〉 ∀g, f ∈ C, and(b) jt extends uniquely to a normal∗-homomorphism fromA into A ⊗ B(0),

(iii) If A is commutative, then the algebra generated by{jt (x)|x ∈ A,0 ≤ t ≤ t0} iscommutative.

(iv) jt (1) = 1 ∀t ∈ [0, t0] if and only if6∗6 = 1h⊗k0.

Proof. (i) We write for1 ⊆ [0,∞),M(1) ≡ aδ(1)+ a†δ (1)+3σ (1)+ IL(1), and

set up an iteration by

J(n+1)t (x ⊗ e(f )) =

∫ t

0J (n)s ◦M(ds)(x ⊗ e(f )), J

(0)t (x ⊗ e(f )) = x ⊗ e(f ),

with x ∈ A andf ∈ C fixed. SinceJ (1)t = M([0, t]), J (1)t is adapted regular andhas the estimate (by the definition ofM(1), estimate (2.14) and Lemma 3.3.1(iii)):||J (1)t (x⊗ e(f ))u||2 ≤ 4et0||e(f )||2 ∫ t0 ds||2(x)(u⊗ f (s))||2||f (s)||2 ≤ 4||e(f )||2et0∫ t

0 ds||f (s)||2||(x ⊗ 1k0)D(u ⊗ f (s))||2. For the givenf , defineE(1)t : h → h ⊗ k

by (E(1)t u)(s) = D(u ⊗ f (s)||ft (s)||), whereft (s) = χ[0,t](s)f (s). Then the aboveestimate reduces to

||J (1)t (x ⊗ e(f ))u||2 ≤ 4||e(f )||2et0||(x ⊗ 1k)E

(1)t u||2. (3.14)

Now, an application of the lifting lemma leads to

||J (1)t (X ⊗ e(f ))u||2 ≤ 4||e(f )||2et0||(X ⊗ 1k)E

(1)t u||2,

for X ∈ A ⊗ k0, where as in the previous proposition,J(1)t = J

(1)t ⊕ ˜

J(1)t . As an

induction hypothesis, assume thatJ (n)t is a regular adapted process having an estimate||J (n)t (x ⊗ e(f ))u||2 ≤ Cn||e(f )||2||(x ⊗ 1

k⊗n )E(n)t u||2, whereC = 4et0, E(n)t : h →

h⊗ k⊗ndefined as:

(E(n)t u)(s1, s2, . . . sn) = (D ⊗ 1

k⊗n−1 )Pn{(E(n−1)s1

u)(s2, . . . sn)⊗ f (s1)||ft (s1)||}.

Furthermore,Pn : h⊗k⊗(n−1)⊗k0 → h⊗k0⊗k⊗(n−1)is the operator which interchanges

the second and third tensor components andE(0)t = 1h. Then by an application of

Proposition 3.3.5 one can verify thatJ (n+1)t also satisfies a similar estimate. Thus, if we

putJt = ∑∞n=0 J

(n)t , then

||Jt (x ⊗ e(f ))u|| ≤∞∑n=0

||J (n)t (x ⊗ e(f ))u||

≤ ||e(f )||∞∑n=0

Cn2 (n!)− 1

4 ||(x ⊗ 1k⊗n )(n!)

14E

(n)t u||

≤ ||e(f )||( ∞∑n=0

Cn√n!

) 12

||(x ⊗ 10f (k)

)Etu||, (3.15)


where we have setEt : h → h ⊗ 0f (k) by Etu = ⊕∞n=0(n!)

14E

(n)t u. It is easy to see

that

||Etu||2 =∞∑n=0

(n!) 12 ||E(n)t u||2

≤ ||u||2∞∑n=0

(n!) 12 ||D||2n

{ ∫0<sn<sn−1<...s1<t

dsn . . . ds1||f (sn)||4 . . . ||f (s1)||4}

= ||u||2∞∑n=0

(n!)− 12 ||D||2nµf (t)n,

whereµf (t) = ∫ t0 ||f (s)||4ds. The estimate (3.15) proves the existence of the solution

of Eq. (3.9), as well as its continuity relative to the strong operator topology inB(h).The uniqueness of the solution follows along standard lines of reasoning.

(ii) First we prove the following identity:

〈Jt (x ⊗ e(f ))u, Jt (y ⊗ e(g))v〉 = 〈ue(f ), Jt (x∗y ⊗ e(g))v〉. (3.16)

For this it is convenient to lift the mapsJt to the moduleA ⊗0f (k0)⊗alg E(C), that is,replaceA by A ⊗ 0f (k0). We defineJt : A ⊗ 0f (k0)⊗alg E(C) → A ⊗ 0 ⊗ 0f (k0)

by Jt = (Jt ⊗ id)P , whereP interchanges the second and third tensor components.Recalling from Sect. 2 that2(x)ζ ∈ B(h, h ⊗ k0) for x ∈ A, ζ ∈ k0, we can define2ζ : A → A ⊗ k0 by2ζ (x) = 2(x)ζ , and extend as above to2ζ : A ⊗ 0f (k0) →A ⊗ 0f (k0) by setting2ζ |A⊗k0

⊗n = 2ζ ⊗ idk0

⊗n . By the lifting lemma, bothJt and

2ζ are well defined and enjoy the estimates as in Theorem 3.3.6 and Lemma 3.3.1(iii)respectively.

Next, note that for fixedf, g ∈ C andx, y ∈ A, one has using Eq. (3.9) forJt andquantum Ito formula (Proposition 2.2.4) and the structure relation in Lemma 3.3.1(ii):

〈Jt (x ⊗ e(f ))u, Jt (y ⊗ e(g))v〉〈xu⊗ e(f ), yv ⊗ e(g)〉+∫ t

0ds{Js(2f (s)(x)⊗ e(f ))u, Js(y ⊗ g(s)⊗ e(g))v〉

+ 〈Js(x ⊗ f (s)⊗ e(f ))u, Js(2g(s)(y)⊗ e(g))v〉+ 〈Js(2f (s)(x)⊗ e(f ))u, Js(2g(s)(y)⊗ e(g))v〉},

(3.17)

wheref (s) andg(s) in k0 are identified with 0⊕f (s) and 0⊕g(s) in k0 respectively. Weclaim that the identity above remains valid even when we replacex, y byX, Y ∈ A ⊗0f (k0) and2ζ (x),2ζ (y) by 2ζ (X), 2ζ (Y ) respectively, whereζ is one of the vectorsf (s), ˆg(s), f (s) andg(s). To see this, it suffices to observe that in the resulting identity,

both left and right-hand sides vanish ifX ∈ A ⊗ k0⊗n

andY ∈ A ⊗ k0⊗m

with m 6= n,

and then use the definition ofJt and2ζ to prove the identity forX, Y ∈ A ⊗alg k0⊗m

.Finally, use Corollary 3.2.2 and strong continuity ofJt obtained from the estimate in (i)


to extend the identity fromX = ∑xα ⊗ eα, Y = ∑

yα ⊗ eα (finite sums) to arbitraryX andY . Thus one has upon setting

8t(X, Y ) ≡ 〈Jt (X ⊗ e(f ))u, Jt (Y ⊗ e(g))v〉 − 〈ue(f ), Jt (〈X, Y 〉 ⊗ e(g))v〉the equation:

8t(X, Y ) =∫ t

0ds{8s(2f (s)(X),Jg(s)(Y ))

+8s(Jf (s)(X), 2g(s)(Y ))+8s(2f (s)(X), 2g(s)(Y ))},(3.18)

where〈X, Y 〉 is the module inner product inA⊗0f (k0) and we have set forζ, η1, . . . ηn

∈ k0 the mapJζ (x ⊗ η1 . . . ⊗ ηn) = x ⊗ η1 . . . ⊗ ηn ⊗ ζ , and extend it naturally asa map fromA ⊗ 0f (k0) to itself. It is clear that the estimates in Lemma 3.3.1(iii) andTheorem 3.3.6(i) extend to

||2ζ (X)u|| ≤ ||(X ⊗ 1k0)D(u⊗ ζ )||

andsup

0≤t≤t0||Jt (X ⊗ e(f ))u|| ≤ C′(f )||(X ⊗ 1

0f (k))Et0u||.

From the above estimates and definition of8t , it is clear that|8t(X, Y )|≤ ||u||||v||||X||||Y ||||Et0||C′(g){||Et0||C′(f ) + ||e(f )||}. This implies, by iteratingthe expression (3.18) sufficient number of times, that8t(X, Y ) = 0 which leadsto 8t(x, y) = 0 for all x, y ∈ A. Since〈ve(g), jt (x)(∑n

i=1 uie(fi))〉 = 〈Jt (x∗ ⊗e(g))v,

∑uie(fi)〉 by the above identity, it follows thatjt (x) is well defined onh⊗alg

E(C), and thus (ii)(a) is proven. The proof of (ii)(b) and (iii) are as in [Ev] and [Par]respectively. For (iv), we note thatjt (1) = 1 for all t if and only if dJt (1⊗ e(f ))u = 0∀ u ∈ h, f ∈ C; and from Eq. (3.9) and (S1) it is clear that this can happen if and onlyif 0 = ∫ t

0 ds〈ue(f ), Js ◦ (36∗6−I (ds)(ve(f ))〉 =− ∫ t0 ds〈ue(f ), Js(〈f (s), (6∗6 −I )f (s)〉⊗e(f ))v〉 for all t , sinceπ(1)δ = δ. But this is possible if and only if〈ζ, (6∗6−I )ζ 〉 = 0 ∀ ζ ∈ k0 which is same as6∗6 = I . ut

4. Dilation of a Quantum Dynamical Semigroup

In this section, first a unitary evolutionUt is constructed inh⊗ 0 such that the vaccumexpectation ofj0

t (x) ≡ Ut(x ⊗ 10)U∗t gives back the q.d.s.Tt that we started with in

Sect. 1. However,j0t (x) in general will not satisfy a flow equation of the Evans–Hudson

type. Here it is also shown that there exists a suitable choice of a partial isometry inh⊗k0such that the above kind of flow equation can be implemented by a partial isometry-valued process inh ⊗ 0. It is to be noted here that in [H-S] an Evans–Hudson typedilation was achieved withA = B(h) for a countably infinite dimensionalh only.

Before proceeding further, we note the following two theorems whose proofs can befound in their respective references .

Theorem 4.0.1 ([C-E]). Let (Tt )t≥0 be a conservative uniformly continuous q.d.s. onA with L as its generator. Then

(i) There is a unital normal∗-representationρ of A in a Hilbert spaceK and aρ-derivationα : A → B(K) such that the setD ≡ {α(x)u|x ∈ A, u ∈ h} is total inK.


(ii) Furthermore, there existsR ∈ B(h,K) such thatα(x) = Rx−ρ(x)R, andL(x) =R∗ρ(x)R − 1

2R∗Rx − 1

2xR∗R + i[H, x] with R∗ρ(x)R ∈ A ∀x ∈ A andH a

selfadjoint element inA.

The property ofρ-derivation has already been introduced in (S2) of Sect. 3 and notethat the above pair(L, α) satisfy the cocycle relation (S3) in Sect. 3.

Theorem 4.0.2 ([Dix] ). Every normal∗-representationρ of a von Neumann algebraAin a Hilbert spaceK is of the formρ(x) = 6∗

1(x ⊗ 1k1)61, wherek1 is a Hilbert spaceand61 is a partial isometry with initial setK and final seth⊗k1 such that the projectionP1 ≡ 616

∗1 commutes withx ⊗ 1k1 for all x ∈ A. If ρ is unital, then6∗

161 = 1K.Moreover, in caseh is separable, one can choosek1 to be separable also.

4.1. Hudson-Parthasarathy (H-P) dilation.Let ρ, α,R be as in Theorem 4.1.1 and61

as in Theorem 4.1.2. Then setR = 61R, R ∈ B(h, h⊗ k1) so thatR∗ = R∗6∗1 and we

have

R∗(x ⊗ 1k1)R = R∗6∗1(x ⊗ 1k1)61R

= R∗ρ(x)R.

Also,R∗R = R∗6∗

161R = R∗R, as6∗161 = 1K.

Now, we take the unitary processUt which satisfies the following q.s.d.e. (as in Sect. 3.1)

dUt = Ut(a†R(dt)− a

R(dt)+ (iH − 1

2R∗R)dt), U0 = I. (4.1)

Let 0 denote0(L2(R+, k1)). Taking j0t (x) = Ut(x ⊗ 10)U

∗t , we see that for each

t, j0t (·) is a∗-homomorphism.We now claim that〈ve(0), j0

t (x)ue(0)〉 = 〈v, Tt (x)u〉.Toprove this, it is enough to show that〈ve(0), d

dtj0t (x)(ue(0))〉 = 〈v, Tt (L(x))u〉, and this

follows from the quantum Ito formula for right integrals as mentioned in Remark 2.2.5.Indeed we have,

〈U∗t (ve(0)), (x ⊗ 10)U

∗t (ue(0))〉 =

t∫0

ds〈ve(0), j0s (L(x))(ue(0))〉,

where

L(x) = R∗ρ(x)R − 1

2R∗Rx − 1

2xR∗R + i[H, x]

= R∗(x ⊗ 1k1)R − 1

2R∗Rx − 1

2xR∗R + i[H, x].

Thus, if we denote byE0 the vacuum expectation map which takes an elementG ofB(h⊗ 0) to an elementE0G in B(h) satisfying〈v, (E0G)u〉 = 〈ve(0), G(ue(0))〉 (foru, v ∈ h), then

d

dtE0j

0t (x) = E0j

0t (L(x)),

which implies (sinceL is bounded), thatE0j0t (x) = Tt (x).


A simple calculation using the quantum Ito formula and Eq. (4.1) shows that

dj0t (x) = Ut [a†

α(x)(dt)− aα(x)(dt)+ L(x)dt]U∗t , (4.2)

whereα(x) = Rx − (x ⊗ 1k1)R = 61[Rx − ρ(x)R], for x ∈ A. In general,α(x) maynot be inA ⊗ k1 and therefore Eq. (4.2) is not a flow equation of the Evans–Hudsontype. However, in caseA = B(h), it is trivially a flow equation.

4.2. Existence of structure maps and Evans–Hudson dilation ofTt . In the context ofTheorems 4.1.1 and 4.1.2, it should be noted that in generalK need not be of the formh⊗ k′ and neitherρ or α be structure maps as defined in Sect. 3, that is,ρ need not bein A ⊗ B(k′) nor α(x) be inA ⊗ k′. However, the following theorem asserts that onecan “rotate” the whole structure appropriately so that the “rotated”ρ andα (denotedπandδ respectively) become structure maps without changingL (see also [P-S]).

Theorem 4.2.1.LetTt be a conservative norm-continuous q.d.s. with generatorL. Thenthere exist a Hilbert spacek0, a normal∗-representationπ : A → A ⊗ B(k0) and aπ -derivationδ ofA intoA⊗k0 such that the hypotheses(S1)–(S3)in Sect. 3 are satisfied.

Proof. (i) Let (ρ, α,D) be as in theorem 4.1.1. We define a mapρ′ : A′ → B(K), whereA′ denotes the commutant ofA in B(h), by

ρ′(a)(α(x)u) = α(x) au, x ∈ A, u ∈ h, a ∈ A′, (4.3)

and extend it linearly to the algebraic span ofD.To show that it is well defined, we need to show that whenever

∑mi=1 α(xi)ui = 0

for xi ∈ A, ui ∈ h, one hasρ′(a)(∑mi=1 α(xi)ui) = 0. Sinceα(xi)∗α(y) = L(x∗

i y) −L(x∗

i )y − x∗i L(y) ∈ A for y ∈ A by the comments at the end of Theorem 4.1.1, we

have fora ∈ A′,

〈ρ′(a)(m∑i=1

α(xi)ui), α(y)v〉 =m∑i=1

〈α(xi)aui, α(y)v〉

=m∑i=1

〈ui, a∗α(xi)∗α(y)v〉

=m∑i=1

〈ui, α(xi)∗α(y)a∗v〉

= 〈m∑i=1

α(xi)ui, α(y)a∗v〉,

(4.4)

thereby proving thatρ′ is well defined. A similar computation gives,

‖ρ′(a)(m∑i=1

α(xi)ui)‖2 =m∑i=1

m∑j=1

〈ui, α(xi)∗α(xj )a∗auj 〉. (4.5)

Denoting the operatorα(xi)∗α(xj ) byAij , and noting thatA ≡ ((Aij ))i,j=1,...m acts asa positive operator onh⊕ . . .⊕ h︸︷︷︸

m copies

, which commutes with the positive operatorC ⊗ Im,


whereC = ‖a‖2.1−a∗a andIm denotes the identity matrix of orderm, we observe thatA(C⊗Im) is also a positive operator.Thus, consideringu1⊕u2⊕. . .⊕um ∈ h⊕ . . .⊕ h︸︷︷︸

m copies

,

the right-hand side of (4.5) can be estimated as:

m∑i=1

m∑j=1

〈ui, α(xi)∗α(xj )a∗auj 〉

≤ ‖a‖2m∑i=1

m∑j=1

〈ui, α(xi)∗α(xj )uj 〉

= ‖m∑i=1

α(xi)ui‖2‖a‖2,

proving that||ρ′(a)|| ≤ ||a|| sinceD is total inK. It is also easy to see from the definitionof ρ′ and (4.4) that it is a unital∗-representation ofA′ in K. Next we show thatρ′ isnormal. For this, take a net{aα} such that 0≤ aα ↑ a, whereaα, a ∈ A′. It is clear fromthe definition ofρ′ thatρ′(aα)α(x)u → ρ′(a)α(x)u for all x ∈ A, u ∈ h, and thus,ρ′(aα) s→ρ′(a) onK by totality ofD in K and since‖ρ′(aα)‖ ≤ ‖aα‖ ≤ ||a|| ∀ α.(ii) By (i), ρ′ : A′ → B(K) is a unital normal *- representation. By Theorem 4.1.2,there exist a Hilbert spacek2, an isometry62 : K → h ⊗ k2 with K2 = Ran62

∼= K,satisfying,

ρ′(a) = 6∗2(a ⊗ 1k2)62, (4.6)

and for alla ∈ A′, a ⊗ 1k2 commutes withP2 ≡ 626∗2. Let us now takeδ(x) =

62α(x), π(x) = 62ρ(x)6∗2. It is clear thatδ is aπ -derivation. Moreover,δ(x∗)∗δ(y) =

α(x∗)∗6∗262α(y) = α(x∗)∗α(y) and henceL(xy) − xL(y) − L(x)y = δ(x∗)∗δ(y)

holds. TakingR = 62R ∈ B(h, h⊗ k2), we observe thatδ(x) = 62α(x) = 62(Rx −ρ(x)R) = Rx − π(x)R. It is also clear thatL(x) = R∗π(x)R − 1

2R∗Rx − 1

2xR∗R +

i[H, x] = R∗ρ(x)R − 12R

∗Rx − 12xR

∗R + i[H, x].To show thatδ(x) ∈ A ⊗ k2 for all x ∈ A, it is enough (by Llemma 3.2.3) to verify

that for anyf ∈ k2, 〈f, δ(x)〉 ∈ A, or equivalently that〈f, δ(x)〉 commutes with alla ∈ A′. Forf ∈ k2, a ∈ A′, u, v ∈ h, x ∈ A, sinceP2 and(a⊗1k2) commute, we have,

〈〈f, δ(x)〉au, v〉 = 〈δ(x)au, v ⊗ f 〉= 〈62α(x)au, v ⊗ f 〉 = 〈62ρ

′(a)(α(x)u), v ⊗ f 〉= 〈626

∗2(a ⊗ 1k2)62α(x)u, v ⊗ f 〉

= 〈P2(a ⊗ 1k2)62α(x)u, v ⊗ f 〉= 〈(a ⊗ 1k2)626

∗262α(x)u, v ⊗ f 〉

= 〈62α(x)u, (a∗v)⊗ f 〉 = 〈〈f, δ(x)〉u, a∗v〉

= 〈a〈f, δ(x)〉u, v〉.Next, we want to show thatπ(x) ∈ A ⊗ B(k2) for x ∈ A; and for this it is enoughto verify π(x)(a ⊗ 1k2) = (a ⊗ 1k2)π(x) for all a ∈ A′. Since6∗

2P⊥2 = 0, and since


62D is total inK2, it suffices to verify thatπ(x)(a ⊗ 1k2)P2 = (a ⊗ 1k2)π(x)P2, orequivalently thatπ(x)(a ⊗ 1k2)62α(y)u = (a ⊗ 1k2)π(x)62α(y)u, for all y ∈ A,u ∈ h. For this, observe that,

π(x)(a ⊗ 1k2)62α(y)u = 62ρ(x)6∗2(a ⊗ 1k2)62α(y)u = 62ρ(x)ρ

′(a)α(y)u= 62ρ(x)α(y)au = 62α(xy)au−62α(x)yau = 62ρ

′(a)(α(xy)− α(x)y)u

= 62ρ′(a)ρ(x)α(y)u = 626

∗2(a ⊗ 1k2)62ρ(x)6

∗2(62α(y)u)

= P2(a ⊗ 1k2)π(x)(62α(y)u) = (a ⊗ 1k2)π(x)(62α(y)u).

(iii) It follows from the above and Theorem 4.1.2 thatπ(x) = 626∗1(x ⊗ 1k1)616

∗2 ≡

6∗(x ⊗ 1k1)6 onh⊗ k2 so that6 is a partial isometry with initial setP2(h⊗ k2) andfinal setP1(h⊗k1). Now setk0 = k1⊕k2 and6 = 6⊕0 : h⊗k0 → h⊗k0 with initialset(0⊕P2)(h⊗k0) and final set(P1⊕0)(h⊗k0) andπ(x) = π(x)⊕0,δ(x)u = δ(x)u

for x ∈ A, u ∈ h. It is clear thatδ(x) ∈ A ⊗ k0, π(x) ∈ A ⊗ B(k0) and (S1)–(S3) aresatisfied. utRemark 4.2.2.Although ρ was assumed to be unital,π chosen by us is not unital.However, in some cases it may be possible to choose6, k0 in such a manner thatπ isunital.

We summarise the main result of this section in form of the following theorem:

Theorem 4.2.3.Let (Tt )t≥0 be a conservative norm-continuous q.d.s. withL as itsgenerator. Then there is a flowJt : A ⊗ E(C) → A ⊗ 0 satisfying an Evans–Hudson type q.s.d.e. (3.9) with structure maps(L, δ, σ ) satisfying (S1)-(S3), where0 ≡ 0(L2(R+, k0)) and C consists of bounded continuous functions inL2(R+, k0),such thatjt (x) defined in Theorem3.3.6is a (not necessarily unital)∗-homomorphismof A into A ⊗ B(k0) andE0jt (x) = Tt (x) ∀x ∈ A.

Proof. The proof is immediate by (i) observing the existence of structure mapsδ andσ satisfying (S1),(S2) from Theorem 4.2.1, (ii) observing thatL satisfies (S3), andfinally (iii) constructing the solutionJt of Eq. (3.9) with structure maps(L, δ, σ ) as inTheorem 3.3.6. ThatE0jt (x) = Tt (x) ∀x ∈ A follows from the q.s.d.e (3.9).utRemark 4.2.4.With reference to the last sentence in the statement of Theorem 4.1.2,it may be noted that both the Hilbert spacesk1 andk2 and hencek0 can be chosen tobe separable if the initial Hilbert spaceh is separable. In such a case, if we choose anorthonormal basis{ej } in k0, then the estimate forδ in (S2) is precisely the coordinate-free form of the condition∑

i=0

||µi0(x)u||2 ≤∑i∈I0

||xDi0u||2

with∑i∈I0

||Di0u||2 ≤ α0||u||2 in [Mo-S]. The similar conditions onµij (j 6= 0) as in

[Mo-S] is trivially satisfied by((µkj ))∞k,j=1 ≡ σ and forµ0

0 ≡ L as can be seen easilyfrom (S1) and (S3). It may also be noted thatjt satisfies the E-H equationdjt (x) =∑i,j jt (µ

ij (x))d3

ji (t), with j0 = id, in the coordinatized form with the appropriate

choices ofµij ’s in terms ofL, δ andσ as above. The flow equation (3.9) is in fact acoordinate-free modification of the old coordinatized E-H equation given above.


4.3. Implementation ofE−H flow . Combining Theorems 4.1.1 and 4.2.3, we see thatfor x ∈ A, π(x) = 6(x ⊗ 1k0)6

∗∈ A ⊗ B(k0), δ(x) = Rx − π(x)R ∈ A ⊗ k0 for a suitable Hilbert spacek0, R ∈B(h, h⊗ k0),6 a partial isometry inh⊗ k0. Now let us consider the H-P type q.s.d.e.:

dVt = Vt (a†R(dt)+36−I (dt)− a6∗R(dt)+ (iH − 1

2R∗R)dt), V0 = I. (4.7)

Then by Proposition 3.1.2, there is a contraction valued unique solutionVt as a regularprocess onh⊗ 0. The following theorem shows that every Evans–Hudson type flowJtsatisfying Eq. (3.9) is actually implemented by a processVt satisfying Eq. (4.7).

Theorem 4.3.1.Every flowJt satisfying Eq. (3.9) is implemented by a partial isometryvalued processVt satisfying (4.7), that is,Jt (x ⊗ e(f ))u = Vt (x ⊗ 10)V ∗

t (ue(f )).

Furthermore, the projection-valued processesPt ≡ VtV∗t andQt ≡ V ∗

t Vt belong toA ⊗ B(0) andA′ ⊗ B(0) respectively.

We need a lemma for the proof of this theorem.

Lemma 4.3.2. If B is a von Neumann algebra inB(H) for some Hilbert spaceH andp is a projection inB(H) such thatB 3 x 7→ pxp is a ∗-homomorphism ofA, thenp ∈ B′.

Proof of the lemma.Let q be any projection inB. We have by the hypothesis that,

pqp = pqnp = (pqp)n ∀n ≥ 1.

But

(pqp)n = pqp · pqp . . .︸︷︷︸n times

= (pq)nps→ (p ∧ q)p = p ∧ q,

by von Neumann’s Theorem, wherep∧q denotes the projection onto Ran(p)⋂

Ran(q).Thus we have,(qp)∗qp = pqp = p ∧ q, which implies thatqp is a partial isometrywith the initial space Ran(p ∧ q) and henceqp.p ∧ q = qp. But qp.p ∧ q = p ∧ q,and thusqp = p ∧ q = (p ∧ q)∗ (since p ∧ q is a projection)= pq. This completesthe proof becauseB is generated by its projections.ut

Proof of the theorem.SettingJ ′t (x⊗ e(f ))u = Vt (x⊗10)V ∗

t (ue(f )) for u ∈ h, f ∈ C,and using Eq. (4.7) we verify easily thatJ ′

0 = id andJ ′t satisfies the same flow equation

(3.9) as doesJt . By the uniqueness of the solution of the initial value problem (3.9) weconclude thatJt = J ′

t . Now, as in Theorem 3.3.6, if we setjt (x)ue(f ) = Jt (x⊗e(f ))u,it follows thatjt (x) = Vt (x⊗10)V ∗

t and thatjt (.) is a∗-homomorphism ofA.Therefore,Vt (xy ⊗ 10)V ∗

t = Vt (x ⊗ 10)Qt(y ⊗ 10)V ∗t for x, y ∈ A. In particularPt = jt (1) =

VtV∗t is a projection, that is,Vt is a partial isometry valued regular process. It also

follows from the same identity thatQt(xy ⊗ 10)Qt = Qt(x ⊗ 10)Qt(y ⊗ 10)Qt ,that is,x ⊗ 10 7→ Qt(x ⊗ 10)Qt is a∗-homomorphism ofA ⊗ 10. ThereforeQt ∈(A ⊗ 10)′ = A′ ⊗ B(0), by the Lemma 4.3.2.ut


5. Weak Markov Process Associated with theE − H Type Flow

Here we consider the solutionJt of Eq. (3.9) or the associatedjt and construct a weakMarkov process (see [B-P]) with respect to the Fock filtration. The next theorem sum-marizes the results:

Theorem 5.0.3. (i) Let jt be as in Theorem 4.2.3. SetFt ≡ jt (1)Et , whereEt is theconditional expectation operator given byEt (ue(f )) = ue(ft ). Then there existsa nonzero projectionj∞(1) such that the family of projections{jt (1)} and {Ft }decreases and increases toj∞(1) respectively.

(ii) The triple {jt , h ⊗ 0,Ft } is a weak Markov process as defined in [B-P], that is,EF0 j0(x) = xF0, jt (x)Ft = Ftjt (x) = Ftjt (x)Ft , EFs jt (x) = js(Tt−s(x))Fs for0 ≤ s ≤ t < ∞, x ∈ A, whereEFs (X) = FsXFs for X ∈ B(h⊗ 0).(iii) If we setkt (x) = jt (x)Ft = jt (x)Et = Et jt (x), then the triple{kt , h⊗0,F } isa conservative weak Markov flow subordinate to{Ft } (see [B-P]), that is,kt satisfiesthe properties listed in (ii) above and alsokt (1) = Ft .

Proof. (i) In the notation of Theorem 4.2.3,jt (1) = VtV∗t and therefore takingWt = V ∗

t

and using the relation (3.5) withT replaced by6 we see that{jt (1)} is a decreasingfamily of projections. On the other hand, a simple computation shows that fort > s > 0,

〈ve(g), (Ft − Fs)(ue(f )〉 =〈Wtve(g),Wtue(f )〉e−

∫∞t 〈g(τ),f (τ )〉dτ − 〈Wsve(g),Wsue(f )〉e−

∫∞s 〈g(τ),f (τ )〉dτ =∫ t

s

exp{−∫ ∞

τ

〈g(τ ′), f (τ ′)〉dτ ′}〈Wτve(g), 〈g(τ),66∗f (τ)〉Wτue(f )〉dτ,

from which it follows that{Ft } is an increasing family of projections. SinceEt increasesto I onh⊗0, and sincejt (1) converges strongly to sayj∞(1), we have thatFt increasesto j∞(1). Thereforej∞(1) cannot be the zero projection.

(ii) Let u, v ∈ h, f, g ∈ k, x ∈ A. Sincejs(1)jt (1) = jt (1) for s ≤ t , we have,

〈Fsjt (x)Fs(ve(g)), ue(f )〉 = 〈js(1)jt (x)js(1)ve(gs), ue(fs)〉= 〈jt (x)ve(gs), ue(fs)〉 = 〈Jt (x ⊗ e(gs))v, ue(fs)〉= 〈Js(x ⊗ e(gs))v, ue(fs)〉 +

∫ t

s

〈Jτ (L(x)e(gs))v, ue(fs)〉dτ

= 〈js(x)Fs(ve(g)), ue(f )〉 +∫ t

s

〈Fsjτ (L(x))Fs(ve(g)), ue(f )〉dτ,

becausefs(τ ) = 0,gs(τ ) = 0 for τ > s. Thus, if we denote by4t the mapA 3 x 7→Fsjt (x)Fs , then d4

dt= 4t ◦ L, for s ≥ t . On the other hand, denoting by5t the

map given by,5t(x) = js(Tt−s(x))Fs , we can easily verify thatd5dt

= 5t ◦ L. Since4s(x) = Esjs(x) = js(x)Fs = 5s(x), the initial values of5 and4 agree. Thus by thestandard uniqueness result of differential equations, we conclude that5t ≡ 4t for allt ≥ s.

(iii) The proof of this part is obvious from the definitions.ut


We have no result with regards to minimality of the above process in the sense of [B-P]. However, if we denote the closed linear span of{jt1(x1)jt2(x2) . . . jtn(xn)ue(0)|xj ∈A, t > t1 ≥ t2 ≥ . . . ≥ tn > 0 } by K′

t for 0 ≤ t ≤ ∞, then it is an easy observationthatK′

t is contained in the range ofFt for eacht < ∞, and thusK′∞ is contained inj∞(1)(h ⊗ 0). We suspect thatK′∞ = j∞(1)(h ⊗ 0), which we have not been able toprove. If this turns out to be true, then the above provides a complete general theory ofstochastic dilation for a uniformly continuous quantum dynamical semigroup on a vonNeumann algebra. It should also be noted that the final weak Markov process(js, Fs)

is actually living inh⊗ 0(k) and its filtration is subordinate to that in the Fock space.

Acknowledgement.The first author would like to acknowledge a helpful discussion with K. R. Parthasarathyregarding the material of the Sect. 5. He also wants to give special thanks to Arup Pal for various discussionson Hilbert modules.

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Communicated by H. Araki

Hilbert Modules and Stochastic Dilation of a Quantum Dynamical Semigroup on a von Neumann Algebra

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