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CHRISTENSEN–EVANS GENERATOR IN A VON NEUMANN ALGEBRA

K. R. PARTHASARATHY K. B. SINHA

1. Introduction

Let be a unital von Neumann algebra of operators on a complex separable Hilbert space!, and letoTt,t0qbe a uniformly continuous quantum dynamical semigroup of completely positive unital maps on . The infinitesimal generator ofoTtqis a bounded linear operator on the Banach space . For any Hilbert space , denote by () the von Neumann algebra of all bounded operators on . Christensen and Evans [3] have shown that has the form

(X)lR*π(X)RjK!XjXK

!, X? , (1.1)

whereπis a representation of in() for some Hilbert space,R:! is a bounded operator satisfying the ‘ minimality ’ condition that the seto(RXkπ(X)R)u, u?!, X? q is total in, and K

! is a fixed element of . The unitality of oTtq implies that (1)l0, and consequently K

!liHk"#R*R, where H is a hermitian

element of . Thus (1.1) can be expressed as

(X)li[H,X]k"#(R*RXjXR*Rk2R*π(X)R), X? . (1.2)

We say that the quadruple (,π,R,H) constitutes the set of Christensen–Evans (CE) parameters which determine the CE generator of the semigroup oTtq. It is quite possible that another set (h,πh,Rh,Hh) of CE parameters may determine the same generator . In such a case, we say that these two sets of CE parameters are equiŠalent. In Section 2 we study this equivalence relation in some detail.

It is known from [1,2] that, corresponding to the quantum dynamical semigroup oTtq, there exists, up to unitary equivalence, a unique minimal Markov flow (,Ft,jt), t0, satisfying the following properties. (1)is a Hilbert space containing!as a subspace. (2) oFtqis an increasing family of projections in , increasing to 1 (the identity projection) in as t _, and F! is the projection on !. (3) jt is a M homomorphism from into () such that j!(X)lXF!, jt(1)lFt, Fsjt(X)Fsl js(Tt−s(X)) for allst, and the mapt jt(X) is strongly continuous for eachXin . (4) The set

ojt

"(X

")jt

#(X

#)(jt

n(Xn)u,u?!,t

"t

#(tn0,nl1, 2,…,Xj? q is total in.

If we drop condition (4) in the preceding paragraph, then we say that (,Ft,jt) is aMarkoŠdilationfor the semigroupoTtqor, equivalently, the generator. In [1,2], the construction of the minimal dilation was achieved on the basis of a full knowledge

Received 20 December 1995 ; revised 25 November 1996 ; transferred fromJ.London Math.Soc.

1991Mathematics Subject Classification81S25, 60J25.

Bull.London Math.Soc. 31 (1999) 616–626

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of the semigroupoTtqand an application of the GNS principle. However, it would be desirable to construct Markov dilations starting fromor some parameters (like the CE parameters) determining . In the simplest case, when l(!), the CE generator assumes the Lindblad form [8] :

(X)li[H,X]k"#

j

(Lj LjXjXLj Ljk2Lj XLj),

whereH,Lj?(!), His hermitian, andjLj Lj is a finite or strongly convergent countable sum. From the methods of quantum stochastic calculus [6,9,11], it is known how to construct Markov dilations of by solving quantum stochastic differential equations (qsde) involvingH and theLjin its ‘ diffusion ’ coefficients [6, 10,11]. However, even in this case, there does not seem to exist a procedure for constructing the minimal dilation starting from the parametersH,Lj. In Section 3 of this paper we start from the CE parameters in (1.2), and construct a Markov dilation for. The Markov process thus obtained turns out to be a Poisson imbedding of a discrete time quantum Markov chain, but looked at in an ‘ interaction ’ picture. The idea of an interaction picture of a quantum diffusion goes back to [4], [5] and [7].

The Markov dilation presented here depends very much on the parameters (,π,R,H) which determine through (1.2). It should be interesting to explore the connection between the dilations determined by different parametrizations for the same generator.

2. An equiŠalence relation for the Christensen–EŠans parameters

Let!, ,be as in Section 1, and let (j,πj,Rj,Hj),jl1, 2, be two quadruples determining the same CE generatorvia (1.2), so that Hj,RjRj? , and

(X)li[Hj,X]k"#(RjRjXjXRjRjk2Rjπ(X)Rj), X? ,jl1, 2. (2.1)

Denote by hthe commutant of in (!).

P2.1. There exists a unitary isomorphismΓ:" #such that,for all X? , the following hold:

(1) Γπ"(X)lπ#(X)Γ;

(2) (Γ*R#kR")Xlπ"(X) (Γ*R#kR").

Proof. Let

δj(X)lRjXj(X)Rj, X? ,jl1, 2. (2.2) By elementary algebra, we have

δj(X)*δj(Y)l(X*Y)kX*(Y)k(X*)Y, X,Y? ,jl1, 2, (2.3) wheresatisfies (2.1). By the definition of the CE parameters, the setoδj(X)u,u?!, X? qis total inj. Hence (2.3) implies that the correspondenceδ"(X)u δ#(X)uis scalar product preserving, and there exists a unique unitary isomorphismΓ:" # satisfying

Γδ"(X)lδ#(X), X? . (2.4)

ReplacingXbyXYand using the relationδj(XY)lδj(X)Yjπj(X)δj(Y) for allX,Y in , we obtain from (2.4) the relation Γπ"(X)δ"(Y)lπ#(X)Γδ"(Y), which proves property (1) of the proposition.

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Substituting for δ",δ# in (2.4) from (2.2), and using property (1), we obtain property (2).

P2.2. LetΓbe as in Proposition2.1.Then there exist C? ,D? h, Z? E h such that:

(1) RR"lCjD;

(2) H#kH"l"#i(C*kC)jZ.

Proof. WriteLlΓ*R#kR". From the remarks at the beginning of this section, we know thatRjπj(X)Rj? ,jl1, 2, for allXin . We have, from Proposition 2.1,

(Γ(R"jL))*π#(X)Γ(R"jL)lR"π"(X)R"jL*LXjR"LXjXL*R",

so

L*LXjR"LXjXL*R

"? for allX? . (2.5)

From (2.1) and Proposition 2.1, we also have

i[H",X]k"#(R"R"XjXR"R"k2R"π"(X)R")

li[H#,X]k"#((R"jL)*(R"jL)XjX(R"jL)*(R"jL)k2(R"jL)*π"(X) (R"jL)),

which simplifies to

i[H"kH

#,X]l"#[R"LkL*R

",X], X? .

Since every derivation of is inner andH"kH#? , it follows that

H#lH"j"#i(R"LkL*R")jB, (2.6)

whereBlB*? h.

Substituting forLin (2.5), we conclude that [R#ΓR

",X]? , and hence, by the

same argument as above,RR

"can be expressed as

RR"lCjD, C? ,D? h. (2.7)

Substituting forLin (2.6), we conclude that

H#kH"k"#ioR"(Γ*R#kR")k(R#ΓkR")R"q? h.

Now (2.7) implies that H

#kH

"k"#i(C*kC)? E h, which together with (2.7)

completes the proof.

T2.3. Two CE quadruples(j,πj,Rj,Hj),jl1, 2,determine the same CE generatorif and only if there exist a unitary isomorphismΓ:" #,and elements C? , D? h,ZlZ*? E hsuch that:

(1) Γπ"(X)lπ#(X)Γ;

(2) (Γ*R

#kR

")Xlπ"(X) (Γ*R

#kR

") ; (3) RR

"lCjD;

(4) H

#kH

"l"#i(C*kC)jZ.

Proof. Propositions 2.1 and 2.2 imply the ‘ only if ’ part. To prove the converse, considerΓ,C,D,Zsatisfying conditions (1)–(4), and the CE generatorsjdefined by

j(X)li[Hj,X]k"#(RjRjXjXRjRjk2Rjπj(X)Rj), X? ,jl1, 2.

WriteLlΓ*R

#kR

", so that LXlπ"(X)Land R

#lΓ(R

"jL). Then, substituting

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forH

#,R

# andπ#from (1)–(4) in#(X), we obtain #(X)li[H

",X]k"#[C*kC,X]

k"#o(R"jL)*(R"jL)XjX(R"jL)*(R"jL)k2(R"jL)*π"(X) (R"jL)q

l"(X)k"#[C*kCkR"LjL*R",X]

l"(X)k"#[C*kCkR"Γ*R

#jRR

",X]

l"(X)

for allX? .

For constructing Markov dilations, it is useful to modify the CE parametrization.

To this end, we prove the following result.

T2.4. Letbe the generator of a conserŠatiŠe and uniformly continuous quantum dynamical semigroup on aŠon Neumann algebra 9(!).Then there exist a unital completely positiŠe mapΨ: ,a positiŠe element K? ,and a hermitian element H? such that

(X)li[H,X]k"#(K#XjXK#k2KΨ(X)K), X? . (2.8)

Proof. In (1.2), putKl(R*R)"/#and consider the polar decompositionRlVK, where V is an isometry from the closure of the range ofKin ! onto the closure of the range ofRin. Denoting byPthe projection on the closure of the range of Kin !, we see that

R*π(X)RlKPV*π(X)VPKlKΨ!(X)K, where

Ψ!(X)lPV*π(X)VP.

Clearly,Ψ!is a contractive completely positive map satisfyingΨ!(1)lP. Nowcan be expressed as

(X)li[H,X]k"#(K#XjXK#k2KΨ!(X)K), X? . (2.9)

Since (X),H,K? , it follows that KΨ!(X)K? for all X in . Hence KmΨ!(X)Kn? form,n1. Thus for any two polynomials p,qsuch that p(0)l q(0)l0, it follows thatp(K)Ψ!(X)q(K)? . Hence for any two continuous functions },ψon [0,_) satisfying}(0)lψ(0)l0, we have}(K)Ψ!(X)ψ(K)? . Define

}n(x)l

(

nx1 if 0ifxx1\n,1\n,

and observe that w:lim

n _}n(K)Ψ!(X)}n(K)lPΨ!(X)PlΨ!(X)? . Define

Ψ(X)lΨ!(X)j(1kP)X(1kP).

ThenΨis a unital completely positive map from into itself, andassumes the form (2.8).

R. Our construction of a Markov dilation for in the next section depends on the discrete time quantum Markov chain defined by the unital completely positive mapΨon . It should be interesting to know the exact relationship between the parameter triples (H,K,Ψ) and (Hh,Kh,Ψh) which determine the same according to (2.8) in Theorem 2.4.

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3. A MarkoŠdilation for the semigroup et

We consider a CE generator expressed in the form (2.8) of Theorem 2.4 in terms of the parametersH,K,Ψ. SinceΨis a unital completely positive map on , it follows from [1,2] that there exists a unique (up to unitary equivalence) minimal discrete time Markov dilation (,Fn,jn),nl0, 1, 2,…, whereis a Hilbert space containing!as a subspace,oFnqis an increasing sequence of projections in,F!is the projection on!,s:limn _Fnl1,

Fmjn(X)Fmljmn−m(X)), X? , 0mn _, j!(X)lXF

!

andojn(Xn)jn−"(Xn−")(j!(X!)u,Xi? ,nl0, 1, 2,…, u?!qis total in.

Our strategy for constructing the dilation forwill be to imbed (,Fn,jn) in a quantum version of the Poisson process and look at it in an appropriate interaction picture. To this end, we introduce the boson Fock spaceΓ(L#(+)), and consider the Poisson processoN(t)q, whereN(t) is a selfadjoint operator realized as the closure of A(t)jΛ(t)jA(t)jt on the domain of exponential vectors, A, Λ, A being the creation, conservation and annihilation processes of quantum stochastic calculus.

We write (forgoing rigour)N(t)lA(t)jΛ(t)jA(t)jt, with the convention that 1 denotes the identity operator, and a scalar times the identity operator is denoted by the scalar itself. We now make the Poisson imbedding of the discrete time chain by puttingg l(L#(+)) and defining

jN(t)(X)B_

n=!

jn(X)"1onq(N(t)),

where 1onqdenotes the indicator of the singletononqin. We have used the fact that N(t) has spectrumo0, 1, 2, …qfort0, andN(0)l0.

P3.1. Let FN(t)ljN(t)(1).Then:

(i) FN(!)lF!"1Γ(L#(

+));

(ii) FN(s)FN(t)for all0st _; (iii) s:limt _FN(t)l1g.

Proof. (i) is obvious since N(0)l0. To prove (ii), we first observe that N(t)lN(s)jN(t)kN(s), whereN(s) andN(t)kN(s) are ampliations of operators in Γ(L#[0,s]) and Γ(L#[s,t]), respectively, in the factorization

Γ(L#(+))lΓ(L#[0,s])"Γ(L#[s,t])"Γ(L#[t,_)).

Thus

FN(t)l_

n=!Fn"1onq(N(t)) l_

n=!Fn"n

j=!1ojq(N(s))"1on−jq(N(t)kN(s)) l

j!,k!

Fj+k"1ojq(N(s))"1okq(N(t)kN(s))

j!,k!

Fj"1ojq(N(s))"1okq(N(t)kN(s)) lFN(s).

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This proves (ii). Finally, FN(t)l_

n=!

Fn"(1on,n+

",q(N(t))k1on+

",n+#,q(N(t)))

l_

n=!(FnkFn−")"(1k1o!,",#,,n−"q(N(t))).

By the isomorphism [11] between Γ(L#(+)) and the L# space with respect to the probability measure of the Poisson process of unit intensity, and the fact thatN(t) viewed as a Poisson random variable tends to _ with probability 1 as t _, it follows that

s:lim

t _

FN(t)l_

n=!

(FnkFn−

")"1Γ(L#(

+))l1g.

In the von Neumann algebra (g ), we consider the Fock vacuum conditional expectationt]which is defined as follows. For anyX?(g ), consider the operator Xt on (L#[0,t]) defined by f},Xtψglf}"Ω[t,Xψ"Ω[tg, where Ω[t is the Fock vacuum vector inΓ(L#[t,_)), and putt]XlXt"1[t, where 1[tis the identity operator inΓ(L#[t,_)).

P3.2. Let FN(t),jN(t) be as in Proposition3.1.Then s]FN(s)jN(t)(X)FN(s)ljN(s)(St−s(X)), 0st _,X? , where

St(X)let(Ψ−id)(X), X? , idbeing the identity map on .

Proof. We have, from properties of the Poisson processoN(t)q, FN(s)jN(t)(X)FN(s)l

n!Fn"1onq(N(s))

n!jn(X)"1onq(N(t))

n!Fn"1onq(N(s)) l

k,n!

Fkjn(X)Fk"1okq(N(s))1onq(N(t)) l

nk!

Fkjn(X)Fk"1okq(N(s))1on−kq(N(t)kN(s))

l

k!,n−k!jkn−k(X))1okq(N(s))1on−kq(N(t)kN(s)).

Now, applying s] on both sides, s]FN(s)jN(t)(X)FN(s)l

k!,%!

jk%(X))1okq(N(s))e−(t−s)(tks)%

%! ljN(s)(e(t−s)(Ψ−id)(X)).

C3.3. Let

jgt(X)ljN(t)(X)"QΩ[t[tQ, Fg

tljgt(1)lFN(t)"QΩ[t[tQ. Then(g ,Fg

t,jg

t), t0,is a MarkoŠdilation for the conserŠatiŠe quantum dynamical semigroupoet(Ψ−id)q,t0.

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Proof. Immediate.

P 3.4. Let H,K be hermitian elements in . Then the quantum stochastic differential equation

dW(t)l ojN(t)(H) (dAkdA)jjN(t)(kiKk"#H#)dtqW(t) (3.1) with W(0)l1admits a unique isometric solution W(t).

Proof. The proof is along the same lines as in Section 4 of [4]. WriteW!(t)1, and define iteratively

Wn(t)l1j

&

!tojN(s)(H) (dAkdA)jjN(s)(kiKk"#H#)dsqWn−"(s).

By the inequality (ii) of Proposition 27.1, page 222 of [11], we conclude that

n

R(Wn(t)kWn−

"(t))fe(u)R _

for allf?and exponential vectorse(u) inΓ(L#(+)). This implies the convergence ofWn(t)fe(u) ing asn _. Denoting this limit byW(t)fe(u), we obtain a solution of (3.1). A routine application of quantum Ito’s formula implies the isometric property ofW(t). Uniqueness follows from the fact that any solution of (3.1) with initial value 0 is identically 0.

P3.5. Let

jN(t)+k(X)l_

n=!

jn+k(X)"1onq(N(t)), k0.

Then

djN(t)+k(X)l(jN(t)+k+"(X)kjN(t)+k(X))dN(t).

Proof. We have

djN(t)+k(X)l

(

n=_!jn+k(X)"(1onq(N(t)j1)k1onq(N(t)))

*

dN(t)

l

(

n=_"jn+k(X)"1on−"q(N(t))kjN(t)+k(X)

*

dN(t)

l(jN(t)+k+"(X)kjN(t)+k(X))dN(t).

P3.6. The isometric process oW(t)qof Proposition3.4is unitary.

Proof. Let X(t)l1kW(t)W(t)*. Then oX(t)q is a projection-valued Fock adapted process with initial value 0. The proposition will be proved if we show that dX(t)l0. By a routine application of quantum Ito’s formula and some algebra, we obtain

dX(t)l[jN(t)(H),X(t)] (dAkdA) (t)

k[ojN(t)(iK),X(t)]j"#[jN(t)(H), [jN(t)(H),X(t)]]qdt. (3.2)

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DefinePn(t)l1onq(N(t)), and observe that dP!(t)l kP

!(t)dN(t), dPn(t)l(Pn−

"(t)kPn(t))dN(t) ifn1.

This, together with (3.2), quantum Ito’s formula and some tedious algebra, implies dPnXPn(t)l(Pn−

"XPn−

"kPnXPn) (t)dN(t)

jPn−

"(t) [jN(t)(H),X(t)]Pn(t)dA(t)jPn(t) [X(t),jN(t)(H)]Pn−

"(t)dA(t)

joPn−

"(t) [jN(t)(H),X(t)]Pn(t)jPn(t) [X(t),jN(t)(H)]Pn−

"(t)

kPn(t) ([jN(t)(iK),X(t)]j"#[jN(t)(H), [jN(t)(H),X(t)]])Pn(t)qdt. (3.3) Note that operators and their ampliations to tensor products have been denoted by the same symbols. SincePk(t) andjN(t)(B) commute with each other, andPk(t)jN(t)(B) ljk(B)Pk(t)lPk(t)jk(B) for any Bin , (3.3) can be expressed as

dPnXPnl(Pn−

"XPn−

"kPnXPn)dN

j(jn−"(H)Pn−

"XPnkPn−"XPnjn(H))dA

j(PnXPn−"jn−"(H)kjn(H)PnXPn−")dA

jojn−"(H)Pn−"XPnkPn−"XPnjn(H)jPnXPn−"jn−"(H)

kjn(H)PnXPn−"j[jn(kiK),PnXPn]

j"#[jn(H), [jn(H),PnXPn]]qdt. (3.4)

Puttingnl0, we obtain

dP!XP!lkP!XP!dNjo[j!(kiK),P!XP!]k"#[j!(H), [j!(H),P!XP!]]qdt.

This is a constant operator coefficient quantum stochastic differential equation (qsde) for P!XP! with initial value 0. Hence (P!XP!) (t)l0. Since X(t) and P!(t) are projections, we conclude that P!(t)X(t)lX(t)P!(t)l0. Let us now make the induction hypothesis thatPn−"(t)X(t)lX(t)Pn−"(t)l0. Then (3.4) becomes

dPnXPnlkPnXPndNjo[jn(kiK),PnXPn]j"#[jn(H), [jn(H),PnXPn]]qdt,

which is once again a constant operator coefficient qsde for PnXPn with initial value 0. Hence (PnXPn) (t)l0, which implies that Pn(t)X(t)lX(t)Pn(t)l0. Thus X(t)Pn(t)l0 for everyn0. Sincen!Pn(t)l1, we conclude thatX(t)0.

P3.7. LetoW(t)qbe the unique unitary solution of the equation(3.1)in Proposition3.4.Then,for any X? ,

dW(t)*jN(t)(X)W(t)

lW(t)*o(jN(t)+"(X)kjN(t)(X))dN(t)j(jN(t)+"(X)jN(t)(H)kjN(t)(HX))dA(t) j(jN(t)(H)jN(t)+"(X)kjN(t)(XH))dA(t)

j(jN(t)(HΨ(X)Hk"#(H#XjXH#)kHXkXHji[K,X])

jjN(t)+

"(X)jN(t)(H)jjN(t)(H)jN(t)+

"(X))dtqW(t). (3.5)

Proof. This is immediate from Proposition 3.5 for the casekl0, equation (3.1), quantum Ito’s formula, and the fact that

jN(t)(H)jN(t)+"(X)jN(t)(H)ljN(t)(H)FN(t)jN(t)+"(X)FN(t)jN(t)(H) ljN(t)(HΨ(X)H).

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P3.8. Let W(t)be as in Proposition3.7.Then FN(t)W(t)lW(t)FN(t).

Proof. PutXl1 in Proposition 3.7. SinceΨ(1)l1 andFN(t)+

"FN(t), we have,

from (3.5),

W(t)*FN(t)W(t)lF

!j

&

!tW(s)*(FN(s)+"kFN(s))W(s)dN(s). (3.6)

On the other hand, the differential equation forWimplies

W(t)l1j

&

!tojN(s)(H) (dAkdA) (s)jjN(s)(kiKk"#H#)dsqW(s)

l1jFN(t)

&

!tojN(s)(H) (dAkdA) (s)jjN(s)(kiKk"#H#)dsqW(s)

l1jFN(t)(W(t)k1),

or W(t)l1kFN(t)jFN(t)W(t). Substituting this in the right-hand side of (3.6), we have

W(t)*FN(t)W(t)lF

!j

&

!t(FN(s)+"kFN(s))dN(s)

lFN(t), by Proposition 3.5.

P3.9.LetoW(t)qbe as in Proposition3.7.Then

FN(s)s](W(t)*jN(t)(X)W(t))FN(s)lW(s)*jN(s)(e(t−s)(X))W(s) for all X? , 0st _,where

(X)li[K,X]k"#((Hj1)#XjX(Hj1)#k2(Hj1)Ψ(X) (Hj1)).

Proof. From Proposition 3.7 and basic quantum stochastic calculus, we have s]W(t)*jN(t)(X)W(t)

lW(s)*jN(s)(X)W(s) j

&

st

s]W(τ)*ojN(τ)(HΨ(X)Hk"#(H#XjXH#)kHXkXHji[K,X])

jjN(τ)+

"(X)jN(τ)(H)jjN(τ)(H)jN(τ)+

"(X)jjN(τ+

")(X)kjN(τ)(X)qW(τ).

Pre- and post-multiplying byFN(s)on both sides, noting thatFN(s)lFN(s)FN(τ)forτs, and using Proposition 3.8, we obtain

FN(s)os]W(t)*jN(t)(X)W(t)qFN(s) lW(s)*jN(s)(X)W(s)j

&

st

FN(s)s]W(τ)*jN(τ)(HΨ(X)Hk"#(H#XjXH#)

kHXkXHji[K,X]jΨ(X)HjHΨ(X)jΨ(X)kX)W(τ)FN(s) lW(s)*jN(s)(X)W(s)j

&

stFN(s)os]W(τ)*jN(τ)((X))W(τ)qFN(s)ds.

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Now the result follows from general principles of ordinary differential equations.

T 3.10. Let be the Christensen–EŠans generator of a uniformly continuous semigroup of unital completely positiŠe maps on a unital Šon Neumann algebra 9(!)giŠen by

(X)li[K,X]k"#(H#XjXH#k2HΨ(X)H), X? ,

where H and K are hermitian elements in , H0, and Ψ is a unital completely positiŠe map on .Let(,Fn,jn),n0,be a MarkoŠdilation of the discrete semigroupnq,n0.Letg l(L#(+)),N(t)lA(t)jΛ(t)jA(t)jt,

Fg(t)lFN(t)(1t]"QΩ[t[tQ),

where1t]is the identity operator in"Γ(L#[0,t])and[t is the FockŠacuumŠector inΓ(L#[t,_)),and

jgt(X)lW(t)*jN(t)(X)W(t) (1t]"QΩ[t[tQ), whereoW(t)qis the unique unitary solution of the qsde

dW(t)l ojN(t)(Hk1) (dAkdA) (t)kjN(t)(iKj"#(Hk1)#)dtqW(t)

with W(0)l1.Then(g ,Fg(t),jgt),t0,is a MarkoŠdilation of the semigroup oetq, t0.

Proof. This is immediate from Proposition 3.9.

R. It is curious that a shift ofHbyk1 is required in the equation forW in order to construct the Poisson imbedding in the interaction picture for obtaining the dilating homomorphismsjg

t. It is also to be noted that we have dealt with the case when no ‘ structure maps ’ in the sense of Evans and Hudson may be available for writing a flow equation for the required dilation.

References

1. B. V. R. Band K. R. P, ‘ Kolmogorov’s existence theorem for Markov processes in C* algebras ’,Proc.Indian Acad.Sci.Math.Sci. 104 (1994) 253–262.

2. B. V. R. Band K. R. P, ‘ Markov dilations of nonconservative quantum dynamical semigroups and a quantum boundary theory ’,Ann.Inst.H.PoincareT Probab.Statist. 31 (1995) 601–652.

3. E. C and D. E. E, ‘ Cohomology of operator algebras and quantum dynamical semigroups ’,J.London Math.Soc. 20 (1979) 358–368.

4. M. P. Eand R. L. H, ‘ Perturbations of quantum diffusions ’,J.London Math.Soc. 41 (1990) 373–384.

5. R. L. H, ‘ Quantum diffusions on the algebra of all bounded operators on a Hilbert space ’, Quantum probability IV, Lecture Notes in Math. 1396 (ed. L. Accardiet al., Springer, New York, 1989) 256–269.

6. R. L. H and K. R. P, ‘ Quantum Ito’s formula and stochastic evolutions ’, Comm.Math.Phys. 93 (1984) 301–323.

7. R. L. Hand P. S, ‘ Stochastic dilation of quantum dynamical semigroups using one- dimensional quantum stochastic calculus ’,Quantum probability V, Lecture Notes in Math. 1442 (ed. L. Accardiet al., Springer, New York, 1992) 216–218.

8. G. L, ‘ On the generators of quantum dynamical semigroups ’,Comm.Math.Phys. 48 (1976) 119–130.

9. P. A. M,Quantum probability for probabilists, Lecture Notes in Math. 1538 (Springer, New York, 1993).

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10. A. Mand K. B. S, ‘ Quantum stochastic flows with infinite degrees of freedom ’,Sankhya_ Ser.A52 (1990) 43–57.

11. K. R. P,An introduction to quantum stochastic calculus(Birkha$user, Basel, 1992).

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References

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