Quantum stochastic processes I

PHYSICS REPORTS (Review Section of Physics Letters) 77, No. 3(1981)339—349. North-Holland Publishing Company

Quantum Stochastic Processes 1* J.T. LEWIS Dublin Institute for Advanced Studies, Dublin 4, Ireland

1. You might well ask: Why on earth should we concern ourselves with quantum stochastic processes? My answer would be: Because that is what the quantum theory of open systems is really about; whenever a system is coupled to a heat bath or reservoir its evolution has a stochastic element which is absent from the Hamiltonian evolution of a closed quantum mechanical system. However, others might give you very different answers because they have other examples in mind. I do not intent to make a proprietary claim to the name “Quantum Stochastic Process”; my talk might be entitled more accurately (though more awkwardly) “A Class of Non-Commutative Stochastic Processes analogous to Classical Stochastic Processes in the sense of Doob”. What I hope to do is to convince you that there is a very strong analogy between the evolution of an open quantum system and the evolution of a classical stochastic process in Doob’s sense, and that it is possible to construct a mathematical theory which embraces them both. Moreover, I hope to persuade you that the exercise is worthwhile by showing you how it clarifies the dilation problem for Quantum Dynamical Semi-groups.

2. Let us begin with Doob’s definition [1] of a stochastic process indexed by the reals: it is a family {X5: t E R} of random variables on a measurable space (.E, .~),which is equipped with a probability measure P. For Doob, the finite dimensional joint distributions are the basic distributions of the theory: processes are classified according to their finite-dimensional distributions. Two processes (X5, li’), (X5, 1~,P) are equivalent if and only if their finite-dimensional joint distributions coincide: .~,

(2.1) for all finite sequences t1, , t~and n E N. As Meyer [2]puts it: “A stochastic process is a mathematical representation of a natural phenomena whose evolution is governed by chance. Suppose that we have observed a very large number of independent realizations of this phenomena. We know then with arbitrary precision (thanks to the laws of large numbers) the expression that figures in the formula (2.1) for an arbitrarily large number of times t1 t~but observation can help us no further. In other words ‘Nature can give us stochastic processes only up to an equivalence’ This is not the place to make a detailed logical analysis of the concept of a stochastic process; nevertheless, I believe it to be instructive to notice that, crudely, we have the following structure: we single out a class of expressions . . .

“.

This article and the one which follows were originally delivered by their respective authors at the Statistical Mechanics Conference. The Open University, Milton Keynes, 11th December 1979. Both parts were covered in the Les Houches talk by J.T. Lewis.

0 370-1573/81/0000—0000/$2.75 © 1981 North-Holland Publishing Company

340

New stochastic methods in physics

P

Mathematical Representation

L~of a Process Basic Objects

Observations as basic (in this case, the finite-dimensional joint-distributions), we choose a mathematical representation of the process (in this case, a family of measurable functions on a measurable space equipped with a normalized measure) from which we can compute the values of the basic expressions; on the other hand, some of the basic expressions can be estimated from observation of the natural phenomenon. Thus the natural phenomenon and its mathematical representation make contact through the basic expressions. If our choice of basic expressions is good the collection of all basic expressions determines the representation up to equivalence (in this case, this is guaranteed by Kolmogorov’s reconstruction theorem). It might seem that the mathematical representation of the process is redundant; on the contrary, a rich mathematical structure is necessary for a fruitful theory. This consideration influences our choice of basic expressions. Indeed, it is the tension between the competing demands of mathematical elegance and observational feasibility which shapes the structure of the theory. 3. Next we apply these considerations to the choice of structure for a quantum theory of stochastic processes. First, in what category should we seek a mathematical representation? Considerations of mathematical elegance and the experience of the past twenty years of mathematical physics conspire: without hesitation I plump for C*~algebrasas objects and completely positive maps as morphisms. Then the state of the system is represented by a positive normalized linear functional w on an algebra ~i; the evolution of a closed system is represented by a group {u~t E R} of automorphisms of the algebra; observables, by self-adjoint elements of the algebra. I am much more hesitant over the choice of basic expression. The quantum constraints on measurement have to be given careful consideration. In the first place, not all elements of the algebra .s~1will be directly accessible to measurement at a fixed time; assume that those which are accessible at time t generate a sub-algebra The group of automorphisms moves the sub-algebras around: ,~.

= u5_5(~);

(3.1)

assume that the collection of all such sub-algebras generates the whole algebra: s~= v{.~’,:tER}.

(3.2)

In the second place, if two observables fail to commute they do not have a joint distribution in the classical sense: a quantum generalization of this concept must take account of the order in which they were measured and the way in which the first measurement was made. This is enshrined in the concept

J.T. Lewis, Quantum stochastic processes 1

341

of filtering: an elementary measurement of a seif-adjoint element a of s.~i ‘filters’ the state w by a mapping co i-~xw, where (xw)(b) = w(x*bx)/w(x*x);

(3.3)

different decompositions of a as x”x correspond to different elementary measurements. Consequently, the basic expressions which are derivable from observation are the time-ordered correlations ~ a1), where a,, is in .s~and t1 t2 ~ t~.If the commutation rules for the algebra ~ . .

.

.

are known then the correlations associated with arbitrary times t1,. , t~are determined by the time-ordered ones, but in general the time-ordered correlations are insufficient to determine (.~1,u,~,w). If we insist that two representations are equivalent if they have the same time-ordered correlations and that the representation be reconstructable, up to equivalence, from the basic expressions then we will have to settle for a much uglier structure than . .

(~=

v{u~(~~): tER},

(3.4)

w).

So let us swallow any scruples we may have about regarding as basic some expressions which have no imaginable correspondence with observation, and investigate the consequence of the following definitions (which describe a mild generalization of the situation we have considered). 4. Let ~ be C*~algebraswith identity, let {j5: t E R} be a family of *~isomorphismsof ~ into i& such that jt(1~)= 14; suppose that is generated as a C*~algebraby the image algebras ~, = j,(~), t E R; let cv be a state on ,s~1such that the GNS representation of stit which it determines is faithful; then the pair (Ut: t E R}, co) is said to be a C*~stochasticprocess over ~ evolving in sti. Let ({j~},w) and ({j}, c~i)be C*~stochasticprocesses over ~ evolving in s~1respectively; they are said to be equivalent if there exists a *~isomorphismu of ~ onto .s&’ such that J’ = u for all tin R and .~‘,

.~‘

-

~‘,

0

U

(0.

The assumption that the GNS representation determined by cv is faithful looks a bit technical; it has the following consequence: 1. may be identified with a concrete C*~algebraof operators acting on a Hilbert space ~‘; there is a normalized vector ll in such that co(a) = (U, all) for all a in ste’. 2. Two stochastic processes ({j~}~ w) and (f31}, ~i) over ~ are equivalent if and only if .~‘

~

w(jt,(ai)*

.

j,~(a~)~’ j5~(b~) . .

.

j~1(bi))= ~(fti(ai)*

.

.

j,~(a~)’~’ j,~(b~) j51(b~)) .

(4.1)

for all t1, , 4, and all a1, , a~,b1, , b~. A W*~stochasticprocess can be defined in a similar way with stit and ~ being W*~algebras,{J~}a family of normal *~isomorphismsof ~ into sti generating as a W*~algebra,and cv a normal state on ~‘; if two W*~stochasticprocesses are equivalent, the *~isomorphismU: ~ is automatically normal. This definition includes classical stochastic processes in the sense of Doob: If stl~ and ~ are Abelian W*~aIgebras,there are measurable spaces (1~,~) and (S, ~) such that sti can be identified with L°°(1, ~), and ~ with L~(S, A normal state cv on defines a probability measure ~con (Z, ~) such that . . .

. . .

. . .

.~‘

~‘).

=

d~

.~‘

for all f in L~(I,~),

(4.2)

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New stochastic methods in physics

and the GNS representation of sti determined by

~ 4a) of n-equivalence classes of bounded measurable functions on (X, ~). A W*~isomorphismI~:L~(S,~‘)—* ~ ~, ~s) is determined by a ti-equivalence class of functions X,: (.X, ~‘)-+ (S. ~‘) such that j~(g)= go X, for all t in R and X,(1~)= S for all t. The argument can be reversed; thus a W*~stochasticprocess for which ~ and ~ are Abelian corresponds to a classical stochastic process in the sense of Doob. Consequently, we say a W*~stochasticprocess over ~ evolving in s~’is classical if and only if both st~’ and ~ are Abelian. j.c

is the space ~

5. The next step is to look for a generalization of Kolmogorov’s reconstruction theorem. To state the theorem I should at this point define an inductive family of correlation kernels over an algebra; to do so would disrupt the exposition by confusing you with unfamiliar notation.t I will be content with telling you that every stochastic process ({j,}, cv) over an algebra ~ gives rise to an inductive family of correlation kernels {w,(~; ): t E R”, n E N} over ~ through the relation w,(a;

b) = w(j51(ai)*

. .

.

j(a)* j5~(b~) j~1(b1)), .

t1

.<

. .

.

<4,,

(5.1)

where t = and a = (a1 ar), b = (b1,. ba). The reconstruction theorem is the converse of this proposition, namely: Given an inductive family {w5(~; )} of correlation kernels over a C*~algebrawith identity, then there exists a C*~stochasticprocess ({j5}, w) over the algebra such that the relation (5.1) holds; the process is unique up to equivalence. The corresponding theorem in the category of W*~algebrasis true, and is a genuine generalization of Kolmogorov’s theorem: a W*~processis classical if and only if its correlation kernels satisfy a symmetry condition; if this is the case, the reconstruction theorem specializes to Kolmogorov’s theorem. Any property of an equivalence class of stochastic processes can be formulated either in terms of the structure ({j,}, cv) or in terms of the correlation kernels {w,(~)}.For example, we say that a C*~ stochastic process ({j~},cv) over ~ evolving in ~ is stationary if there exists a group {u~: E R} of automorphisms of s&~such that (t~

t~)

. . ,

t

Jg+s = U, Oj,,

(0 =

(00

U,,

(5.2)

for all s and t in II; equivalently, the process is stationary if and only if its correlation kernels are invariant under time-translation. For the rest of this talk I shall confine myself to stationary processes. 6. Although the Kolmogorov reconstruction theorem is of great theoretical importance, it is not of much use in practice: we cannot list all the joint probability distributions. Consequently, attention centres on two classes of processes for which there are compact ways of describing the hierarchy of joint probability distributions: 1.. Markov processes, where the joint probability distributions are determined by a semi-group. 2. Gaussian processes, where the joint probability distributions are determined by the mean and the covariance functions. It remains to be seen what classes of quantum stochastic processes turn out to be important. The first step is to look critically at analogues of classical Markov and Gaussian processes; in tThe details are given in an Appendix.

J.T. Lewis, Quantum stochastic processes I

343

the talk which follows, A. Frigerio describes some quasi-free processes which have some features in common with Gaussian processes. I am going to tell you something about non-commutative Markov processes. To discuss the Markov property it is necessary to introduce some sub-algebras of the algebra s~’in which a process ({j,}, cv) over ~ evolves. Let ~, s4, and ~ be the past, present and future algebras, relative to the time t, defined by s~=v{j5(b):bE~& st},

(6.1)

sti= vU,(b):bE~},

(6.2)

=

V

{j~(b):b E ~,

s

t}.

(6.3)

To discuss questions of independence we would like to have projection maps onto these algebras, with several good properties. In the classical case, conditional expectations onto a sub-algebra always exist: given an algebra a state cv and a sub-algebra ~, these exists a unique map E: stI-+ ~ such that .~‘,

CE1: E(a1E(a2)) = E(E(a1)a2) for all a1, a2 in

E(a1) E(a2)

.~,

CE2: E(14) = 1~, CE3: E(a)0 whenever a furthermore, E is compatible with cv in the sense that CEC: wcvl’cc’E. Then the Markov property can be stated for a stationary process as Ml: E01st110 = When the Markov property holds, there is a state Co0 on ~ and a semi-group {Z~:t into itself such that cvo=

for all t0,

C0o~Z,

0} of maps of ~

(6.4)

the multi-time correlations are given by the regression relation cv(jri(ai)*.

.

j,ja~)*j,,,(b~).

. j0(b1))

=

cvo(a~Z0_,1(a~. Z,~_,~_1(a~b~) b2)b1), .

. .

(6.5)

and Z is given by Z,=j~oEo1ou,ojo,

t0.

(6.6)

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New stochastic methods in physics

Moreover, the conditional expectation E,1 onto sti,~satisfies the relations E,1 = U, ° E01 ° U_, E~1E,1= ESA,l

for all t,

(6.7)

for all s, t.

(6.8)

The question arises: how much of this carries over to the non-commutative situation? Conditional expectations can be defined by CE1, CE2, CE3 and they turn out to be completely positive whenever they exist; the partial trace is a familiar example of a conditional expectation on a non-commutative algebra. The fact is, however, that conditional expectations which are compatible with a given state are relatively rare in the non-commutative situation. For example, suppose ~ is a sub-algebra of st~’ and ~ fl ‘~‘ consists of multiples of the identity; if there exists a conditional expectation E: ~ ~ compatible with a state cv on then w(ac) = cv(a) w(c) for all a in the relative commutant s.~/fl of ~ and c in This means that if corresponds to an open system and ~ fl represents a reservoir, the existence of a conditional expectation onto the open system, compatible with a given state cv, places a severe restriction on o: with respect to that state, system and resevoir are independent. On the other hand, the evolution of an open system in physics is usually described by a differential equation; in other words, by a semi-group. (I have the impression that an evolution equation with a memory term is regarded as belonging to technology rather than to pure physics.) The differential equation is phenomenological: it is based on ad hoc assumptions, rather than on a systematic derivation from the basic laws of physics. Such phenomenological equations are often highly successful in describing the approach to thermal equilibrium. The challenge to the theorist is to derive these equations from the fundamental laws of physics; that is, from the evolution of a closed system: a group of automorphisms of an algebra with an invariant state. In the .theory of quantum stochastic processes we are concerned with the general features of such derivations, with existence theorems and ‘no-go’ theorems. An open system whose evolution is described by a semi-group is loosely described as having a Markov evolution; we would like to know in what precise sense this may be the case. Sometimes the regression relation (6.5) is taken as a definition of the Markov property: how is it related to ‘absence of memory’? ~

~.

~‘

~‘

~‘

7. We choose the ‘absence of memory’ definition of the Markov property for a quantum stochastic process. Let (Ut}~cv) be a stationary C*~stochasticprocess over an algebra ~ evolving in sti with automorphism group {U,: tE R}; we say that the process is Markov if there exists a family {E,1} of conditional expectations E,1: sti—* compatible with cv and such that conditions (Ml-3) hold: ~

(Ml):

Eo1.~’[() =

(M2):

E,1 =

(M3):

E~1E,1= ~

~s,oEo1

°

U~..,

for all

for all s,

t;

t.

Let (Ut}~cv) be a stationary process; if the process is Markov, the family of maps {Z,: t Z,=j~oEo]oU,ojo

0} defined by (7.1)

I T Lewis

is a semi-group leaving invariant the state Coo

Quantum

stochastic processes I

34S

w 010; and the regression relation holds:

w,(a; b) = cvo(a~Z0_,,(a~ . Z,~_,~_,(a~b~) b~)b1) .

.

(7.2)

.

for all t = (t1, ,t~),a = (a1, ,a~),b = (b1, ,b~)In the converse direction we have the following result: Let ({j,}, cv) be a stationary W*~stochasticprocess over ~ evolving in ~ and assume that ~ is the weak closure of the linear span of the finite time-ordered products j,1(b1)~. . ~ t1 t2 ~ tn, and that the cyclic vector U,, is also separating. If there is a semi-group {Z,: t 0} on ~ such that the regression relation (7.2) holds, then a family {E,1} of compatible conditional expectations exists and the semi-group {Z,} is given by the formula (7.1). If, in addition, a semi-group {Z~:t 0} exists such that the reversed regression relation holds for t1 t2 ~ . 4,, then there is a conditional expectation N: ~i—*~ compatible with cv so that .

Zt=Nou,ojol

(7.3)

>

)

Z_,=Nou_,ojoJ

(7.

cvo(Z,(b1) b2) = wo(bi Z,(b2))

(7.5)

and

for all b1, b2 in The map Z, is completely positive (since it is the composition of completely positive maps) and identity-preserving; a semi-group of such maps is called a quantum dynamical semi-group. The dilation problem for a quantum dynamical semi-group is this: given a qds {Z,: t 0} and an invariant state W~On 9~can we find an algebra with automorphism group {U,: t E R} and an invariant state cv, together with an embedding Jo: ~ and a compatible conditional expectation N: sti—~~ such that the relation (7.3) holds? Presumably the conditions that ~ be generated as a vector space by the time-ordered products, and that the cyclic vector D,~be separating, can be translated into conditions on an inductive family of correlation kernels; in which case, we could formulate sufficient conditions for a solution of the dilation problem in terms of regression relations. The underlying process would then be Markov. We have seen that there are circumstances in which the regression relation for all correlation kernels implies the Markov property. The existence of a semi-group evolution for an open system is not enough to ensure the Markov property; an example due to Lindblad, inspired by the phenomenon of spin-echoes in nuclear induction, shows that a C*~stochasticprocess can have its two-time correlations given by a quantum dynamical semi-group according to the relation ~.

.~

~

—~

I .1. 1.\_ I *‘7I w0,,~a1,a2, Ui~02) w0~a ~ —

*I,\l. 202)01

without the process being Markov. Another subtlety of quantum stochastic processes is the phenomenon of quantum thermal memory. In the Ford—Kac—Mazur model at inverse temperature f~we have an embedding j: ~ .s~,an automorphism group {U~} of .cil and a state cv~which satisfies the KMS condition at inverse temperature ,6 with respect to the automorphism group {u~};putting J, = U, oj we have a C*~stochasticprocess ({J,}, Co~~) which is non-Markov for 0< f3 ~ In the zero-temperature limit (/3 cc) the process is a non-commutative Markov process; in the infinite-temperature limit (13 —*0) the —~

—~

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New stochastic methods in physics

process is a classical Markov process. The memory does not originate in the dynamics; it is a consequence of the KMS state, which is why I have called it ‘quantum thermal memory’. We have seen that non-commutative Markov processes are relatively rare because of the strong conditions imposed on a state by the existence of a family of compatible conditional expectations. The next question to be answered is: which processes are ‘close’ to Markov processes? Before we attempt to answer this, we must agree on a sense for ‘close’. Here again the basic expressions are our guide: two processes are close to one another on a fixed time-interval if their correlation kernels are numerically close on that interval.

Bibliographical notes 1. Results reported in this talk were obtained in a collaboration with L. Accardi and A. Frigerio; see [3] and the talk by A. Frigerio which follows this one. The algebraic structure we use was introduced by Accardi [4] in his papers on non-commutative Markov processes; his motivation differs in important respects from that which I have described in this talk. For other aspects of non-commutative stochastic processes, see the work of Hudson and his collaborators [5] on analogues of Wiener processes, the pioneering investigations of Davies [6] into waiting times in quantum theory, as well as the series of papers by Lindblad [7]. 3. Segal [8] was the first to point out the importance of abstract C*~algebrasin the quantum theory of infinite systems. Haag and Kastler [9] provided the first satisfactory physical interpretation of the algebraic formalism; the filtered state representation of measurement is given in their paper. The point of view which gives primacy to ‘state’ and ‘operation’, while relegating ‘observable’ to the status of a derived concept, is older; I learned it from the papers of Schwinger [10]. Statistical concepts were developed systematically from this standpoint in [11]; it now seems to be standard in the quantum theory of signal detection [12]. The idea that multi-time correlations are basic we owe to Glauber [13]. 4 and 5. The proofs of results stated in these sections are given in Accardi, Frigerio and Lewis [3]; the proof of the reconstruction theorem is similar to the Dubin—Sewell construction [14] of the dynamics of an infinitely extended quantum system. 6 and 7. Again the proofs are given in [3]. I should have said that I assume that the map t—*j,(b) is continuous for each b, so that the algebras ~ and = V {j~(b):b E ~, s < t} coincide. Apart from the simplification which flows from this assumption, our definition of the Markov property is the one given by Accardi in [4]. Roughly speaking, it states that the future, given the present is independent of the past. There is another result which confirms the view that compatible conditional expectations are rare in the non-commutative situation. Takesaki [151proved if ~ and S&’t] are W*~algebrasand cv is a faithful normal state then a conditional expectation E,1 of stci onto which is compatible with cv exists ~

~

if and only if st~ is stable under the modular automorphism group of s~~’ associated with cv. Noncommutative Gaussian processes are discussed in [161. The proofs are given in [3]. For the dilation problem for quantum dynamical semi-groups see [16] and the review by Evans [17].For the quantum regression theorem see Haake [18]and literature cited therein; Lindblad [7] uses the regression property as a definition of a Markov process. His ‘spin-echo’ example is given in [7] and discussed from our point of view in [3]. The quantum Ford—Kac—Mazur process [19]is discussed in [15].

J. T. Lewis, Quantum stochastic processes 1

347

Appendix In order to state the reconstruction theorem we need some notation: LetT = U,,EJ R~anelementtofTisinR~forsomen = n(t)andisoftheformt =

(t

1, , t~).Foreach n(t)copiesof~anddenote elementsof ~&byb= (b1,.. , ba), . . .

tin T, let PIJ, be the Cartesian product ofn n = n(t).

=

.

Definition. For each t in T the correlation kernel ‘w, of the stochastic process (Ut}~w) over ~ is the functional on ~, X ~, with values in C given by w,(a; b) = cv(1,(al)*.

.

j,,(b~)~ . j,(b1)) for all t =

. J,(~)*

(t1,

..

.

, t~)

in T, a = (a1, , an), b = (l,~ b~)in For s in R, tin R~,a in ~, b in we write s, t for (Si, , Sm, t1, , t~)and a, b for (a1, , am, b~)in Ps,,. For 1 ~ k n, let A: R~—*R”~be defined by At = (t1, . , tk_1, tk+1, , ta), tE R”, . . .

~,.

~,,

. . .

. . .

. . .

. .

. . .

and A:~,—*~iby f~b=(bl,...,b~,bk_l,bk+i,...,b~),bE~,.ForsinRm,tinR1,wesaystifmn and s

=

A

A

At

..

with {k1

kr}

C {1

n}. This defines a partial order on T; for s ~t we can

write t uniquely as 5m~Um 1, t

=

~

. . .

, UOk(O), Si, U~, .

. .

, ~

. . .

, Umk(m)),

Un,

. . .

, Ur~,) ~ Sr

,

. . .

forallr=1,...,m. Then we define an embedding f of

~.

into

~,

by

,1,b

f~b~(1

1,1 k (0) times

Then f~is the identity map of

k (1) times

~,

and f~’of~ = f~for all s

k(m) times

t ~cu in T.

Proposition. The family {w,(~; ): t E T} of correlation kernels of a stochastic process over ~ satisfies the following conditions: CK1 (sesquilinearity): For all tin T and a, b in the map bk i-~w,(a; b)from ~ to C is linear, and the map ak w,(a; b) from ~ to C is conjugate-linear, for each k such that 1 k CK2 (positivity): For all tin T and for all finite sequences {c1} in C, {b~}in ~,

~-‘~

~,,

Ië~c1w,(b1b1)

0;

CK3 (normalization): w,(l; 1) = 1 for all tin T where 1 = (1,. 1) E ~,; CK4(*~condition):For all tin T and s in R, the map a, b -* w,,5(a, a; b, b)of ~ x ~ to C factorsthrough the map a,bl~*a*bof ~ CK5 (faithfulness): For all s in R, ~ 1; b, b) = 0 for all a, b in ~, and tin T, if and only if b = 0; . .,

348

New stochastic methods in physics

CK6 (multiplicativity): Whenever tin T is such that

tk = tk_ ~, then

w,(a; b)

=

w A ~(Aa; Ab) for all a, b in

CK7 (inductivity): For all s, tin T, such that s t, w,(a; b) = w,(f~a;f~b) for all a, b in P1~,. The proof is a straightforward verification. Notice that, because of the automatic continuity of positive linear functionals on a C*~algebra,it follows from CK1, CK2 and CK4 that the map b F-* w,,5(a, 1; a, b) is a continuous linear functional on ~ for all a in ~ tin T and s in R. For a W*~stochasticprocess we have in addition NCK (normality): For all a in P/i,, tin T and s in R, the map bi—* w,,5(a, 1; a, b) is a normal functional. We are now in a position to frame the Definition. Let ~ be a C*~algebrawith identity; a family {w,(~~): t E T} is said to be an inductive system of correlation kernels over ~ if, for each t in T, w~is a functional on P13, x P~,with values in C such that conditions CK1 to CK7 hold. If, in addition, ~ is a W*~algebraand NCK holds, then the family is said to be an inductive system of normal correlation kernels. The Reconstruction Theorem. Let ~ be a C*~algebrawith identity, and let {w,: t E T} be an inductive system of correlation kernels over Then there exists a C*~algebra~ with identity, a C*~stochastic process ({j,}, cv) over P13 evolving in sil and having {w,: t E R} as its family of correlation kernels; the process is unique up to equivalence. Moreover if ~ is a W*~algebraand the w, satisfy the NCK condition, then the j, are normal maps from into and ({i~}~ cv) is a W*~stochasticprocess over evolving in Sketch of Proof (the details are given in Accardi, Frigerio and Lewis [3]). Let X be the inductive limit lim,{~,,f~:s t} in the category of sets. By CK2 and CK7 there exists a positive-definite kernel w on Xx X such that w(i,a; i~b)= w,(a; b) for all a, b in s/i,. ~.

~

~“

.~“.

X

‘,/_~\ ).,glJ

__

).

f~

Let

(LW,

v) be the minimal Kolmogorov decomposition (Theorem 1.9 of [17]) of w:

v:X—*~/1’,(v(x),v(y))=w(x,y)

and

~C=v{v(x):xEX}.

Define j,(b), for each b in ~ and tin R, by j,(b) v(i,(a)) = v(i,,,(a, b)); define U in ~W’ to be the common value of v(i,(1,)) for all tin T; let ~ be the C*~algebraV {j~(b): b E P/i, t E l~}and put w(a) = (U, all) for all a in ste. The rest of the proof is a verification using the properties CK1 to CK7.

References [I] iL. Doob, 5tochastic Processes (New York, Wiley, 1953). [21 PA. Meyer, Probability and Potentials (Waltham. Mass., Blaisdell, i966). [3] L. Accardi, A. Frigerio and J.T. Lewis, Quantum Stochastic Processes, PubI. RIMS. Kyoto 18(1981) No. I. [41L. Accardi, On the non-commutative Markov property, Funct. Anal. AppI. (in Russian) 9 (1975) 1—8; Non-relativistic quantum mechanics as a non-commutative Markov process, Advances in Math. 20 (1976) 239—366.

J. T. Lewis,

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