Table Of ContentQUANTUM STOCHASTIC INTEGRALS AS
OPERATORS
ANDRZEJ L UCZAK
Abstract. We constructquantumstochasticintegralsforthe in-
0 tegrator being a martingale in a von Neumann algebra, and the
1 integrand—asuitable processwith valuesinthe samealgebra,as
0 densely defined operators affiliated with the algebra. In the case
2 of a finite algebra we allow the integrator to be an L2–martingale
n in which case the integrals are L2–martingales too.
a
J
2
1
Introduction
]
A
Thetheoryof‘general’quantumstochasticintegrals(i.e. notfounded
F
. on Fock space) deals mainly with the setup which can be roughly de-
h
scribed as follows: For a von Neumann algebra A with a filtration
t
a
{A : t > 0} we have a corresponding process (X(t) : t > 0) with val-
m t
ues in Lp(A), and a corresponding process (f(t) : t > 0) with values
[
in Lq(A), 1/p+1/q = 1/2, 2 6 p, q 6 +∞, Lp(A) and Lq(A) being
1
appropriate noncommutative Lp-spaces. Then under various specific
v
9 assumptions about X and f, among which the most natural is that X
5 b b
isa martingale, one candefine stochastic integrals f dX and dXf
9 a a
1 as elements of L2(A) (cf. [1, 4, 5, 7, 8, 9, 10]). TRhe advantageRof this
.
1 is that, for any reasonable definition of the integral, it may be approx-
0 imated by integral sums of the form f(t )[X(t ) − X(t )] (for
0 i−1 i i−1
b
1 f dX, the other one being definedPin complete analogy). Now a
a
v: pRroduct of elements from Lp(A) and Lq(A), is an element from Lr(A),
i where 1/p+1/q = 1/r. In particular, the approximation holds in the
X
case r = 2 (in a Hilbert space). However, this approach can be ex-
r
a ploited also in the following quite natural setting. If we assume that
the algebra A acts in a Hilbert space H, and if we let f and X take
their values in A, then the integral sum belongs to A too, and we may
ask aboutitsbehavior onelements of H. This againleadsus to approx-
imation of the integral evaluated at some points of H, i.e., we come to
b b
the notion of integrals f dX and dXf as operators on H. This
a a
idea has been carried oRut in [3] forRa particular class of martingales
1991 Mathematics Subject Classification. Primary: 81S25;Secondary: 46L53.
Key words and phrases. Quantum stochastic integrals, quantum martingales,
adapted processes.
Work supported by KBN grant 2 P03A 03024.
1
2 ANDRZEJL UCZAK
X defined ‘canonically’ in quasi-free representations of the CCR and
CAR.
Itturnsoutthatitispossibletoapplythispointofviewinthefollow-
ingsituation: Astheintegratorwetakeamonotoneornorm-continuous
A-valued process. Then we show that there exists a Riemann-Stieltjes
b
type integral f dX as a densely defined operator affiliated with the
a
algebra A. InRthe case when A is finite, we show the existence of both
b b
the integrals f dX and dXf as densely defined closed operators
a a
affiliated withR A. MoreovRer, we can weaken the assumptions about
(X(t)) and allow it to be a martingale in L2(A), in which case the
integrals above will be elements from L2(A) too.
Finally, let us say a few words about quantum stochastic integrals
in general. In the existing theories of quantum integration, especially
those of Lebesgue type, the classes of ‘theoretically admissible’ inte-
grands are rather narrow and lack any concrete examples of processes
that can be integrated. On the other hand for integrals of Riemann-
Stieltjes type such examples have been provided, and, for example, it
was shown how one can integrate predictable processes and how inte-
gration with respect to a quantum random time can be performed (cf.
[8,9]). Inthisnotewetakeasimilarapproachshowingthepossibilityof
integrating either a monotone or norm-continuous process with respect
to a martingale. It is worth noting that in the case of a monotone pro-
cess, apparently nothing can be said about the existence of Lebesgue
type integrals, while in the case of a norm-continuous process its exis-
tence can be proved only under some additional assumption.
1. Preliminaries and notation
Throughout the paper, we assume that A is a von Neumann algebra
acting in a Hilbert space H with inner product h·,·i, and that ω is a
normal faithful state given by a cyclic and separating unit vector Ω
in H, i.e., A is represented in standard form. We suppose, further,
that we have an increasing family {A : t ∈ [0,+∞)} of von Neumann
t
subalgebras of A (A ⊂ A for s 6 t), called a filtration, and a corre-
s t
sponding family {E } of normal conditional expectations from A onto
t
A leaving ω invariant.
t
A process in A or L2(A) is a function defined on [0,+∞) with val-
ues in A or L2(A), respectively. We shall denote by f, processes in
A, and by X, processes either in A or L2(A). Following the no-
tation of probability theory, we shall sometimes denote a process by
(X(t) : t > 0) (a family of ‘random variables’), and the same applies
to f.
The norms in H and L2(A) will be denoted k·k and k·k , respec-
H 2
tively, while k·k will stand for the operator norm.
QUANTUM STOCHASTIC INTEGRALS 3
Define on (a dense subspace of) H operators P given by:
t
(1) P (xΩ) = (E x)Ω, x ∈ A.
t t
It is well–known (cf. e.g. [6] Propositions 1.1 and 1.2) that
(P : t > 0) is an increasing family of orthogonal projections in H with
t
ranges H = A Ω. Moreover, P ∈ A′, where A is the commutant of
t t t t t
A .
t
A process in A or L2(A) will be called adapted if its value at each
point t > 0 belongs to A or L2(A ), respectively. We call processes f
t t
in A, and X in L2(A) martingales if for any s,t ∈ [0,+∞), s 6 t, the
equalities
E f(t) = f(s), E X(t) = X(s)
s s
hold, where in the L2-case we use the same symbol E to denote the
s
extension of the conditional expectation to L2(A). It follows that a
martingale is an adapted process.
Let (X(t) : t ∈ [0,+∞)) be a process, and let 0 6 t 6 t 6 ··· 6
0 1
t < +∞ be a sequence of points. To simplify the notation we put
m
∆X(t ) = X(t )−X(t ), k = 1,...,m.
k k k−1
Let (X(t) : t ∈ [0,+∞)), (f(t) : t ∈ [0,+∞)) be arbitrary processes,
and let [a,b] be a subinterval of [0,+∞). For a partition
θ = {a = t < t < ··· < t = b} of [a,b] we form left and right
0 1 m
integral sums
m m
Sl = ∆X(t )f(t ), Sr = f(t )∆X(t ).
θ k k−1 θ k−1 k
X X
k=1 k=1
If there exist limits (in any sense) of the above sums as θ is refined, we
call them respectively the left and right stochastic integrals of f with
respect to (X(t)), and denote
b b
limSl = dXf, limSr = f dX.
θ θ
θ Z θ Z
a a
This notion of integral is a weaker one than defining the integrals as
the limits
b b
dXf = lim Sl, f dX = lim Sr,
θ θ
Za kθk→0 Za kθk→0
where kθk stands for the mesh of the partition θ. A definition of this
kind is standard in the classical theory of Riemann-Stieltjes as well as
the theory of stochastic integrals. It is worth noticing that in non-
commutative integration theory, whenever this Riemann-Stieltjes type
integral is considered, its definition refers to the weaker form of the
limit with the refining net of partitions (cf. [2, 8, 9]). However, un-
der additional assumptions we shall be able to obtain the integral also
in the stronger sense thus making it similar to the classical stochastic
integral.
4 ANDRZEJL UCZAK
2. Integrals as operators on H — the non-tracial case
Our construction of the integral is given by the following
Theorem 1. Let (X(t) : t > 0) be a martingale in A, and let
f: [0,∞) → A be a hermitian adapted process such that f is mono-
tone. Then for each t > 0 there exists a Riemann–Stieltjes type integral
tf dX, which is a densely defined operator on H affiliated with A.
0
R
Proof. Fix t > 0,and let θ = {0 = t < ... < t = t} be a partition of
0 m
[0,t]. Put
m
Sr = f(t )[X(t )−X(t )].
θ k−1 k k−1
X
k=1
We want to define tf dX as an operator on A′Ω (where A′ is the
0
commutant of A) byR
t
f dX(x′Ω) = limSr(x′Ω), x′ ∈ A′,
Z θ θ
0
as θ is refined. For this it is sufficient to show the existence of the limit
lim SrΩ, since
θ θ
Srx′ = x′Sr.
θ θ
We have
m
SrΩ = f(t )[X(t )−X(t )]Ω
θ k−1 k k−1
X
k=1
m
= f(t )[E X(t)−E X(t)]Ω
k−1 tk tk−1
X
k=1
m
= f(t )(P −P )X(t)Ω,
k−1 tk tk−1
X
k=1
where the P are projections defined by (1). Consider the operator sum
t
m m
(2) σr = f(t )(P −P ) = (P −P )f(t ) = (σr)∗.
θ k−1 tk tk−1 tk tk−1 k−1 θ
X X
k=1 k=1
Tofixattention, assumethatf isincreasing(i.e., f(s) 6 f(t)fors 6 t),
and let θ′ = θ ∪{t′} for some t < t′ < t be a one-point refinement
j j+1
of θ. Then
σθr′ −σθr = f(tj)(Pt′ −Ptj)+f(t′)(Ptj+1 −Pt′)−f(tj)(Ptj+1 −Ptj)
= [f(t′)−f(tj)](Ptj+1 −Pt′)
= (Ptj+1 −Pt′)[f(t′)−f(tj)](Ptj+1 −Pt′) > 0,
because Ptj+1−Pt′ is a projection commuting with f(tj) and f(t′), and
f(t′) −f(t ) > 0. It follows that the net {σr} is increasing. Further-
j θ
more, for each k = 1,...,m we have
(P −P )f(t )2(P −P ) 6 kf(t )k2(P −P ).
tk tk−1 k−1 tk tk−1 k−1 tk tk−1
QUANTUM STOCHASTIC INTEGRALS 5
Hence
m m
(P −P )f(t )2(P −P ) 6 kf(t )k2(P −P )
tk tk−1 k−1 tk tk−1 k−1 tk tk−1
X X
k=1 k=1
m
6 c2 (P −P ) = c2(P −P ),
tk tk−1 t 0
X
k=1
where c = sup kf(s)k = max{kf(0)k,kf(t)k}. Consequently,
06s6t
m
kσrk2 = k(σr)2k = (P −P )f(t )f(t )(P −P )
θ θ tj tj−1 j−1 i−1 ti ti−1
(cid:13)(cid:13)iX,j=1 (cid:13)(cid:13)
m (cid:13) (cid:13)
= (P −P )f(t )2(P −P ) 6 c2kP −P k = c2,
ti ti−1 i−1 ti ti−1 t 0
(cid:13)(cid:13)Xi=1 (cid:13)(cid:13)
(cid:13) (cid:13)
which means that {σr} is norm-bounded. This, together with the fact
θ
that the net is increasing, yields the existence of lim σr in the strong
θ θ
operator topology, in particular, there exists
limσr(X(t)Ω) = limSrΩ.
θ θ
θ θ
It is clear that for each x′,y′ ∈ A′ we have
t
f dX y′(x′Ω) = limSry′(x′Ω) =
(cid:18)Z (cid:19) θ θ
0
t
y′limSr(x′Ω) = y′ f dX (x′Ω),
θ θ (cid:18)Z (cid:19)
0
thus tf dX is affiliated with A′′ = A. (cid:3)
0
R
It turns out that even a stronger form of integral can be obtained
for norm-continuous processes.
Theorem 2. Let (X(t) : t > 0) be a martingale in A, and let
f: [0,∞) → A be a norm-continuous adapted process. Then for each
t > 0 there exists a Riemann-Stieltjes type integral tf dX which is a
0
densely defined operator on H affiliated with A. MorReover, this integral
is given as the limit
m
lim f(t )∆X(t )
k−1 k
kθk→0X
k=1
on A′Ω.
Proof. Fix t > 0. We shall show that the net {SrΩ} is Cauchy as the
θ
mesh kθk of the partition θ tends to 0. Take an arbitrary ε > 0, and
let δ > 0 be such that for each t′,t′′ ∈ [0,t] with |t′ −t′′| < δ we have
ε
kf(t′)−f(t′′)k < .
2kX(t)Ωk
H
6 ANDRZEJL UCZAK
Let θ′ = {0 = t < t < ··· < t = t} be an arbitrary partition of [0,t]
0 1 m
with kθ′k < δ, and let θ′′ be a partition of [0,t] finer than θ′. Denote by
t(k),t(k),...,t(k) the points of θ′′ lying between t and t , such that
0 1 lk k−1 k
(k) (k) (k)
t = t < t < ··· < t = t . We then have
k−1 0 1 lk k
m lk
Sr = f(t(k))∆X(t(k))
θ′′ i−1 i
Xk=1Xi=1
m m lk
Sr = f(t )∆X(t ) = f(t )∆X(t(k)),
θ′ k−1 k k−1 i
Xk=1 Xk=1Xi=1
so that
m lk
Sr −Sr = [f(t(k) )−f(t )]∆X(t(k)).
θ′′ θ′ i−1 k−1 i
Xk=1Xi=1
As in the proof of Theorem 1 we have
m lk
(Sr −Sr)Ω = f(t(k))−f(t ) ∆X(t(k))Ω
θ′′ θ′ i−1 k−1 i
Xk=1Xi=1 h i
m lk
(k)
= f(t )−f(t ) (P −P )X(t)Ω
Xk=1Xi=1 h i−1 k−1 i t(ik) t(i−k)1
=(σr −σr )X(t)Ω,
θ′′ θ′
and thus
m lk
k(Sr −Sr)Ωk2 = k(P −P ) f(t(k))−f(t ) X(t)Ωk2
θ′′ θ′ 2 Xk=1Xi=1 t(ik) t(i−k)1 (cid:2) i−1 k−1 (cid:3) H
by the orthogonality of P −P and P −P for i 6= j. Further-
more t(ik) t(i−k)1 t(jk) t(jk−)1
k(P −P ) f(t(k) )−f(t ) X(t)Ωk2
t(ik) t(i−k)1 i−1 k−1 H
(cid:2) (cid:3)
=h(P −P )|f(t(k))−f(t )|2(P −P )X(t)Ω,X(t)Ωi,
t(ik) t(i−k)1 i−1 k−1 t(ik) t(i−k)1
(k)
and since |t −t | < δ, we have
i−1 k−1
ε2
|f(t(k))−f(t )|2 6 1,
i−1 k−1 4kX(t)Ωk2
H
giving
ε2
(P −P )|f(t(k))−f(t )|2(P −P ) 6 (P −P ).
t(ik) t(i−k)1 i−1 k−1 t(ik) t(i−k)1 4kX(t)Ωk2H t(ik) t(i−k)1
QUANTUM STOCHASTIC INTEGRALS 7
This yields the estimate
k(P −P ) f(t(k))−f(t ) X(t)Ωk2
t(ik) t(i−k)1 i−1 k−1 H
(cid:2) (cid:3)
=h(P −P )|f(t(k))−f(t )|2(P −P )X(t)Ω,X(t)Ωi
t(ik) t(i−k)1 i−1 k−1 t(ik) t(i−k)1
ε2
6 k(P −P )X(t)Ωk2 .
4kX(t)Ωk2H t(ik) t(i−k)1 H
Consequently, we obtain
(3)
m lk
k(Sr −Sr)Ωk2 = k(P −P ) f(t(k) )−f(t ) X(t)Ωk2
θ′′ θ′ H Xk=1Xi=1 t(ik) t(i−k)1 (cid:2) i−1 k−1 (cid:3) H
ε2 m lk
6 k(P −P )X(t)Ωk2
4kX(t)Ωk2H Xk=1Xi=1 t(ik) t(i−k)1 H
ε2 m lk 2
= (P −P )X(t)Ω
4kX(t)Ωk2H(cid:13)(cid:13)Xk=1Xi=1 t(ik) t(i−k)1 (cid:13)(cid:13)H
ε2 (cid:13) (cid:13)
= k(P −P )X(t)Ωk2
4kX(t)Ωk2 t 0 H
H
ε2 ε2
6 kX(t)Ωk2 = .
4kX(t)Ωk2 H 4
H
Let now θ and θ be arbitrary partitions of [0,t] such that
1 2
kθ k < δ, kθ k < δ, and let θ′′ = θ ∪θ . Then we have by (3)
1 2 1 2
ε ε
k(Sr −Sr )Ωk < and k(Sr −Sr )Ωk < ,
θ′′ θ1 H 2 θ′′ θ2 H 2
so
k(Sr −Sr )Ωk < ε,
θ1 θ2 H
showing that the net {SrΩ} is Cauchy. Thus there exists lim SrΩ
θ kθk→0 θ
and the rest of the proof is the same as that of Theorem 1. (cid:3)
3. Integrals as operators on H — the tracial case
Let us now assume that ω is a normal tracial state. Recall that the
Lebesgue space L2(A,ω) is formally defined as the completion of A
with respect to the norm
kxk = [ω(x∗x)]1/2 = kxΩk ,
2 H
and may be realized as a space of densely defined closed operators
affiliated with A such that Ω belongs to their domains.
For the von Neumann subalgebras A the normal ω-invariant con-
t
ditional expectations E : A → A extend to orthogonal projections
t t
8 ANDRZEJL UCZAK
(denoted by the same letter) from L2(A,ω) onto L2(A ,ω). If we de-
t
fine an operator P on H by
t
(4) P (XΩ) = (E X)Ω, X ∈ L2(A,ω),
t t
then P is an orthogonal projection from A′. (The definition above is
t t
thesameasthatgivenby (1)forX ∈ A.) Foranyx ∈ A, A ∈ L2(A,ω)
the operators xA and Ax belong to L2(A,ω), consequently we may
again consider the integral sums
m m
f(t )∆X(t ), ∆X(t )f(t ),
k−1 k k k−1
X X
k=1 k=1
where f: [0,∞) → A, X: [0,∞) → L2(A,ω), as operators on H. We
shall use the notation
m m
Sr(f,X) = f(t )∆X(t ), Sl(f,X) = ∆X(t )f(t ),
θ k−1 k θ k k−1
X X
k=1 k=1
for the integral sums, to indicate their dependence on f and X. Then
since Sr(f,X) and Sl(f,X) are affiliated with A, we have an explicit
θ θ
description of their actions on A′Ω as
Sr(f,X)(x′Ω) = x′Sr(f,X)Ω, Sl(f,X)(x′Ω) = x′Sl(f,X)Ω.
θ θ θ θ
Theorem 3. Let (X(t) : t > 0) be a martingale in L2(A,ω) and let
f: [0,∞) → A be either monotone or norm continuous. Then for
each t > 0 there exist integrals tf dX and tdXf as elements of
0 0
L2(A,ω). Moreover, the L2(A,ω)R-processes (YR(t) : t > 0),
(Z(t) : t > 0) defined by
t t
Y(t) = dXf, Z(t) = f dX
Z Z
0 0
are martingales.
Proof. The existence of the integral tf dX is proved exactly as in
0
Theorems 1 and 2 upon observing thRat according to formula (4) we
have
[X(t )−X(t )]Ω = E X(t)−E X(t) Ω = (P −P )X(t)Ω.
k k−1 tk tk−1 tk tk−1
It follows that the net {(cid:2)Sr(f,X)Ω} is Cauch(cid:3)y, and since
θ
kSr(f,X)k = kSr(f,X)k ,
θ H θ 2
{Sr(f,X)} converges in k·k -norm, and thus its limit is an element of
θ 2
L2(A,ω).
For Sl(f,X) we have
θ
Sl(f,X) = [Sr(f∗,X∗)]∗;
θ θ
so we obtain
kSl(f,X)k = k[Sr(f∗,X∗)]∗k = kSr(f∗,X∗)k ,
θ 2 θ 2 θ 2
QUANTUM STOCHASTIC INTEGRALS 9
because Ω is tracial. Since f∗ satisfies the same assumptions as f,
and (X(t)∗ : t > 0) is also an L2(A,ω)–martingale, we obtain the
convergence of {Sl(f,X)} in k·k -norm.
θ 2
Now we shall show that (Y(t) : t > 0) is a martingale. Fix t > 0 and
take an arbitrary s < t. We have
t
dXf = lim Sl.
θ
Z0 kθk→0
We may assume that s is one of the points of each partition
θ = {0 = t < t < ··· < t = t}, say s = t . Then we have
0 1 m k
k m
E Sl = E ( [X(t )−X(t )]f(t )+ [X(t )−X(t )]f(t ))
s θ s i i−1 i−1 i i−1 i−1
Xi=1 i=Xk+1
k m
= E [X(t )−X(t )]f(t )+ E [X(t )−X(t )]f(t ).
s i i−1 i−1 s i i−1 i−1
Xi=1 i=Xk+1
For i 6 k we have t 6 s, and thus
i
E [X(t )−X(t )]f(t ) = [X(t )−X(t )]f(t ),
s i i−1 i−1 i i−1 i−1
while for i > k we have t > s, and thus
i−1
E [X(t )−X(t )]f(t ) = E E [X(t )−X(t )]f(t )
s i i−1 i−1 s ti−1 i i−1 i−1
=E (E [X(t )−X(t )])f(t ) = 0
s ti−1 i i−1 i−1
by the martingale property. Consequently,
k
(5) E Sl = [X(t )−X(t )]f(t ).
s θ i i−1 i−1
Xi=1
But the sum on the right hand side of (5) is an integral sum for the
s
integral dXf, and passing to the limit in (5) yields
0
R
t s
E dXf = dX f,
s
Z Z
0 0
which shows that (Y(t)) is a martingale. Analogously for
(Z(t) : t > 0). (cid:3)
Remarks. 1. The formulas
t t
f dX (x′Ω) = x′ limSrΩ, dXf (x′Ω) = x′ limSlΩ
(cid:18)Z (cid:19) θ θ (cid:18)Z (cid:19) θ θ
0 0
describe explicitly the actions of the integrals on A′Ω.
2. Let us notice that an attempt to define Lebesgue type integrals
above, e.g., along the lines of [4] or [10], would be successful only in the
simple case of norm-continuous f, and even then under the additional
assumption of left- or right-continuity in k·k -norm of the martingale
2
(X(t)). The reason for this is that this type of integral is defined for
10 ANDRZEJL UCZAK
µ–measurable functions f ∈ L2([0,∞),µ,A), where µ is a measure
defined by
µ([a,b)) or µ((a,b]) = ω(|X(b)−X(a)|2) = ω(|X(b)|2)−ω(|X(a)|2).
The continuity of the martingale allows then the extension of µ from
the intervals to the Borel sets. For the case of monotone f the failure
of the definition of the integral as Lebesgue type is most strikingly
seen when the function f is increasing projection-valued. To be more
concrete, assume that e is a spectral measure with support [0,∞), and
put f(t) = e([0,t]). Then for s < t, f(t)−f(s) is a non-zero projection,
so kf(t)−f(s)k = 1, which means that f is not norm continuous either
from the left or from the right at any point, while a µ–measurable
function must be norm-continuous on some compact set.
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Faculty of Mathematics and Computer Science, L o´d´z University,
ul. Banacha 22, 90–238 L o´d´z, Poland
E-mail address: [email protected]
