数理解析研究所講究録975 巻 1996 年 65-84

65
ON A TOY FOCK SPACE GENERATEDBY PERMUTATIONSNAOFUMI MURAKIABSTRACT. An example of independence, conditional expectation andmartingales in quantum probability theory is constructed on the permutational Fock space which is a kind of discrete “Fock space” generated byan the permutations from the natural numbers. Besides a discrete timeanalogue of quantum Ito’s formula for the permutational Fock space isobtained.

1. INTRODUCTION
The notion of “Fock space” is a nice tool for the purpose of constructing various
examples of “independence,” “white noise,” and “quantum stochastic calculus” inquantum probability theory or
probability theory.1,2,3,4 For example, the followings are known.
For a oneparticle Hilbert space 7#, let
be the r-th symmetric tensor power,
the r-th antisymmetric tensor power, andthe r-th tensor power. Let
be the boson Fock space,
the fermion Fock space,andthe free Fock space. Let us specialize the one-particle Hilbert
space to be the complex Hilbert spaceof all -functions of time $t\geq 0$ .
Then the three Fock spaces are equipped with very rich probabilistic structures. The
boson Fock spaceleads to the commuting independence, the noncommuting
pair of classical Brownian motions, and the bosonic stochastic calculus of R. L.
Hudson and K. R. Parthasarathy5 with quantum Ito’s formula. The fermion Fockspaceleads to the anticommuting independence, the noncommuting pair of
fermion Brownian motions, and the fermionic stochastic calculus of D. Applebaumand Hudson6 with fermion Ito’s formula. The free Fock spaceleads to the freeindependence of D. Voiculescu,7 the noncommuting pair of free Brownian motionsof R. Speicher,8 and the free stochastic calculus of B. K\"ummerer and Speicher9with free Ito’s formula. There have been also known several discrete models of Fockspace.1,2 For example, the toy Fock space
introduced by J. L. Journ\’e and further
studied by P. A.
is an elementary but interesting discrete model for thebosonic stochastic calculus and the fermionic stochastic calculus. Journ\’e’s toy Fockspace
is connected with theory of spin systems.12

nonc.ommutative$\mathcal{H}^{\mathrm{o}r}$$\mathcal{H}^{\wedge r}$$\mathcal{H}^{\otimes r}$$\Phi_{boson}=\oplus_{r}\mathcal{H}^{\mathrm{o}r}$$\Phi_{fermi}on=\oplus_{r}\mathcal{H}^{\wedge r}$$\Phi_{free}=\oplus_{r}\mathcal{H}^{\otimes r}$$L^{2}(\mathrm{R}_{+})$$\mathcal{H}$$L^{2}$$\Phi_{boson}$$\Phi_{fermi}on$$\Phi_{free}$$\Phi_{toy}$$\mathrm{M}\mathrm{e}\mathrm{y}\mathrm{e}\mathrm{r}^{2,10,11}$$\Phi_{toy}$

1991 Mathematics Subject Classification. Primary$60\mathrm{G}48$.

$81\mathrm{S}25$; Secondary$46\mathrm{L}89,46\mathrm{L}50,60\mathrm{G}42$,

Key words and phrases. quantum probability theory, Fock space, K\"u mmerer independence,
conditional expectation, operator martingales, quantum Ito’s formula.
This work was supported by-Aid for Scientific Research No. 07740157, the Ministryof Education, Science and Culture, Japan.
$\mathrm{c}_{\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{t}-}\mathrm{i}\mathrm{n}$66

In this paper, for the purpose of constructing an example of probabilistic notions
in discrete time quantum probability theory, we introduce the permutational Fockspace which is a kind of discrete “Fock space” generated by all the permutations fromthe natural numbers. This “Fock space” can be viewed as a variation of Journ\’e’s.
toy Fock space
The motivation of finding such Fock space is explained as follows. Let us consider
the discrete and finite time situation. That is, we take the set $T_{n}=\{1,2, \cdots , n\}$ asofthe set of times, and let the one-particle Hilbert space to be the -space
all functions of time $k\in T_{n}$ . Let us calculate the dimension of the -particle space.
, and free
, fermionfor each cases of boson
ily getWe$\Phi_{toy}$$l^{2}$$\mathcal{H}$$l^{2}(\tau_{n})$$r$$(\mathcal{H}_{r}=\mathcal{H}^{\otimes r})$$(\mathcal{H}_{r}=\mathcal{H}^{\wedge r})$$(\mathcal{H}_{r}=\mathcal{H}^{\mathrm{o}r})$$\mathcal{H}_{r}$$\mathrm{e}\mathrm{a}s$${}_{n}\mathrm{H}_{r}=\mathrm{t}\mathrm{h}\mathrm{e}$$\dim(\mathcal{H}_{r})=$${}_{n}\mathrm{C}_{r}=\mathrm{t}\mathrm{h}\mathrm{e}$$n^{\Pi}r=\mathrm{t}\mathrm{h}\mathrm{e}$number of repeated combinations (in the boson case)(in the fermion case)number of combinations
number of repeated permutations (in the free case)This encourages us to guess the existence of “Fock space”
satisfies-particle space$r$$\Phi=\oplus_{r}\mathcal{H}_{r}$such that its$\mathcal{H}_{r}$$\dim(\mathcal{H}_{r})={}_{n}\mathrm{P}_{r}=\mathrm{t}\mathrm{h}\mathrm{e}$number of permutations.

Our aim in this paper is, (i) the construction of such “Fock space” which we callthe permutational Fock space, and (ii) the construction, over such “Fock space” ,of an example of probabilistic notions in discrete time quantum probability theory,
i.e. “independence,” “filtration,” “conditional expectation,” “martingales,” “processwith independently and identically distributed subalgebras,” etc.
The paper is organized as follows. In \S 2, we construct on the set of all naturalic operators, i.e. the creation
numbers the permutational Fock space with theoperators, annihilation operators, conservation operators, and exclusion operators.
The pair $(A, \phi)$ consisting of the -algebra $A$ generated by all the creation and theannihilation operators with identity and the vacuum state is a quantum probability space to be studied in this paper. In \S 3, we obtain the weak Fock expansion
theorem for bounded linear operators on the permutational Fock space . In \S 4, we
examine the independence structure arising from the permutational Fock space .
with independently and identically distributed subalWe obtain a processwith respect to the vacuum state . In \S 5, we examine the filtration
gebrasis the -algebra generated byof the -algebra $A$ where
structure
all the creation and annihilation operators up to time with identity. In \S 6, wefrom the -algebra $A$ onto theconstruct the natural conditional expectation
with respect to the vacuum state . In \S 7, we obtain the-subalgebra-algebra $A$ . We examine
uniform Fock expansion theorem for operators in thethe predictable representation theorem for operator martingales on the quantum
probability space $(A, \phi)$ . In \S 8, we examine a discrete time analogue of quantum
Ito’s formula for discrete time operator processes on the quantum probability space. The last \S 9, contains some complementary remarks.
Before closing this introduction, we remark that the independence of the diswith independently and identicallycrete time quantum stochastic process
obtained in this paper is neither the commuting indistributed subalgebras
dependence, nor the anticommuting independence, nor the free independence. But
it is an example of the K\"ummerer independence.8,14$\Phi$$\Phi$$T$$\Phi$$\mathrm{b}\mathrm{a}s$$C^{*}$$\phi$$\Phi$$\Phi$$\{A_{k}\}_{k\in}\tau$$A_{k}\subset A$$\phi$$\{A_{k]}\}_{k\in}\tau$$C^{*}$$C^{*}$$A_{k]}$$k$$C^{*}$$\epsilon_{k\mathrm{l}}$$C^{*}$$A_{k]}\subset A$$\phi$$C^{*}$$(A, \phi)$$\{A_{k}\}_{k\in}\tau$$A_{k}\subset A$

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2. DEFINITION OF PERMUTATIONAL FOCK SPACEIn this section, we give the definition of permutational Fock space.
Let $T=\{1,2,3, \cdots\}$ be the set of all positive natural numbers which is interthe set ofpreted as time. For a natural number $r=0,1,2,$ , we denote by
all permutations of length from . We also denote by Per $(T)$ the disjoint union
the singleton where A denotes the nullunder the convention thatthe complex Hilbert space
permutaion, i.e. the null string from . Denote by
the -particle space, and their. We callof all -functions over the set
the permutational Fock space over . It contains the natural
direct sumlabelled by all the permutations .
complete orthonormal system
is called the vacuum vector, and denoted by .
A special vector, we denote the associated set
For any permutation. For any pair of perif and only if
by . We wright.
mutationsa new permutation $(i_{1}, \cdots, i_{r},j1, \cdots,js)$ obtained from com, we denote byposition of and . Furthermore, when $\tau=(j)$ a permutation of length 1, we, and so on. We define thewright in the short notation
mutiplication of basis vectors by$\tau^{\mathrm{P}_{r}}$$\cdots$$T$$r$$\sigma$$\tau \mathrm{P}_{0}=\{\Lambda\}$$\bigcup_{rT}\mathrm{p}_{r}$$T$$l^{2}$$l^{2}(_{T}\mathrm{P}_{r})$$\mathcal{H}_{r}$$r$$\mathcal{H}_{r}$$\tau \mathrm{P}_{r}$$T$$\Phi=\oplus_{r}\mathcal{H}_{r}$$\{e_{\sigma}|\sigma\in \mathrm{P}\mathrm{e}\mathrm{r}(T)\}$$\sigma$$\Omega$$e_{\Lambda}$$\sigma=$$\{i_{1}, \cdots , i_{r}\}$$(i_{1}, \cdots , i_{r})\in \mathrm{P}\mathrm{e}\mathrm{r}(T)$$[\sigma]\cap[\tau]=\emptyset$$[\sigma]\perp[\tau]$$[\sigma]$$\sigma=(i_{1}, \cdots,i_{r}),$$\tau=(j_{1}, \cdots , j_{s})\in \mathrm{P}\mathrm{e}\mathrm{r}(T)\mathrm{S}.\mathrm{t}$$\{i_{1}, \cdots , i_{r}\}\cap\{j_{1}, \cdots , j_{s}\}=$$(\sigma, \tau)$$\emptyset$$\sigma$$\tau$$(\sigma, \tau)=(\sigma,j),$$(\tau, \sigma)=(j, \sigma)$),(if(otherwise).
$[\sigma]\perp[\tau]$$e_{(\sigma,\tau)}$$e_{\sigma}e_{\tau}=\{$$0$This multiplication is extended to the dense subspace $\Phi_{0,0}=$ “the linearspan ofthrough the bilinearity. Furthermore it can be still extended“the algebraic direct sum of the -particle spaces”
to the dense subspacebecause of the boundedness of multiplication of vectors with the fixed particle. This multiplication
andnumber, i.e. $||uv||\leq||u||||v||$ fora structure of associative algebra.
gives tois defined by
For each time $i\in T$ , the creation operator$\{e_{\sigma}|\sigma\in \mathrm{P}\mathrm{e}\mathrm{r}(T)\}"\subset\Phi$$\Phi_{0}=$$r$$\subset\Phi$$v\in \mathcal{H}_{s}\cap\Phi_{0,0}$$u\in \mathcal{H}_{r}\cap\Phi 0,0$$\Phi_{0}$$d_{i}^{+}$(if(otherwise).

$i\not\in\{_{\dot{i}_{1},i_{2}},$$e_{(i,i_{1},i_{2},\cdots,i_{r}})$$d_{i(i_{1},i_{2}}^{+_{e=}},\cdots,ir)\{$$0$$\cdots,$$i_{r}\}$

is just the left multiplication operator
The creation operator.
respect to the above mentioned multiplication in,is defined by
The annihilation operator$d_{i}^{+}$),$e_{\sigma}\mapsto e_{i}e_{\sigma}$with$\Phi_{0}$$d_{i}^{-}$(if $r\geq 1$ and(otherwise).

$e_{(i_{2},\cdots,i\mathrm{r})}$$d_{i}^{-_{e_{(ii}}}\ldots,i_{r})=1,2,\{$$0$$i=\dot{i}_{1}$),These operators are bounded linear operators on , and $||d_{i}^{+}||=||d_{i}^{-}||=1$ . The. The
annihilation operator is just the adjoint of creation operator:$d_{j}^{-}(i,j\inT)$relations:followingthesatisfy
operators
creation and annihilation$\Phi$$d_{h}^{-}=(d_{h}^{+})^{*}$$d_{i}^{+},$$(d_{i}^{+})^{2}=(d_{i}^{-})^{2}=0$$\{$$d_{i}^{-}d_{j}^{+}=0$

(for

,

$\dot{i}\neq j$),$d_{i}^{-}d_{i}^{+}=I-\Sigma_{\sigma}\in \mathrm{P}\mathrm{e}\mathrm{f}(\tau\backslash \{i\})d^{+}d(\sigma,i)(-\sigma,i)$.

68
Here denotes the identity operator on . The infinite series in the righthandside of the last equality converges in the weak topology of $B(\Phi)$ which is the spaceof all bounded linear operators on . These relations may be compared with the
CCR (canonical commutation relations)5 in the boson case, the CAR (canonicalanticonlmutation relations)6 in the fermion case, and the relations for the Cuntz.
algebra9 in the free case. The conservation operator is defined by, put
For any permutation$\Phi$$I$$\Phi$$d_{i}^{\mathrm{O}}=d_{i}^{+_{d_{i}}}-$$d_{i}^{\mathrm{o}}$$\sigma=(i_{1}, \cdots,\dot{i}_{r})\in \mathrm{P}\mathrm{e}\mathrm{r}(T)$$d_{\sigma}^{+}=d_{i_{1}}^{+}\cdots d_{ir}^{+}$,$d_{\sigma}^{-}=d_{i_{r1}}^{-\ldots d_{i}^{-}}$,$d_{\sigma}^{\mathrm{o}}=d_{\sigma}^{+}d_{\sigma}^{-}$

Then, the creation and annihilation operators
following relations:$=$$d_{\sigma \mathcal{T}}^{-d^{+}}$$d_{\sigma}^{+},$

and$d_{\Lambda}^{+}=d_{\Lambda}^{-}=d_{\Lambda}^{\mathrm{O}}=I$$d_{\tau}^{-}(\sigma, \tau\in \mathrm{P}\mathrm{e}\mathrm{r}(T))$.

satisfy the$|d_{\beta}-d_{\beta}^{+}-0^{-}[ \gamma]\perp[\mathcal{T}][\gamma]\perp[(\mathrm{i}\mathrm{f}(\mathrm{i}\mathrm{f}\exists(\mathrm{o}\mathrm{t}\mathrm{h}’ \mathrm{e}\mathrm{r}\sigma\exists,\sum\sum_{)}^{\gamma}]\alpha,\beta\in\alpha,\beta\in \mathrm{w}\mathrm{i}i\in[\alpha.]i\in[\mathrm{s}\mathrm{e}\alpha]\mathrm{P}\mathrm{P}\mathrm{e}\mathrm{r}(T)\mathrm{e}\mathrm{r}(Tdd_{(\gamma,i}^{\mathrm{o}}\beta+_{d_{(,i}^{\mathrm{O}}})\mathrm{s}.\mathrm{t})d^{-}\mathrm{S}.\mathrm{t}\beta)..[\alpha]\perp[\alpha]\perp[\beta[\beta],\sigma=\alpha],\sigma=(\alpha’,\beta\tau=(\alpha, \beta))),$$\mathcal{T}=\alpha)’$,The serieses in the righthand side converge in the weak topology of $B(\Phi)$ .
For the discussion in the following sections, it is convenient to introduce a new. For any permutation
operator. The $exclI\iota sion$ operator is defined by. Then we have
, put
$d_{i}=d_{i}^{-}d_{i}^{+}$$d_{i}$$\sigma\in \mathrm{P}\mathrm{e}\mathrm{r}(T)$$d_{\sigma}=d_{\sigma}^{-}d_{\sigma}^{+}$),(if(otherwise).
$[\sigma]\perp[\tau]$$e_{\tau}$$d_{\sigma}.e_{\mathcal{T}}$

$=$

$\{$

$0$

.

. So we can define, for any finite set $U\subset T$ , anwheneverNote thatis a projection. $[\sigma]=U$ . The exclusion operator
using
operator
operator and satisfies the following relations:$d_{\sigma}=d_{\tau}$$d_{\dot{U}}:=d_{\sigma}$$[\sigma]=[\tau]$$d_{\dot{U}}$$\sigma \mathrm{s}.\mathrm{t}$and$d_{\dot{U}}d_{\sigma}^{+}=d_{\sigma}^{+}d_{\dot{U}}$$\{$$d_{\dot{U}}d_{\dot{V}}=d_{\dot{U}\cup V}$$d_{\dot{U}}d_{\sigma}^{-}=d_{\sigma}^{-}d_{\dot{U}}$(if$U\perp[\sigma]$),.

Let $A=C^{*}(I, d_{i}+, d_{i}^{-|i}\in T)$ be the -algebra generated by all the creation-algebra $A$ has a special state
and annihilation operators with identity. The$\phi(\cdot)=<\Omega|\cdot|\Omega>$ called the vacuum state.
The pair $(A, \phi)$ is interpreted as aquantum probability space.
$C^{*}$$C^{*}$

69
3. FocK EXPANSIONIn this section, we prove an expansion theorem for bounded linear operators on
the permutational Fock space .
Denote by $B(\Phi)$ the von Neumann algebra of all bounded linear operators on. Let us show that the space $B(\Phi)$ has a “ is” {, in the “weak” topology,
consisting of normally ordered products of the creation and annihilation operators.
The expansion theorem like the following Theorem 3.1 and Theorem 3.2 which assertthat every operator in a sufficiently large class can be expanded to the infinite linear
combinantion (or integration) of normally ordered products of the basic operators
is sometimes called the Fock expansion theorem.13
At first let us show the following purely “algebraic” Fock expansion. Denote by$L(\Phi 0,0)$ the set of all linear operators $A:\Phi_{0,0}arrow\Phi_{0,0}$ which is possibly
unbounded.
$\Phi$$\Phi$$\mathrm{b}\mathrm{a}s$$d_{\sigma}^{+}d_{\mathcal{T}}^{-\}}$

Theorem 3.1: Every operator $A\in L(\Phi_{0,0})$ has the unique empansion of theform$A= \sum_{\sigma,\mathcal{T}}a_{\sigma,\mathcal{T}}d\sigma \mathcal{T}+_{d^{-}}$

where$a_{\sigma,\tau}$

are scalars, and$\sigma,$$\tau$,

mm over all permutaions from$T$.

Proof. At first let us show the existence of the expansion. For a given$\cdot$$L(\Phi_{0,0})$$A\in$, put$’<e_{\sigma}|Ae_{\mathcal{T}}>-<e_{\sigma’}|Ae_{\tau’}>$$a_{\sigma_{)}\tau}$

(if

$=$

$-<e_{\sigma}|Ae_{\tau}>$$\sigma=(i_{1},$$\cdots,$$i_{r})\neq\Lambda,$$\tau=(j_{1},$ $\cdots,j_{S})\neq\Lambda$, and$\dot{i}_{r}=j_{1}$),(otherwise),

where $\sigma’=(i_{1}, \cdots, i_{r-1})$ and. Then, the family of scalars
satisfy the condition that, for eachexcept for finitely many
number of indices . By this condition, the operator
be welldefined as an element in. Let us calculate the matrix element, then we have$\tau’=(j_{2}, \cdots , j_{s})$$\in \mathrm{P}\mathrm{e}\mathrm{r}(T)\}$$\{a_{\sigma,\tau}|\sigma,$$\tau,$$a_{\sigma,\tau}=0$$\sigma$$\Sigma_{\sigma,\tau}a_{\sigma,\tau}d_{\sigma}+d_{\mathcal{T}}-\mathrm{c}\mathrm{a}\mathrm{n}$$L(\Phi_{0,0})$$<e_{\lambda}|\Sigma\sigma,\tau d^{+_{d}}a_{\sigma,\mathcal{T}}\mathcal{T}\sigma\mu-_{e}>,$ $\lambda,\mu\in \mathrm{P}\mathrm{e}\mathrm{r}(\tau)$$<e_{\lambda}| \sum\sigma,\tau a_{\sigma,\tau}d_{\sigma}^{+_{d_{\tau}e_{\mu}>}}-$$= \sum_{\sigma,\tau}a_{\sigma,\tau}<e\lambda|d_{\sigma \mathcal{T}}^{+}d-_{e_{\mu}>}$$=$$<e_{\sigma}|Ae_{\tau\lambda}><e|d_{\sigma}^{+}d_{\tau\mu}-e>$$\sigma=\Lambda\circ \mathrm{r}\sum_{\tau=\Lambda}$$\tau=(j\mathrm{o}\mathrm{r}<1<,\cdot\sigma..=,j(i_{1}\mathit{8})’\neq..\Lambda^{\cdot},’ i_{r}\neq ir)\neq\Lambda j_{1}’\gg$$+ \mathcal{T}=(j1\sigma=,\cdot(.i.1,’\cdots,ijs)\neq,rj=1\sum_{i}r_{\Lambda})\neq\Lambda,(<e_{\sigma}|Ae\tau>-<e\sigma’|Ae_{\mathcal{T}’}>)<e_{\lambda}|d_{\sigma}^{+}d_{\tau\mu}-e>$$= \sum_{\sigma,\tau}<e_{\sigma}|Ae\tau><e_{\lambda}|d_{\sigma}^{+}d_{\tau\mu}^{-}e>$$- \sum_{\Lambda\sigma\neq\Lambda,\tau\neq,i_{r}=j_{1}}<e_{\sigma}’|Ae_{\tau}’><e_{\lambda}|d^{+}d_{\mathcal{T}}-e_{\mu}>\sigma$$\tau$

70$= \sum_{\sigma,\tau}<e_{\sigma}|Ae_{\mathcal{T}}><e_{\lambda}|d^{+_{d_{\tau}^{-}}}e_{\mu}>\sigma$$- \sum_{)\sigma,\tau\in \mathrm{p}_{\mathrm{e}}\mathrm{r}(\tau,i\in\tau}<e_{\sigma}|Ae_{\mathcal{T}}><e\lambda|d_{\sigma}+d^{\mathrm{O}}di\tau-_{e_{\mu}>}$.

$= \sum_{\sigma,\tau}<e_{\sigma}|Ae_{\mathcal{T}}><e_{\lambda}|d_{\sigma}^{+}(I-\sum d_{i}^{\circ})di\tau-e_{\mu}>$

Here the operator
the sense that

$E_{\sigma,\tau}=d_{\sigma}^{+}(I-\Sigma_{ii}d^{\circ})d_{\mathcal{T}}-\mathrm{i}\mathrm{s}$(if $\rho=\tau$ )$E_{\sigma,\tau}e_{\rho}=e_{\sigma}$

$=0$

$,$

(if

shown to be an elementary operator in). So we get$\rho\neq\tau$.

$<e_{\lambda}| \sum a_{\sigma},-d+d-e_{\mu}>=<e_{\lambda}|7\sigma\tau Ae_{\mu}>\sigma,\tau$.
Hence we get the Fock expansion
For the proof of the unieque.
oftheexpansion.
uniqueness
Now, let us show theindependence” of the family
ness, we only have to show the, where the family of scalars. Assume thatexcept for finitely many
the condition that, for each
. By
for all
have
.
we
Then
number of indicesand hence we get, we havespecialization, we have
from
Next,
.
specializationbyfor all$A=\Sigma_{\sigma,\tau\tau}a_{\sigma},d_{\sigma}+_{d_{\tau}^{-}}$$‘(\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{a}\mathrm{r}$$\{d_{\sigma}^{+}d_{\tau}-|\sigma,$$\tau\in$$\{a_{\sigma,\tau}|\sigma,$$\tau\in$$\Sigma_{\sigma,\tau}a_{\sigma,\tau}d_{\sigma}+_{d_{\tau}^{-}}=0$$\mathrm{P}\mathrm{e}\mathrm{r}(T)\}$$\mathrm{P}\mathrm{e}\mathrm{r}(T)\}\mathrm{s}\mathrm{a}\mathrm{t}\mathrm{i}_{\mathrm{S}}\mathrm{f}\mathrm{y}$$\tau,$$a_{\sigma,\tau}=0$$\rho:=\Lambda$$\Sigma_{\sigma}a_{\sigma,\Lambda}d_{\sigma}^{+}d_{\Lambda}-e_{\Lambda}=\Sigma_{\sigma}a_{\sigma,\Lambda\sigma}e=0$$\rho:=(\dot{i})$$\sigma\in \mathrm{P}\mathrm{e}\mathrm{r}(T)$$a_{\sigma,\Lambda}=0$$\rho\in \mathrm{P}\mathrm{e}\mathrm{r}(T)$$\Sigma_{\sigma,\tau\rho}a_{\sigma,\tau}d\sigma\tau+d-_{e}=0$$\sigma$$\tau^{\mathrm{P}_{1}}$,$\sum_{\sigma}a\sigma,\Lambda d^{+}\sigma\Lambda e(d^{-}i)+\sum a\sigma,(i)d_{\sigma}^{+}d-e(i)=\sigma)(i0$and hence we get$a_{\sigma,(i)}=0$

for all$\sigma\in \mathrm{P}\mathrm{e}\mathrm{r}(T)$. Similarly, by specialization$\rho:=(\dot{i},j)$$\mathrm{p}\mathrm{e}\mathrm{r}(\tau),\mathrm{a}_{\mathrm{d}\mathrm{e}}\mathrm{n}\mathrm{d}\mathrm{S}\mathrm{o}\mathrm{o}\mathrm{n}.\mathrm{F}\mathrm{i},\mathrm{n}\mathrm{a}11\mathrm{y}\mathrm{w}\mathrm{e}_{\mathrm{f}}\mathrm{g}\mathrm{a}\sigma,(j)\mathrm{S}\mathrm{t}\mathrm{h}\mathrm{e}1\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{a}\mathrm{r}\mathrm{i}\mathrm{n}\mathrm{p}\mathrm{e}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{n}\mathrm{c}\mathrm{e}’ \mathrm{o}\mathrm{f}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{m}\mathrm{t}\mathrm{e}a\sigma,\tau \mathrm{o}_{\frac{}{\tau}}\mathrm{f}\mathrm{o}=\Gamma \mathrm{i}1\mathrm{y}\{d+d\sigma\}$$\mathrm{f}\mathrm{r}\mathrm{o}\mathrm{m}_{T}\mathrm{a}11\sigma,\tau \mathrm{P}2,\mathrm{w}\mathrm{e}\mathrm{g}\mathrm{e}\in \mathrm{p}\mathrm{e}\mathrm{r}(\tau^{\mathrm{t}a\mathrm{o}\mathrm{f}_{0}\mathrm{a}}).\mathrm{T}\mathrm{h}\mathrm{i}\mathrm{s}\mathrm{C}\mathrm{o}\mathrm{n}\mathrm{C}1\mathrm{u}\mathrm{d}\mathrm{e}1i,=\mathrm{r}1\sigma\in$,“and hence the uniqueness result.

$\square$Let us investigate the “topological” Fock expansion for operators in the settingof complexof the “rapidly decreasing sequences.” We call a family
ifitsatisfies
familynumbers a rapidly decreasing$x=\{x_{\sigma}\}_{\sigma}\in \mathrm{p}\mathrm{e}\mathrm{r}(T)$$\sum_{\sigma}(\max[\sigma])^{k}|X_{\sigma}|<\infty$

for all $k=1,2,3,$$\cdots$.

. Then naturally
Denote by the set of all rapidly decreasing families
be the dual space of . Denotehas a structure of countably Hilbert space.13 Let. The weak
the space of all continuous linear operators from tobymeans $<u|A_{n}|v>arrow<u|A|v>$ for
convergence operators $A_{n}arrow A$ in$v\in
S$. In this setting, the following “topological” Fock exapnsion theorem
allis easily obtained by the similar discussion in the proof of Theorem 3.1 and some
simple evaluations.
$S$$x=\{x_{\sigma}\}$

$S$

$S$

$S^{*}$$S$$\mathcal{L}(S, S^{*})$$S^{*}$$\mathcal{L}(S, S^{*})$$u,$

Theorem 3.2 (Weak Fock Expansion for
the unique expansion of the form

$B(\Phi)$$A= \sum_{\sigma,\tau}a\sigma,\tau d_{\sigma}^{+_{d_{\tau}^{-}}}$as an elementfrom .
$T$$of\mathcal{L}(S, s^{*})$, where$a_{\sigma,\tau}$):

Every operator $A\in B(\Phi)$ has,

are scalars, and$\sigma,$$\tau$

runs over all permutaions

71
of the von Neumann algebra $B(\Phi)$ can be
This means that each operator. Contrary to
expanded to the Fock expansion in the weak topology ofTheorem 3.2, we will show in \S 5 (resp. \S 7) that each operator of the -algebra(resp. $A$) has the another form of “Fock expansion” which is useful in the later
discussion.
$A$$\mathcal{L}(S, S^{*})$$A$$C^{*}$$A_{k]}$

4. INDEPENDENCE
In this section we examine the independence structure in the quantum probability
space $(A, \phi)$ .
The axiomatic theory of “independence” and “white noise” in quantum probability theory was studied by K\"ummerer.14 The “independence” in the sense of
K\"ummerer was stated as follows.

of subalgebras of a unital algebra
Definition 4.1: A time indexed family
is independent in the sense of the K\"ummerer with respect to a state of , if the$\{\mathcal{B}_{k}\}_{k\in}\tau$

$B$

$B$

$\varphi$factorization for time ordered products holds, i.e.

$\varphi(B_{1}\cdots B_{m})=\varphi(B_{1})\cdots\varphi(B_{m})$wheneverand $k_{1}<k_{2}<\cdots<k_{m}$ .

$B_{i}\in \mathcal{B}_{k_{i}}$Such form of independence is referred to as the K\"ummerer independence. The
important examples of the K\"ummerer independence are the commuting independence,3 the anticommuting independence, 3 and the free independence.7,8
is independent in the sense of thecommuting independence if the algebras commute with each other (i.e.
and $i\neq j$ implies
whenever
if $k\neq l$ ) and.

Definition 4.2: A family of subalgebras,$\mathcal{B}_{k}\subset \mathcal{B}$$B_{i}\in B_{k}$$\varphi(B_{1}\cdots B_{m})=\varphi(B_{1})\cdots\varphi(B_{m})$.

$[\mathcal{B}_{k},\mathcal{B}_{l}]=0$$k_{i}\neq k_{j}$Definition 4.3:

A family of unital subalgebras,$\varphi(B_{1}\cdots B_{m})=0$ whenever$\mathcal{B}_{k}\subset B$$B_{i}\in B_{k_{i}},$ $k_{1}\neq k_{2}\neq\cdots\neq k_{m}$andis freely independent if$\varphi(B_{i})=0\forall\dot{i}$.

We omit the definition of anticommuting independence.3 The commuting inde.
. The anticommuting independence
pendence appears in the boson Fock space. The free independence appears in theappears in the fermion Fock space.
free Fock space
Let us investigate the independence structure in the permutational Fock space
be the -subalgebra of $A$ generated. For each time $k\in T$ , let $A_{k}=C^{*}(I,$with identity . Then the time indexed family of subalgebras
andbyis “independent” with respect to the vacuum state in the following sense.
$\Phi_{boson}$$\Phi_{fermi}on$$\Phi_{free}$$C^{*}$$d_{k’ k}^{+}d^{-)}$$\Phi$$d_{k}^{+}$$\{A_{k}\}_{k\in}\tau$$I$$d_{k}^{-}$$\phi$

Theorem 4.4:.
implies$\emptyset(A_{1}\cdots A_{n})=\phi(A_{1})\cdots\emptyset(An)$whenever$A_{i}\in A_{k_{i}}$and $i\neq j$$k_{i}\neq k_{j}$For the proof of this theorem, we prepare a lemma which is easily proved.

Lemma 4.5:

The algebra$A_{k}$

has a basis {I,$d_{k}^{+},$$d_{k[mathring]_{k}}^{-,d},$$d_{\dot{k}}$} as a vector space.

It is sufficient for the proof to restrict ourselves to, , let, $k_{n}=n$ . For each $i=1,$
consider the special case that $k_{1}=1,$ $k_{2}=2,$

Proof of Theorem 4.4:$\cdots$$A_{i}= \sigma_{i},\mathcal{T}_{i}\in^{\mathrm{p}}\sum_{\mathrm{e}\mathrm{r}(\{i\})}a_{\sigma_{i},\mathcal{T}}d_{\sigma}^{+}(i)d^{-}+a^{(}ii\mathcal{T}ii)d_{i}$.

$\cdots$$n$

72
be the expansion ofby the basis
singleton consisting of only one element . Since
expansion$A_{i}$

of$\{1, d_{i’ i}^{+}d-, d_{iii}^{+}d^{-}, d\}$$i$$\lambda$$A_{i}\in \mathcal{B}(\Phi),$, whereis thehas the another$\{i\}$$A_{i}$$A_{i}= \sum_{)\sigma,\tau\in \mathrm{P}\mathrm{e}\Gamma(\tau}b^{(i})d_{\sigma}+d_{\tau}^{-}\sigma,\mathcal{T}$by theby:“$\mathrm{b}\mathrm{a}s$is”$\{d_{\sigma}^{+}d^{-}|\mathcal{T}\sigma, \tau\in \mathrm{P}\mathrm{e}\mathrm{r}(T)\}$of$\mathcal{B}(\Phi)$. The coefficients$b’ \mathrm{s}$can be represented$a’ \mathrm{s}$$b_{\sigma,\tau}^{(i)}=|a_{(i)}^{(i)}a\Lambda,’\Lambda^{+a}(i)(i)(i)--a_{\sigma,\tau}^{(i)}aa^{(}(i)i\mathrm{o})$$(_{\mathrm{o}\mathrm{t}}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{W}\mathrm{i}\mathrm{s}\mathrm{e})(\mathrm{f}\mathrm{o}\mathrm{r}\sigma(\mathrm{f}_{\mathrm{o}\mathrm{r}}\sigma=\mathcal{T}=.(\dot{i})’)(\mathrm{f}\mathrm{o}\mathrm{r}(\sigma, \tau)(\mathrm{f}\mathrm{o}\mathrm{r}\sigma=\tau=\Lambda)(=\tau=\rho,i’)_{\mathrm{W}\mathrm{h}}\mathrm{e}\mathrm{r}\mathrm{e}=(\Lambda,(_{\dot{i}}))\mathrm{o}\mathrm{r}(\sigma, \tau)\rho\in=((i\mathrm{P}\mathrm{e}\mathrm{r}(T\backslash ),\Lambda))\{i\}’)$

andNote that$b_{\sigma,\tau}^{(i)}=0$$\ll\tau\neq\Lambda$),

if, and the right terminal of$\mathrm{o}\mathrm{r}\ll\tau\neq\Lambda$$\rho\neq\Lambda$, the right terminal of$\tau$$\tau$

is not equals $i>>$

is , and$i$$\sigma\neq\tau\gg$.

Let us consider the vacuum expectation$< \Omega|A_{1}\cdots A_{n}|\Omega>=\sum_{\sigma_{1^{\mathcal{T}}},1,\cdots,\sigma n\mathcal{T}n},b^{(}1)\ldots b^{()}n<\Omega|d+_{d\cdot\cdot d_{\sigma}}-.+d^{-}|\sigma_{1^{\mathcal{T}}},1\sigma_{n},\tau_{n}\sigma_{11}n\mathcal{T}n>\Omega$

For the vector
the term$d_{\tau_{n}}^{-}\Omega$

not to vanish,$\tau_{n}$.

must be equal to the null permutation A. For$b_{\sigma_{1}}^{(1)},b^{(2)}\cdots b_{\sigma_{n}}^{(n},<\Omega \mathcal{T}1\sigma 2,\tau 2)\Lambda|\cdots|\Omega>$

or $\tau_{n-1}=(n)$ . By the way, for the term
not to vanish, it is necessary that, and$\tau_{n-1}=\Lambda$...

$b_{\sigma_{11}}^{(1)},\mathcal{T}b(2)$$\mathrm{h}\mathrm{e}\mathrm{n}\mathrm{c}\mathrm{e}\mathrm{i}\mathrm{t}\sigma_{2},\mathcal{T}2$$b_{\sigma_{n-}}^{(n-1)}1^{\mathcal{T}_{n}},-1\neq 0$$\ll\tau_{n-1}=$

A

$>>$

,
, and the right terminal of$\mathrm{o}\mathrm{r}\ll\tau_{n-1}\neq\Lambda$$\tau_{n-1}$$b_{\sigma_{n},\Lambda}^{(n)}<\Omega|\cdots|\Omega>$is necessary that

equals $n-1\gg$ .

)Therefore, for the term
not to vanish, it is necessary that $\tau_{n-1}=$ A. Repeating this discussion, we can see that, for the term
to vanish, it must be hold that$b_{\sigma_{1}}^{(1)},T1\sigma_{2},\mathcal{T}_{2}b^{(}2$$\ldots b_{\sigma_{n},\Lambda}(n)<\Omega|\cdots|\Omega>$$b^{(1})\sigma_{1},\tau 1\sigma b^{(2})\ldots b2,\mathcal{T}2\sigma_{n}(n),\Lambda<\Omega|\cdots|\Omega>\mathrm{n}\mathrm{o}\mathrm{t}$$\tau_{12n}=\tau=\cdots=\tau=\Lambda$

This implies that$\sigma_{i}=$A or$<\Omega|d_{\sigma_{11}}^{+_{d}}\mathcal{T}-\ldots d^{+}\sigma_{n}\mathcal{T}d^{-}n|\Omega>=0$

term which survives is.

for each $i=1,$
if there exists such that$\sigma_{i}=(i)$)$b_{\Lambda,\Lambda}^{(1)}\cdots b^{(n}\Lambda,\Lambda$. In such case we have
. Therefore the only
. So we have$\cdots,$ $n$$i$$\sigma_{i}=(i)$$<\Omega|d_{\Lambda}^{+}d_{\Lambda\Lambda\Lambda}^{-}\ldots d+d^{-}|\Omega>$$<\Omega|A_{1}\cdots A_{n}|\Omega>=b_{\Lambda}(1,)\ldots b\Lambda(n)=<\Lambda,\Lambda\Omega|A_{1}|\Omega>,$ $.,$

This implies the factorization:$<\Omega|A_{n}|\Omega>$$\emptyset(A_{1}\cdots A_{n})=\emptyset(A_{1})\cdots\phi(A_{n})$.

$\square$.

73
Corollary 4.6: The family of -subatgebras
space $(A, \phi)$ is independent in the sense of K\"ummerer.
$C^{*}$$\{A_{k}\}_{k\in}\tau$in the quantum probability

can be viewd as a discrete time
The time indexed family of subalgebras
quantum i.i.d. process, i.e. a process with independently and identically distributed
subalgebras, in the K\"ummerer independence.
$\{A_{k}\}_{k\in}\tau$Remark 4.7: The following simple example shows that the independence arisingfrom the permutational Fock space is not the free independence. Put $A_{1}=d_{12}^{-,A=}$$(i=1, \cdots , 5)$ with
$A_{3}=I-d\mathrm{i},$, then we haveand$k_{1}\neq k_{2}\neq k_{3}\neq k_{4}\neq k_{5}$ and $\emptyset(A_{i})=0(i=1,2, \cdots , 5)$ but.
to see that the independence in the permutational Fock space is
Besides it isneither the commuting independence nor the anticommuting independence.
$A_{5}=d_{1}^{+}$$A_{4}=d_{2}^{+}$$d_{2}^{-},$$A_{i}\in A_{k_{i}}$$\emptyset(A_{12}AA_{\mathrm{s}}A4A_{5})=1$$\mathrm{e}\mathrm{a}s\mathrm{y}$5. FILTRATIONIn this section, we examine the filtration structure of the quantum probability
space $(A, \phi)$ .
Put $T_{k}=\{1,2, \cdots , k\}$ , and let $A_{k]}=C^{*}(I, d+, d^{-}ii|i\in T_{k})$ be the -subalgebraof $A$ generated by all the creation and annihilation operators up to time , withof -subalgebras of $A$ can be interidentity. Then the increasing family. Putpreted as the filtration of $A$ . Let us find the canonical basis of the algebra. Then we have$C^{*}$$k$$\{A_{k]}\}_{k\in}\tau$$C^{*}$$A_{k]}$$L_{k}=\{(\sigma, U, \tau)|\sigma, \tau\in \mathrm{p}_{\mathrm{e}\mathrm{r}}(T_{k}), U\subset T_{k}, U\perp[\sigma]\cup[\tau]\}$Theorem 5.1 (Fock Expansion for
unique expansion of the form

$A_{k]}$):

Every operator $A\in A_{k]}$ has the$A= \sum_{k(\sigma,U,\mathcal{T})\in L}a\sigma,U,\tau d+d\cdot d^{-}\sigma U\mathcal{T}$where$a_{\sigma,U,\tau}$’

are scalars.
under$Proof.\cdot \mathrm{A}\mathrm{t}\mathrm{f}\mathrm{i}\mathrm{r}\mathrm{s}4\mathrm{j}1\mathrm{e}\mathrm{t}\mathrm{u}\mathrm{S}\mathrm{s}\mathrm{h}_{\mathrm{o}\mathrm{w}\mathrm{t}}\mathrm{h}\mathrm{a}\mathrm{t}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{s}\mathrm{e}\mathrm{t}\{d^{+_{d}}d-\}\sigma\dot{U}\tau(\sigma,\tau)\in Lk\{\mathrm{o}_{\mathcal{T}}\mathrm{m}\mathrm{u}1\mathrm{t}\mathrm{i}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{C}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}.\mathrm{L}\mathrm{e}\mathrm{t}\mathrm{u}\mathrm{S}\mathrm{C}\mathrm{o}\mathrm{m}\mathrm{P}^{\mathrm{u}}\mathrm{t}\mathrm{e}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{d}\mathrm{u}\mathrm{c}\mathrm{t}(d_{\sigma\dot{U}}^{+_{dd^{\frac{U}{\tau’}})(d}}d_{\sigma}^{+},d_{\dot{U}}^{\cup},-,),\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{n}\}\mathrm{i}\mathrm{s}\mathrm{S}\mathrm{t}\mathrm{a}\mathrm{b}\mathrm{l}\mathrm{e}$the$(d_{\sigma}^{+}d_{U}.d_{\tau}^{-})(d_{\sigma}^{+},d_{U}.,d_{\tau}^{-},)$$=$$|d_{\sigma}^{+}d\cdot(Ud^{+_{d)}}d_{\sigma_{\mathrm{i}}\dot{U}()d}^{+_{d}}\mathrm{o}(_{0}\mathrm{t}\mathrm{h}\mathrm{e}.\mathrm{r}\mathrm{W}\mathrm{i}\mathrm{s}\mathrm{e}(\mathrm{i}\mathrm{f}\exists\alpha,\beta\in \mathrm{p}\mathrm{e}\mathrm{r}_{d_{\mathcal{T}}^{-}}((\mathrm{f}\exists\alpha,\beta d\cdot d^{-}\beta\alpha\alpha\beta\in \mathrm{P}\mathrm{e}\mathrm{r}(.Tk)\mathrm{S}.\mathrm{t}dl\dot{U}\tau’\dot{U}’d)-Tk)_{\mathrm{S}}’.\mathrm{t}..[\alpha]\perp[[\alpha]\perp[\beta\beta],\tau=],\mathcal{T}=(\alpha,\beta\alpha,\sigma’=(,\alpha), \sigma=’\alpha\beta)))’$,

Here we used the multiplication formula given in \S 2. Furthermore the factors in theabove expression can be rewritten as$d_{\beta}^{+}d_{\dot{U}}$(for$[\beta]\perp U$),$d_{U},d_{\beta}^{-}$$d_{\beta}^{-}d_{U}.’=\{$$d_{U}.d_{\beta}^{+}=\{$$0$$(\mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{w}\mathrm{i}\mathrm{S}\mathrm{e})_{?}$

$0$

),
(for
(otherwise).
$[\beta\}\perp U’$

74, is equal to either

This implies that$(d_{\sigma\dot{U}\tau}^{+}dd^{-})(d_{\sigma}^{+},d_{\dot{U}},d_{7^{-)}}^{-}$

the vector space.
Let us show the linear independence of the family
that. Using, or$d_{\sigma}^{+}d^{+_{d_{\dot{U}}d_{\alpha}}}\beta d_{\dot{U}},d_{\mathcal{T}}^{-}$$A_{k]}$$\{d_{\sigma}^{+}d_{\dot{U}}d\tau-\}_{(U,\mathcal{T}}\sigma,)\in Lk$. Assume$( \sigma,U,\mathcal{T})\sum_{L\in k}a_{\sigma},U,\tau d_{\sigma}+dUd_{\tau}-=0$.

$d_{U}$$d_{(\gamma,i}^{\mathrm{o}})$’$=I-[ \gamma]\perp U,\gamma\in.\mathrm{p}\mathrm{e}\mathrm{r}(\tau)(\gamma,i)i\in\sum_{s}Ut.,$

we have$d_{\sigma}^{+}d_{U}.d_{\mathcal{T}}-$$a_{\sigma,U,\tau}$$( \sigma,U,\tau\sum_{)\in L_{k}}$$=$$\sum_{\sigma,\tau\in \mathrm{p}\mathrm{e}\mathrm{f}(\tau k\rangle}a\sigma,\emptyset,\tau d+_{d_{\mathcal{T}}^{-}}+\sigma U\subset T_{k}\sum_{U\neq\emptyset}$$\sigma,\mathcal{T}\in[\sigma][\mathcal{T}]\perp\sum_{T\mathrm{p}_{\mathrm{e}}}\perp UU\mathrm{r}(k)a_{\sigma,U,\tau}d_{\sigma}+_{d_{U\tau}}\cdot d^{-}$$=$$( \sigma,U,\mathcal{T}\sum_{k)\in L}a_{\sigma)}U,\tau d^{+_{d-}}\sigma \mathcal{T}\sum_{U\neq\emptyset}-[\sigma]\sum_{[\tau]U}\perp U\perp [\gamma]\perp U,i\in U\gamma\in \mathrm{P}\mathrm{e}\mathrm{r}()(\gamma,i)s.t\sum_{T}.a_{\sigma,U,\gamma}\tau d_{\sigma}^{+}d_{(,i}^{\circ})\tau d^{-}$$=$$a_{\sigma,U,\tau}d^{+_{d_{()}^{\circ}}}\sigma\gamma,id_{\mathcal{T}}^{-}$$( \sigma,U,\mathcal{T})\in\sum_{k}a\sigma,U,\tau d+_{d-}\tau\sigma\sum_{UL\neq\emptyset}-[\sigma]\perp\sum_{[\tau]\perp U}U$$[ \gamma]\perp U\gamma\in(\gamma,i)\sum_{\mathrm{e}\mathrm{p}’ \mathrm{r}(T_{k})}.tsi\in.U,$.
$- \sum_{U\neq\emptyset}[\sigma]\perp U\sum_{[\tau]1U}$$\gamma\in \mathrm{P}\mathrm{e}\Gamma[\gamma]\perp U,i(\gamma,i)s(\tau\sum_{))\backslash \mathrm{p}\mathrm{r}(\tau}.t\in.U\mathrm{e}’ ka_{\sigma,U,\tau}d_{\sigma}+d_{(\gamma,i}\circ d_{\tau}^{-})$The third term of the last equality can be rewritten as($\sum$–$(\lambda, \mu, i, \nu)s.t$$\lambda,$.

$\nu\in \mathrm{P}\mathrm{e}\mathrm{r}(\tau_{k})$$i\in T_{k}$,$\sum$,$i\in U\subset T_{k}$$\sigma,$,“the left terminal of: mutually disjoint$\mu"\in T\backslash T_{k}$$[\mu],$$\exists\rho\in T_{k}s.t$$\tau\in \mathrm{P}\mathrm{e}\mathrm{r}(\tau_{k})$.

)$d_{\lambda}^{+}d_{()}^{\mathrm{o}}\mu,id^{-}\nu$

.

.

,

$U\perp[\lambda]\cup[\mu]\cup[\nu]$,$\mu\in \mathrm{P}\mathrm{e}\mathrm{r}(T)\backslash \mathrm{P}\mathrm{e}\mathrm{r}(\tau_{k})$$[\lambda]\cup[\nu],$$a_{\sigma,U,\tau}$$(\sigma, U, \mathcal{T})s.t$$\lambda=(\sigma, \rho),$,,$\nu=(\tau, \rho)$$\{i\}$

By the “linear independence” of$\{d_{\alpha\beta}^{+_{d}}-|\alpha,\beta\in \mathrm{P}\mathrm{e}\mathrm{r}(T)\}$, we get$a_{\sigma,U,\mathcal{T}}=0$$\sum_{(\sigma,U,\tau)S.t}$.

$i\in U\subset T_{k}$,$\exists\rho\in T_{k}S.t.\lambda=(U\perp[\sigma,\tau\lambda]\in^{\mathrm{p}_{\mathrm{e}}\mathrm{r}(\tau^{\cup}}\cup[\mu\sigma],[\nu]\rho))\nu k),’=(\tau,\rho)$

for all quadrupple
left terminal of$(\lambda,\mu,i, \iota \text{ノ})$$\mu"\in T\backslash T_{k}$s.

$\mathrm{t}$.

$\lambda,\nu\in \mathrm{P}\mathrm{e}\mathrm{r}(\tau k),$and the triple$i\in T_{k},$$[\lambda]\cup[\nu],$$[\mu],$$\mu\in \mathrm{P}\mathrm{e}\mathrm{r}(T)\backslash ^{\mathrm{p}\mathrm{e}\Gamma}(T_{k})$$\{i\}$, “theis mutually disjoint.

75$\lambda,$

Let us show that
and$i\in T_{k}$$\nu\in T_{k},$$a_{\sigma,U,\tau}=0$

for all, put$(\sigma, U, \tau)\in L_{k}$. For each triple$(\lambda,i, \nu)\mathrm{s}.\mathrm{t}$.

$i\not\in[\lambda]\cup[\nu]$$l( \lambda, \nu)=\max\{1\mathrm{e}\mathrm{n}\mathrm{g}\mathrm{t}\mathrm{h}(\rho)|\exists\lambda’, \nu’, \rho\in T_{k}s.t.

\lambda=(\lambda’, \rho), \nu=(\nu’,\rho)\}$,. Let us first cinsider the case of $l(\lambda, \nu)=0$ . Put
where lengh $(\rho)=r$ foris any permutation . .
$\mu:=\mu_{1}$ , where$\rho\in\tau^{\mathrm{p}_{r}}$$\mathrm{t}$$\mathrm{s}$$\mu_{1}$$[\mu_{1}]=\tau_{k+1}\backslash ([\lambda]\cup[\nu]\cup\{i\})$then we get$a_{\lambda,\{i\},\nu}=0$. Put$\mu:=\mu_{2}$, where,is any permutation$\mu_{2}$. .
$\mathrm{t}$$\mathrm{s}$$[\mu_{2}]=T_{k}+1\backslash ([\lambda]\cup[\nu]\cup\{i,j\})$with $i\neq j$ and$\{\dot{i},j\}\subset T_{k}\backslash ([\lambda]\cup[\nu])$, then we get$a_{\lambda,\{i\},\nu}+a_{\lambda},\{i,j\},\nu=0$

and hence$a_{\lambda,\{i,j\})\mathcal{U}}=0$,. Repeating this discussion, we have$a_{\lambda,\{i\},\nu}=0$,$a_{\lambda,\{i,j\},\nu}=0$$a_{\lambda,\{i,k\},\nu}=0$

,
,

$\{$

$a_{\lambda,\{i,j,k}\},\nu=0$,.

$a_{\lambda,\tau_{k\backslash }}([\lambda]\cup[\nu]),\nu=0$. $l(\lambda, \nu)=0$ and $U\ni i$ . Next
for all triple
Hence we get
and $\nu=(j_{1}, \cdots,j_{S})$ ,let us consider the case of $l(\lambda, \nu)=1$ . For$\nu’=(j_{1},\cdots,j_{s-1})$. By the discussion similar to theandputpreceeding one, we get$(\lambda, U, \nu)\in L_{k}\mathrm{s}.\mathrm{t}$$a_{\lambda,U,\nu}=0$$\lambda=(i_{1}, \cdots, i_{r})$$\lambda’=(i_{1}, \cdots,\dot{i}_{r-1})$$a_{\lambda,\{i\},\nu}+a_{\lambda’},\{i\},\nu^{\prime=0}$’$a\lambda,\{i\},\nu+a\lambda’,\{i\},\nu’+a\lambda,\{i,j\},\nu+a\lambda^{;},\{i,j\},\nu’=0$$a_{\lambda,\{i\}_{\mathcal{U}}},+a\lambda’,\{i\},\nu’+a\lambda,\{i,k\},\nu+a_{\lambda’,\{k}i,\},\nu’=0$,,$a_{\lambda,\{i\},\nu}+a_{\lambda’,\{i}\},\nu’+a\lambda,\{i,j\},\nu+a\lambda’,\mathrm{t}^{i},j\},\nu l+a\lambda,\mathrm{t}i,k\},\nu+a\lambda’,\mathrm{t}^{i},k\},\nu’$$\{$$+a_{\lambda,\{i,j,\},\nu}k+a_{\lambda}’,\{i,j,k\},\nu’0=$,$a_{\lambda’,U,\nu}’=0$$a_{\lambda,U,\nu}+$$\sum_{Us.t}$$\sum_{Us.t}$

.

.

.

$U\perp i\in U\subset\tau_{k}[\lambda]\cup[\nu \mathrm{i}, U\perp i\in U\subset\tau_{k}[\lambda]\cup[\nu \mathrm{i}$,Hence we get$a\lambda,\{i\},\nu=a\lambda,\{i,j\},\nu=a\lambda,\{i,k\},\nu=\ldots=a\lambda,\{i,j,k\},\nu\lambda,T_{k}\backslash =\cdots=a([\lambda]\cup[\nu]),\nu=0$.

76Here we used the as sumption $l(\lambda’, \nu’)=0$ . So we have
fcr all triple. $l(\lambda, \nu)=1$ and $U\ni i$ . By the induction on the number $l=l(\lambda, \nu)$ ,we finally get$a_{\lambda,U,\nu}=0$$(\lambda, U, v)\in L_{k}\mathrm{s}.\mathrm{t}$(for$a_{\lambda,U,\nu}=0$$(\lambda,$$U,$ $\nu)\in L_{k},$$[\lambda]\cup[\nu]\perp\{i\},$$U\ni\dot{i}$).

So we have$a_{\sigma,U,\tau}=0$(for$(\sigma,$$U,$ $\tau)\in L_{k},$$U\neq\emptyset$).

.
that this family is a$\mathrm{b}\mathrm{a}s$

is of$A_{k]}$$\square$

6. CONDITIONAL EXPECTATION:In this section, we construct a kind of “conditional expectation”
from the algebra $A$ onto the subalgebrawith respect to the vacuum state .
The conditional expectation
will be used in the next section to define the notionof operator martingales.
Before constructing the conditional expectation, let us first examine the decom, then the. Put
position of the algebra
can be decomposed to the following form as a vector space.
algebra$\epsilon_{k\mathrm{l}}$$A\mapsto\epsilon_{k\mathrm{l}}(A)$$\phi$$A_{k]}$$\epsilon_{k\mathrm{l}}$$B_{k]}=C^{*}(d^{+}, d^{-|\sigma}\sigma\sigma\in \mathrm{p}_{\mathrm{e}\mathrm{r}}(T_{k}),$ $[\sigma]\ni$A_{k]}$k)$$\mathcal{B}_{k]}$Proposition 6. 1:$\mathcal{B}_{k]}$$=$$\sigma,\tau\in \mathrm{P}\mathrm{e}\mathrm{r}(\tau_{k})-\bigoplus_{-1}d_{(}+Ak-1]d^{-}\sigma,k)(\mathcal{T},k)\oplus\bigoplus_{\sigma\in \mathrm{P}\mathrm{e}\mathrm{r}(Tk-1)}d+A_{k-1}(\sigma,k)]$$\oplus\bigoplus_{-}Ak-1]d^{-}\tau\in \mathrm{P}\mathrm{e}\mathrm{r}(T_{k}1)k(\tau,)\oplus A_{k-1]}d_{k}.$.

Proof. At first let us show that the set$\cdot$$\{d_{(\sigma,k)}^{+}Ad_{(\mathcal{T},k)}-, d^{+}B, Cd-Dd(\sigma,k)(\mathcal{T},k)’ k|A, B, C, D\in A_{k-1]}, \sigma, \mathcal{T}\in \mathrm{p}_{\mathrm{e}}\mathrm{r}(\tau_{k-1})\}\cup\{0\}$is stable under the multiplication. Using the fact that$d_{\sigma}^{+}d_{k}$$=$$d_{k}d_{\sigma}^{+}$$d_{k}^{+}d_{k}$$=$$d_{k}^{+}$

,

(for

$k\not\in[\sigma]$$d_{k}d_{k}^{-}=d_{k}-$),$d_{\dot{k}}$is a projection and formulas(for$d_{\sigma}^{-}d_{k}=d_{k}d_{\sigma}^{-}$$k\not\in[\sigma]$),,

we get the multiplication rule as follows:$(_{\mathcal{T}=\sigma’})$$0$(otherwise),,,$(d_{(\sigma,k)}^{+}B)(d_{(}^{+}A\prime d_{(\tau\cdot k}^{-}’,))\sigma’,k)=0$,$d_{(\sigma,k)}^{+}(Ad_{\mathcal{T}}.B’)$$(_{\mathcal{T}=\sigma’})$$0$(otherwise),$(d_{(\sigma,k)(\mathcal{T}}^{+}Ad-,)k)(d(\sigma kl)B^{;}+,)=\{$$(d_{(\sigma,k)(\tau}^{+}Ad-,k1)(C\prime d_{(}-)\tau k’,)=0$$(d_{(\sigma,k)}^{+}Ad^{-})(\tau,k)(D\prime d_{k}.)=0$,,$d_{(\sigma,k)}^{+}(Ad_{\mathcal{T}}.A’)d_{()}^{-}\mathcal{T}k’$$(d_{(\sigma,k}^{+}A)(\mathcal{T},k)d-)(d_{(,)}^{+},A’d_{(,)}-,)\sigma k\mathcal{T}k=\{$$(d_{(\sigma,k)}^{+}B)(d_{\mathrm{t}^{\sigma,k})}^{+}\prime B’)=0$$(d_{(\sigma,k)}^{+}B)(c’d_{(k)}^{-}\tau’,)=d_{(k}+(\sigma,\rangle Bo’)d^{-}(\tau k’,)$

’

,

,

77

$(d^{+}B)(\sigma,k)(D\prime d_{k}.)=d(\sigma,k)(+BD’)$,,(otherwise),$(_{\mathcal{T}=\sigma’})$$(CA’)d_{(,)}^{-}\mathcal{T}kl$$(Cd_{(\mathcal{T}}^{-},)k)(d+,A’(\sigma k)\prime d_{()}^{-};,)\mathcal{T}k=\{$$0$$(Cd_{(\tau}^{-},)k)(c’d-,,)(\tau k)=0$,$(Cd_{\mathcal{T}}.B’)d_{\dot{k}}$$(\tau=\sigma)’$$0$(otherwise),$(Cd_{(\tau}^{-},)k)(d(\sigma^{l},k+)B’)=\{$,$(Cd_{(\tau_{\mathrm{C}}k)}^{-},)(DJd\cdot)k=0$$(Dd_{k}.)(d+A’(\sigma’,k)d_{(k)}^{-})\tau’,=0$,,$(Dd_{k}.)(d_{()}^{+}\sigma^{l},kB’)=0$$(Dd_{k})(c\prime d_{(}-,,)\mathcal{T}k)=(Dc’)d^{-}(\tau" k)$’,$(Dd_{k})(D’ d\cdot)k=(DD’)d_{k}$,where $A,$ $B,$ $C,$ $D,$ $A’,$ $B’,$ $c’,$ $D’\in A_{k-1]}$ . This implies that vector space $C=$
forms an algebra.
The equality $B_{k]}=C$ is obvious. We also get the direct sum property ffom the.
linear independence of the finite family of operators$\Sigma_{\sigma,\tau}d_{(\sigma,k}^{+}A_{k1]}-d)(\tau,k)-+\Sigma_{\sigma}d_{(\sigma,k}^{+}A_{k-1})]+\Sigma_{\tau}A_{k-1]k}d(\mathcal{T}-,)+A_{k-1]}d_{\dot{k}}$$\square$$\{d_{\sigma}^{+}d_{\dot{U}}d\tau-\}_{(U,\tau)\in}\sigma,Lk$

Besides the algebrawith CI.

Proposition 6.2:

$A_{k]}$

can be decomposed to the direct sum of$A_{k_{\rfloor}^{\rceil}}=\mathrm{C}I\oplus\beta_{1]}\oplus\cdots\oplus\beta_{k]}$$B_{j]}’ \mathrm{s}(j\leq k)$.

Corollary 6.3:
$A_{k]}$$=$$A_{k-1]} \oplus\bigoplus_{\mathrm{r}\sigma,,\mathcal{T}\in \mathrm{P}\mathrm{e}(Tk-1)}d_{(k}^{+}A_{k}\sigma,)-1]d_{(\rangle}-\tau,k\oplus\bigoplus_{\sigma\in \mathrm{p}\mathrm{e}\mathrm{r}(Tk-1)}d_{(}^{+}A_{k1}\sigma,k)-]$$\oplus\bigoplus_{\mathrm{r}\tau\in \mathrm{P}\mathrm{e}(\tau_{k-1})}A_{k}-1]d_{(}-k\mathcal{T},)\oplus A_{k-1]}d_{k}.$.

, let us construct a conditional exUsing the decomposition of the algebra$Ak$Let: ] $arrow A_{k-1]}$ be a linear map defined by$A_{k]}$$\epsilon_{k-1,k}$$\mathrm{p}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{i}_{\mathrm{o}\mathrm{n}}..\cdot$

$A$

$0$

$\mathcal{E}_{k-1,k}(A)=\{$$A\in\oplus_{\sigma,\mathcal{T}\in \mathrm{p}}\mathrm{e}\mathrm{r}(\tau_{k-}1)d^{+}A_{k}k)-(\sigma,1]d(-\mathcal{T},k))$$0$$A\in\oplus_{\sigma\in \mathrm{p}_{\mathrm{e}\mathrm{r}}(}\tau_{k1}-$$0$) $d^{+}A_{k-1}](\sigma,k))$$A\in\oplus_{\mathcal{T}\in \mathrm{P}\mathrm{r}}\mathrm{e}(\tau_{k-}1)Ak-1]d(\mathcal{T},k)-$$B$

For $j<k$ , put(if $A\in A_{k-1]}$ ),
(if
(if
(if
(if

$A=Bd_{\dot{k}}\in A_{k-1]}d_{\dot{k}}$$\epsilon_{j,k}=\epsilon\circ\cdots\circ\epsilon j,j+1k-1,k$

’

,

,

),

).

then the following holds.

$j<k$ , satisfies the followings for:
Proposition 6.4: The map:
all $A\in A_{k]}$ and all $B,$$(A*)=\mathcal{E}(j,k)^{*}A;$;(3)(1) $\epsilon_{j,k}(B)=B$ ; (2)
. (5)
;
(4)
.
(6) $\phi(\epsilon_{j,k}(A))=\phi(A)$ ; (7)$\epsilon_{j,k}$$B_{1},$$A_{k]}arrow A_{j]},$$B_{2}\in A_{j]}$$\epsilon_{j,k}(A^{*}A)\geq 0$$(A^{*}A)\geq\epsilon_{\mathrm{j},k}(A)^{*}\epsilon jk(A)$$||_{\mathcal{E}_{j,k}}(A)||\leq||A||)$$\epsilon_{j,k}(B_{1}AB_{2})=B_{1}\epsilon_{\mathrm{j},k}(’ A)B_{2}$Proof. We only have to show that the map$\cdot$$\epsilon_{k-1,k}.\mathrm{S}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{S}\mathrm{f}\mathrm{i}\mathrm{e}\mathrm{S}(1:.),$$\cdots,$(7).

(1) Obvious from the definition.
(2) Let$A=A_{0}+ \sum_{\tau\sigma},d+A_{\sigma},\mathcal{T}d^{-}+(\sigma,k)(\tau,k)\sum_{\sigma}d^{+}B_{\sigma}+(\sigma,k)\sum_{\mathcal{T}}C_{\mathcal{T}}d-+Dd_{k}(_{\mathcal{T},k}).$

,

78
where

$A_{0},$$A_{\sigma,\mathcal{T}},$$B\sigma’ C\tau’ D\in A_{k-1]}$, then$A^{*}=A_{0}^{*}+ \sum\sigma,\tau d_{(_{\mathcal{T}},)\sigma}+kA^{*},d^{-}+k\sigma,)\sum_{\sigma}\tau(d_{(k)}^{-}B_{\sigma}*+\sigma,\sum d+C_{\tau}*+k)D*d_{k}\mathcal{T}(\mathcal{T},\cdot$,

and hence $\epsilon_{k-1k}(A^{*})=A_{0}^{*}+D^{*}=\epsilon_{k-1,k}(A)^{*}$ .
(3) By the muitiplication rule discussed in Proof of Theorem 6.1, we get$\epsilon_{k-1,k}(A^{*}A)$$=$$A_{0^{A}0+}^{*}+A_{0^{D}0+\sum_{(}B_{\sigma}}*D*A+D*D\sigma\in \mathrm{p}_{\mathrm{e}}\mathrm{r}Tk-1)*B_{\sigma}$$=$$(A_{0}+D)*(A0+D)+ \sum B^{*}B_{\sigma}\geq\sigma 0\sigma$.

(4) From $\epsilon_{k-1,k}(I)=I$ and (3), we get $||\epsilon_{k-1,k}||=1$ . Hence $||\epsilon_{k-1,k}(A)||\leq||A||$ .
(5) Since $\epsilon_{k-1,k}(A)^{*}\epsilon-1,k(k)A=(A_{0}+D)^{*}(A_{0}+D)$ , we get$\epsilon_{k-1,k}(A^{*}A)-\mathcal{E}_{k-1},(kA)^{*}\mathcal{E}_{k-1},(kA)=\sum_{\sigma}B_{\sigma}^{*}B_{\sigma}\geq 0$.

(6) We have$<\Omega|A|\Omega>$$=< \Omega|A_{0}+\sum_{\sigma,\mathcal{T}}d_{(\sigma,k}^{+}A\sigma,\mathcal{T}d_{(k)}-+\sum_{\sigma})\tau,\sigma,)+d_{(k}+B_{\sigma}\sum_{\mathcal{T}}c_{T}d-+k)Dd\cdot|(\tau,k\Omega>$$=<\Omega|(A_{0}+Dd_{k})|\Omega>=<\Omega|(A_{0}+D)|\Omega>=<\Omega|\epsilon_{k-1},(kA)|\Omega>$.

(7) We have$B_{1}AB_{2}$$=$$B_{1}(A_{0+} \sum_{\sigma,\mathcal{T}}d_{(\sigma,k}^{+}A\sigma,\mathcal{T}d_{(k)}-+\sum_{\sigma})\mathcal{T},(\sigma,k)\sigma+d^{+}B\sum_{\mathcal{T}}c_{\tau}d-+k)D(\mathcal{T},kd\cdot)B_{2}$$=$$B_{1}A_{0}B_{2}+(B_{1}DB_{2})dk$$+ \{\sum_{\sigma,\mathcal{T}}B_{1}d^{+}A_{\sigma},\tau d-B2+(\sigma,k)(\tau,k)\sum_{\sigma}B_{1}d+B_{\sigma}B_{2}+(\sigma,k)\sum B_{1}c\mathcal{T}d_{(\mathcal{T},k)}^{-}B_{2}\tau\}$

Since the last term$\{\cdots\}$

vanishes under$\epsilon_{k-1,k}$.

, we get$\epsilon_{k-1},(kB1AB_{2})=B1A_{02}B+B_{1}DB_{2}=B_{1}(A_{0}+D)B_{2}=B_{1}\mathcal{E}_{k-1,k}(A)B_{2}$.

$\square$

The system of maps
is consistent, i.e.
for $j<k<l$ .
So we can define in the natural way a map
as the extension of:each. Furthermore a map ; $Aarrow A_{j]}$ can be defined in the natural way as thecontinuous extension of because of the norm continuity of . Then the following
is easily obtained.
$\{\epsilon_{j,k}|j<k\}$$\epsilon_{j,k}0\epsilon_{k},l=\epsilon_{j,l}$$\tilde{\epsilon}_{j\mathrm{l}}$$\epsilon_{j,k}$$\bigcup_{k}A_{k]}arrow A_{j]}$$\epsilon_{j\mathrm{l}}$$\tilde{\epsilon}_{J\mathrm{l}}$$\tilde{\epsilon}_{j\mathrm{l}}$Theorem 6.5: The map
and all $B,$
:
(1)
; (2)
(4)
; (5)
$B_{1},$:$Aarrow A_{j]}$satisfies the followings for all $A\in A$$B_{2}\in A_{j]}$$\epsilon_{j\mathrm{l}}(B)=B$$||\epsilon_{j\mathrm{l}}(A)||\leq||A||$(6)$\epsilon_{j\mathrm{l}}$$\epsilon_{j\mathrm{l}}(A^{*})=\epsilon_{j\mathrm{l}}(A)^{*};$(3)$\epsilon_{j\mathrm{l}}(A^{*}A)\geq 0,\cdot$$\epsilon_{j\mathrm{l}}(A^{*}A)\geq\epsilon_{j\mathrm{l}}(A)^{*}\epsilon_{j\mathrm{l}}(A),\cdot$$\phi(\epsilon_{j\mathrm{l}}(A))=\emptyset(A);(7)\epsilon_{j\mathrm{l}}(B_{1}AB_{2})=B_{1}\epsilon_{j\mathrm{l}}(A)B_{2}$.

Hence it is natural to interpret the map
as a conditional expectation from thewith respect to the vacuum state .
algebra $A$ onto the subalgebra$\epsilon_{k\mathrm{l}}$$A_{k]}$$\phi$

79
Remark 6.6: Conditional expectation for an operator algebra was originallyintroduced by H. Umegaki15 in the theory of von Neumann algebras. Note that themapsatisfies the properties of conditional expectation in the sense of Umegaki,
up to the faithfulness of the state and the weak continuity of the map. The map
is rather a conditional expectation in the -algebraic setting.
$\epsilon_{k\mathrm{l}}$$\phi$$C^{*}$$\epsilon_{k\mathrm{l}}$7. OPERATOR MATINGALESIn this section, we discuss about operator martingales on the quantum probability space $(A, \phi)$ .
Let us introduce the notion of adapted operators, previsible operators and operator martingales. These notions are defined in the parallel way to those in the Journ\’es
toy Fock space.2 An operator $A\in A$ is called a -adapted operator if $A\in A_{k]}$ . An
be a time
operator $A\in A$ is called a -previsible operator if $A\in A_{k-1]}$ . Let
indexed family of adapted operators $M_{k}\in A_{k]}$ . We call it an operator martingde ifof operators is called afor $k\leq k’$ . A time indexed family
is a -previsible
rator.
previsible process if each
$k$

$k$

$\{M_{k}\}_{k\in}\tau$$\{u_{k}\}_{k\in}\tau$$\epsilon_{k\mathrm{l}}(M_{k’})=M_{k}$$k$$u_{k}$$\mathrm{o}\mathrm{p}\mathrm{e}\llcorner$

The creation process
annihdation process$D_{k}^{+}= \sum_{j=1}^{k}d_{j}+$$D_{k}^{-}= \sum_{j=1}^{k}d_{j}-\mathrm{a}\mathrm{r}\mathrm{e}$Note that the process$D_{k}= \sum_{j=1}^{k}d_{j}$submartingale) but the process$incl\mathrm{t}/sion$

operator, the conservation processby$e_{\dot{k}}$$D_{k}^{\mathrm{o}}= \sum_{j=1}^{k}d_{j}^{\circ}$

and theimportant examples of operator martingale.

is not an operator martingale (it is rather a$\prod_{j=1}d_{j}k$$e_{\dot{k}}=\backslash I-d_{\dot{k}}$is an operator martingale. Let us define the, then the process

martingale. For a finite subset $U\subset T$ , put
inclusion operatoris$e_{\dot{U}}=\Pi_{k\in U}e_{\dot{k}}$$E_{k}= \sum_{j=1}^{k}e_{j}$is an operator. The explicit action of the$e_{U}$$e_{\sigma}$$e_{U}.e_{\sigma}=\{$
$0$

),
(if
(otherwise).
$U\subset[\sigma]$

We get two expansions:$d_{U}= \sum_{V\subset U}(-1)\#(V)e_{V}$and$e_{U}= \sum_{V\subset U}(-1)\#(V)d_{V}$.

becomes another
Using this expansion, we can see that the family
expansion for
Fock. We also get the following uniform
basis of the algebra
operators in the -algebra $A$ because of the the norm continuity of the conditional
expectation: $Aarrow A_{j]}$ .
$\{d_{\sigma}^{+}e_{\dot{U}}d_{\tau}^{-}\}_{(U,\tau)\in}\sigma,Lk$$A_{k]}$$C^{*}$$\epsilon_{j\mathrm{l}}$Theorem 7.1 (Uniform Fock Expansion): Each operator $M\in A$ has the uniqueexpansion of the form
$M= \lim_{jarrow\infty(\sigma,U,\tau}\sum_{j)\in L}m_{\sigma},U,\tau d_{\sigma}^{+}e.d_{\mathcal{T}}U-$80
in the uniform topology.

In Journ\’e’s toy Fock space, the following predictable representation theorem
for operator martingales holds.2 Let(resp.
) be the creation operator (resp.
annihilation operator, conservation operator) in Journ\’e’s toy Fock space.
$\Phi_{toy}$$\mathrm{O}$$a_{j}^{+}$$a_{j}^{-},$$a_{j}$$\Phi_{toy^{2}}$Theorem 7.2 (Predictable): Each operator martingale
on
can be uniquely expanded to the form$\mathrm{R}e\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}^{2}$$\{M_{k}\}_{k\in}\tau$$\Phi_{toy}$$(k=1,2, \cdots)$ ,$M_{k}=mI+ \sum_{j=1}u_{jj}^{++}ak+\sum_{j=1}u_{jj}^{--}ak+\sum_{j=1}^{k}u_{j}^{\mathrm{O}}a^{\mathrm{o}}j$where $m$ is a scalar,$\{u_{k}^{+}\}_{k\in T},$$\{u_{k}^{-}\}_{k}\in T$andare previsible processes on$\{u_{k}^{\mathrm{o}}\}_{k}\in\tau$$\Phi_{toy}$.

This means that each operator martingale on
has the canonical expansion asa sum of triple of discrete time “stochastic integrals” of previsible processes where the$\Phi_{toy}$“integrators” are the three basic martingales, i.e. the creation process
the annihilation process$A_{k}^{-}= \sum_{j=1}^{k}a_{j}-\mathrm{a}\mathrm{n}\mathrm{d}$the conservation process$A_{k}^{+}= \sum_{j=1}^{k}a_{j}+$$A_{k}^{\mathrm{O}}= \sum_{j=1}^{k}a^{\mathrm{o}}j$,.

In the case of permutational Fock space , the similar representation holds for
only a special class of martingales. Denote by $D(T_{k})$ the set of all triples.
with $i_{1}>\cdots>i_{r}$ and $\tau=(j_{1}, \cdots,j_{S})$ with $j_{1}>\cdots>j_{s}$ .
Then the following is easily obtained.
$\Phi$$(\sigma, U, \tau)\in$$L_{k}\mathrm{s}.\mathrm{t}$$\sigma=(i_{1}, \cdots,\dot{i}_{r})$Proposition 7.3: Let
the Fock expansion of the form$\{M_{k}\}_{k\in}\tau$be an operator martingale such that each$M_{k}= \sum_{k(\sigma,U,\mathcal{T})\in D(}\tau)m_{\sigma,U}^{(},d^{+}ek)\tau\sigma.Ud_{\tau}-$then$\{M_{k}\}_{k\in}\tau$can be expanded to the form$\{u_{j}^{+}\}_{j\in}T,$$\{u_{j}^{-}\}_{j}\in T,$.

$\{u_{[mathring]_{j}}\}_{j\in}\tau$

and

has

,

$M_{k}=mI+ \sum_{1j=}^{k}d_{j}++u^{+}\sum^{k}ju_{jj}j=1-d-+\sum d^{+_{u_{j}^{\mathrm{O}}}}d_{j}-+j=1k\sum ju_{j}.ej=1kj$

where $m$ is a scalar,in the algebra $A$ .

$M_{k}$$\{u_{\dot{j}}\}_{j\in}\tau$$(k=1,2, \cdots)$ ,are previsible processes

For a general martingale, another form of predictable representation holds.
Proposition 7.4:

Each operator martingale

form$\{M_{k}\}_{k\in}\tau$can be $e\varphi anded$ to the$M_{k}=mI+ \sum_{j=1\sigma\in Pe}^{k}\sum dr(T_{j-}1)+(\sigma,j)u_{(}+\sigma,j)+\sum_{j=1\tau\in P}ke\sum_{r(T_{j-1})}u^{-d_{(j)}}(\tau,j)-\tau$

,

$+$

where

$m$

is a scalar,$j \mathcal{T}\in\sum_{=1\sigma}^{k},\sum d+\mathrm{O}d_{(\mathcal{T},j)}u_{(\sigma}),(\tau,j)\sum(\sigma,j),ju_{j}-+\cdot ePer(\tau_{j-1})j=1kj.$$u_{(\sigma,j)}^{+},$$u-u_{(\sigma}(_{\mathcal{T},j)}’ \mathrm{O},j),(_{\mathcal{T},j)}$and$u_{\dot{j}}$.

are -previsible operatoYs.
$\gamma$

81
is not an operator
Though the process with independent increments
is useful to describe the discrete time analogue ofmartingale, the process
quantum Ito’s formmula in the next section because it allow us to make quantum
Ito table simple.
$\{D_{\dot{k}}\}_{k\in}\tau$$\{D_{\dot{k}}\}_{k\in}\tau$

8. ANALOGUE OF QUANTUM$\mathrm{I}\mathrm{T}\mathrm{O}’ \mathrm{S}$FORMULAIn this section, we examine in the quantum probability space $(A, \phi)$ a discrete
time analogue of quantum Ito’s foumula.
Generally speaking, quantum Ito’s formula which is given in the continuous timeof twois the rule to compute the productquantum stochastic. In a differential form, quantum Ito’s formula canandstochastic integrals
be viewed as the rule to compute the product $dM_{t}dN_{t}$ of two stochastic differentials, appearing in the following equation:and$M_{t}N_{t}$$\mathrm{c}\mathrm{a}\mathrm{l}\mathrm{c}\mathrm{u}\mathrm{l}\mathrm{u}\mathrm{S}^{5,6,9}$$M_{t}$$dM_{t}$$N_{t}$$dN_{t}$$d(M_{t}N_{t})=(dM_{t})N_{t}+M_{t}(dN_{t})+dM_{t}dN_{t}$.

The corresponding multiplication table described by the stochastic differentials of
the basic operator processes is often called the quantum Ito table.
Well, in our case of discrete time situation, the increment of the productis equal toandof two discrete time “stochastic integrals”$M_{k}N_{k}$$M_{k}$$N_{k}$$M_{k}N_{k}-M_{k-1k-1}N=\xi_{k}N_{k-1}+M_{k-1}\eta_{k}+\xi_{k}\eta_{k}$Here $\xi_{k}=M_{k}-M_{k-1}$ and $\eta_{k}=N_{k}-N_{k-1}$ . Hence we interpret the rule for computing
andof increments of two discrete time “stochastic integrals”
the productas a discrete time quantum $Ito’ s$ formula.
For simplicity, we restrict ourselves to consider a special class of operator processes ( discrete time analogue of stochastic integrals) which is stable under the
pointwise multiplication and which contains the basic operator processes, i.e. the, the annihilation, the conservation processcreation process.
, and the exclusion process
cessbe two operator processes of the following form:
andLet$M_{k}$$\xi_{k}\eta_{k}$$N_{k}$$=$$\{D_{k}^{\mathrm{o}}\}_{k}\in\tau$$\{D_{k}^{+}\}_{k}\in\tau$$\mathrm{p}\mathrm{r}\mathrm{c}\succ$$\{D_{\dot{k}}\}_{k\in}\tau$$\{D_{k}^{-}\}_{k}\in T$$\{N_{k}\}_{k\in}\tau$$\{M_{k}\}_{k\in}\tau$$M_{k}$$=$$mI+ \sum_{j=1}^{k}Ajd^{+}Bjjdjj-_{C}+\sum_{J=1}^{k}D_{j}d+E_{j}j+\sum_{j=1}^{k}F_{j}d-c_{j}j+\sum_{j=1}^{k}Hjdj$’$N_{k}$$=$$nI+ \sum_{j=1}^{k}A_{j}\prime d_{j}+B_{j}’d^{-_{C}}jj’+\sum_{J=1}^{k}D’jjd^{+}E’j+.\sum_{j=1}^{k}F’d_{j}j-G’j+\sum_{j=1}^{k}H_{j}’d_{j}$,,are scalars, andproduct
pointwise. Then theare -previsible operators frombecomes to a finite
andof two operator processesof the above
processsum of operator processes where each summand is an operator
form. So the set of all such operator processes is stable under the multipliCatio.n.
Now let us define a continuous linear map $\delta:Aarrow A$ by
where $m,$
$E_{j}’,$$F_{j}’,$$A_{j},$$n$$G_{j}’,$$H_{j}’$$B_{j},$$C_{j},$$D_{j},$$E_{j},$$F_{j},$$G_{j},$$H_{j},$$A_{j}’,$$B_{j}’,$$C_{j}’,$$D_{j}’$$A_{j-1]}$$j$$\{M_{k}\}_{k\in}\tau$$\{M_{k}N_{k}\}_{k}\in\tau$$\{N_{k}\}_{k\in}\tau$$\cdot$.

$e_{U}$$\delta(d_{\sigma U}^{+_{ed_{\mathcal{T}}^{-}}}.)=\{$$0$(if $\sigma=\tau=\Lambda$ ),(otherwise),82
then we get(for $A\in A_{k-1]}$ ).

$d_{kk}^{-A}d^{+}=\delta(A)dk$

Using the map , we have the following.
$\delta$

Theorem 8.1 (Analogue of Quantum Ito’s Formula): Letandbe two operator processes given above, then the pointwise product $\{M_{k}N_{k}\}k\in T$ is calculated based on the followoning rvle.
$\{M_{k}\}_{k\in}\tau$$(Ad_{j}+Bd_{j}^{-}C)(A’d_{j}^{+}B\prime d_{j}^{-}c’)=Ad_{j}^{+}(B\delta(CA’)B’)d^{-c’}j$$(Ad_{j}^{+}Bd_{j}^{-}C)(D’d+E’)j=Ad_{j}^{+}(B\delta(cD’)E’)$$(Ad_{j}^{+}Bdj-C)(F’d_{j}^{-}G’)=0$,$(Dd_{j}^{+}E)(A\prime d^{+}jB\prime d_{j}^{-}c’)=0$’,$(Ad_{j}^{+}Bd_{j}^{-}C)(H’d\cdot)j=0$,$(Dd_{j}^{+_{E}})(D^{J}d^{+_{E)}}j=0$$(Dd_{j}^{+}E)(F^{;}d-G’)j=Dd_{j}^{+}(EF’)d_{j}-c’$$(Dd_{j}^{+}E)(H’d\cdot)j=Dd_{j}^{+}(EH^{J})$$\{N_{k}\}_{k\in}\tau$

,

,

,

,

$(Fd_{j}^{-}G)(A’d_{j}^{+}B’d^{-}C)j=(F\delta(GA’)B’)d^{-}C’j$’$(Fd_{j}^{-}G)(D’d_{j}^{+_{E^{J}}})=(F\delta(GD’)E’)dj$$(Fd_{j}^{-}G)(F’d_{j}^{-}G’)=0$,$(Fd_{j}^{-}G)(D^{;}d_{j})=0$,$(Hd_{j})(A\prime d^{+}jB\prime d_{j}^{-_{C’)}}=0, (Hd_{j})(D’d^{+}jE’)=0$$(Hd_{j})(F’dj-c^{;})=(HF’)d_{j}^{-_{G^{J}}}$$(Hd_{j})(H’d\cdot)j=(HH’)d_{j}$

,

,

,

$C,$ $D,$ $E,$ $F,$ $G’,$ $H’$ are -previsible operators
where $A,$ $B,$ $C,$ $D,$ $E,$ $F,$ $G,$ $H,$ $A’,$from. This rule is summarized in a quantum $Ito$ table:$B^{\prime\prime\prime\prime\prime},$$j$$A_{j-1]}$$A’d. B’d^{-}C’ D’dE’ F’d^{-}G’ H’d_{j}$$Ad_{j}Bd_{j}^{-}C$$Ad_{j}(B\delta(cA’)B’)d^{-}jc’ Ad_{j}(B\delta(CD’)E^{;}) 0 0$

$Dd_{j}^{+}E$

$-$

$-$

$Fd_{j}^{-}G$$–j+(EF’\mathrm{o})_{-_{j}}-’- Dd_{j}^{+}(EH’)0$$(F\delta(GA’)B’)d_{j}^{-}C’ (F\delta(GD’)E’)d_{j}$$0 0 (HF’)d_{\mathrm{j}}^{-c’} (HH’)d_{\mathrm{j}}.$$Hd_{\dot{j}}$This is a discrete time analogue of quantum Ito’s formula.

9. ADDITIONAL REMARKS
Finally we close this paper with some complementary remarks in the quantum
probability space $(A, \phi)$ .
Remark 9.1 : We can naturally define the canonical pair$q_{k}(k\in T)$ by$p_{k}=i(d_{kk}^{+}-d^{-})$and. Here is the imaginary unit. The spectrumof the self-adjoint operator(resp.
) is $\mathrm{S}\mathrm{p}(q_{k})=\{-1,0, +1\}$ (resp.
$\{-1,0, +1\})$ and the probability distribution on the spectrum of operator(resp.
$\mathrm{P}(\{-1\})=\mathrm{P}(\{+1\})=1/2$
under the vacuum expectation isand.
(resp.
So the operator) takesalmost surely in the vacuum state .
Hence the operator(resp. ) can be interpreted as a quantum Bernoulli randomvariable. Besides the operator process $\{q_{1}+\cdots+q_{k}\}_{k\in}\tau$ (resp. $\{p_{1}+\cdots+p_{k}\}_{kT}\in$)can be interpreted as a quantum random walk because it is a process with stationary
independent increments in the K\"ummerer independence. It is easily proved that the$p_{k},$$q_{k}=d_{k}^{+}+d_{k}^{-}$$i$$p_{k}$$q_{k}$$\mathrm{S}\mathrm{p}(p_{k})=$$q_{k}$$p_{k})$$\phi$$\mathrm{v}\mathrm{a}\mathrm{l}\mathrm{u}\mathrm{e}\mathrm{s}\pm 1$$p_{k}$$q_{k}$$q_{k}$$p_{k}$$\mathrm{P}(\{\mathrm{o}\})=0$$\phi$

83

limit distribution of central limit type for this random walk is the Wigner semicircle
law with mean and variance .
$1^{16}$$0$Remark 9.2: In the permutational Fock space , there exists naturally a unitary
operator $F$ defined by. The operator can be interpreted as a kindof Fourier transform because of the relations:$\Phi$$Fe_{\sigma}=i^{\#([\sigma])}e_{\sigma}$$F^{-1}d_{k}^{+}F=-id^{+}k$$\mathcal{F}^{-1}q_{k}F=-p_{k}$,’$\mathcal{F}$$F^{-1}d_{k}^{-}F=id^{-}k$$F^{-1}p_{k}F=q_{k}$$F^{-1}d^{\mathrm{o}}\mathcal{F}k=d^{\circ}k$’’$F^{-}1d_{k}F=d_{k}$,.

$(k\in T)$ . Then, for eachRemark 9.3: Put $X_{k}=q_{k},$ $Y_{k}=p_{k}$ and$k\in T$ , the triple
is a realization of the angular momentum commutation
relations up to the factor 2. That is$Z_{k}=d_{\dot{k}}-d_{[mathring]_{k}},$$(X_{k}, Y_{k}, Z_{k})$$[X_{k}, Y_{k}]=2iZ_{k}$,$[Y_{k}, z_{k}]=2_{\dot{i}}X_{k}$,$[z_{k}, x_{k}]=2iY_{k}$.

is the discrete time quantum i.i.d. process
in the K\"ummerer independence, this representation can be interpreted as a noncommuting, non anticommuting, non freely independent, but “independent” system ofwhere the corspins. Compare this with the case of Journ\’e’s toy Fock space
two
representations of angular momentum commutation relations form
responding
the commuting spins and the anticommuting spins.2

Since the family of subalgebras$\{A_{k}\}_{k\in}\tau$$\Phi_{toy}$

Further discussions and generalizations of the permutational Fock space
be presented elsewhere.17,18

1$\Phi$

will

K. R. Parthasarathy, An Introduction to Quantum Stochastic Calculus (Birkha\"user, Basel,
1992).

2

P. A. Meyer, Quantum Probability for Probabilists, Lecture Notes in Mathematics, Vol. 1538
(Springer-Verlag, Berlin, 1993).
3
M. Sch\"urmann, White noise on Bialgebras, Lecture Notes in Mathematics, Vol. 1544 (SpringerVerlag, Berlin, 1993).
4 M.
Ohya and D. Petz, Quantum Entropy and its Use (Springer-Verlag, Berlin, 1993).
5
R. L. Hudson and K. R. Parthasarathy, “Quantum Ito’s formula and stochastic evolution,”
Commun. Math. Phys. 93, 301-323 (1984).
’
6 D. B. Applebaum and
ComR. L. Hudson, “Fermion Ito’s formula and stochastic
mun. Math. Phys. 96, 473496 (1986).
7 D. Voiculescu, “Symmetries of some reduced free product-algebras,” Operator Algebras andtheir Connections with Topology and Ergodic Threory (H. Araki, ed.), 556-588, Lecture Notes
in Mathematics, Vol. 1132 (Springer-Verlag, Berlin, 1985).
8
R. Speicher, “A new example of independence and white noise,” Prob. Th. Rel. Fields, 84,
141-159 (1990).
”
9 B. K\"ummerer andJ. Funct.
R. Speicher, “Stochastic integration on the Cuntz-algebra
Anal. 103, 372-408 (1992).
10 P. A.
Meyer, “A finite approximation to boson Fock space,” Stochastic Processes in Classicaland Quantum Systems (Eds: S. Albeverio, G. Casati, D. Merlini), 405-410, Lecture Notes inPhysics, Vol. 262 (Springer-Verlag, Berlin, 1986).
11
P. A. Meyer, “Fock space and probability theory,” Stochastic Processes -Mathematics andPhysics II (Eds: S. Albeverio, Ph. Blanchard, L. Streit), 161-170, Lecture Notes in Mathematics, Vol. 1250 (Springer-Verlag, Berlin, 1987).
$\mathrm{e}.\mathrm{v}\mathrm{o}\mathrm{l}\mathrm{u}\mathrm{t}\mathrm{i}_{\mathrm{o}\mathrm{n}’},$$C^{*}$$\mathcal{O}_{\infty},$84

Ph. Combe, R. Rodriguez, M. Sirugue Collin and M. Sirugue, ‘Weyl quantization of classical: S. Albeverio
spin systems, quantum spins and Fermi systems,” Feynman Path Integmls (1978).
et al.), 105-119, Lecture Notes in Physics, Vol. 106 (Springer-Verlag, Berlin,
13 N. Obata,White Noise Calculus and Fock Space, Lecture Notes in Mathematics, Vol. 1577
(Springer-Verlag, Berlin, 1994).
14 B. K\"ummerer, “Markov dilations and non-conunutative Poisson processes,” preprint.
15 H. Umegaki, “Conditional expectation in an operator algebra,” T\^ohoku Math. J. 6, 177-181

12$\mathrm{E}\mathrm{d}s$16(1954).
N. Muraki, “Examples of independence, limit theorem and Brownian motion in noncommu-

tative probability theory,” Proceedings of the Fourteenth Symposium on Applied Fhnctional
Analysis, 66-81 (1995).
17 N. Muraki, “A new example of noncommutative de Moivre-Laplace theorem,” to appear.
18 N. Muraki, “Noncommutative Brownian motion in monotone Fock space,” to appear.

SCIENCE, FACULTY OF ENGINEERING, YAMAGUCHI UNIVERSITY,
DEPARTMENTUBE CITY, YAMAGUCHI 755, JAPAN. yamaguchi-u.
-mail address: muraki$\mathrm{o}\mathrm{F}\mathrm{A}_{\mathrm{P}\mathrm{p}\mathrm{L}}1\mathrm{E}\mathrm{D}$$E$$\mathrm{Q}\mathrm{p}\mathrm{o}.\mathrm{c}\mathrm{c}$$\mathrm{a}\mathrm{c}.\mathrm{j}\mathrm{p}$