Volume 2007, Article ID 80152,22pages doi:10.1155/2007/80152

*Research Article*

**Conditional Expectations for Unbounded Operator Algebras**

Atsushi Inoue, Hidekazu Ogi, and Mayumi Takakura
Received 18 December 2006; Revised 20 March 2007; Accepted 19 May 2007 Recommended by Manfred H. Moller

Two conditional expectations in unbounded operator algebras (O* ^{∗}*-algebras) are dis-
cussed. One is a vector conditional expectation defined by a linear map of an O

*-algebra into the Hilbert space on which the O*

^{∗}*-algebra acts. This has the usual properties of conditional expectations. This was defined by Gudder and Hudson. Another is an un- bounded conditional expectation which is a positive linear mapᏱof an O*

^{∗}*-algebraᏹ onto a given O*

^{∗}*-subalgebraᏺofᏹ. Here the domain*

^{∗}*D(Ᏹ) of*Ᏹdoes not equal toᏹin general, and so such a conditional expectation is called unbounded.

Copyright © 2007 Atsushi Inoue et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

**1. Introduction**

In probability theory, conditional expectations play a fundamental role. A noncommuta- tive analogue of conditional expectations in von Neumann algebras has been studied in [2–4]. A typical feature of probability in von Neumann algebras is that the observables permitted are usually bounded and some finiteness is imposed. But, unbounded observ- ables occur naturally in quantum mechanics and quantum probability theory [1,5–8]

and so it is natural to consider conditional expectations in algebras of unbounded ob-
servables (O* ^{∗}*-algebras). The first study of conditional expectations in O

*-algebras was done by Gudder and Hudson [1]. Letᏹbe an O*

^{∗}*-algebra on a dense subspaceᏰin a Hilbert spaceᏴwith a strongly cyclic and separating vector*

^{∗}*ξ*0andᏺan O

*-subalgebra ofᏹ. These notions are defined inSection 2. Gudder and Hudson have defined a condi- tional expectation given by (ᏺ,*

^{∗}*ξ*0) by the map

*A*

*→*

*P*ᏺ

*Aξ*0ofᏹinto the closed subspace Ᏼᏺ

*≡*ᏺξ0ofᏴ, which has the usual properties of a conditional expectation, where

*P*ᏺ

is the projection ofᏴontoᏴᏺ*. We call this the vector conditional expectation given by*
(ᏺ,ξ0). On the other hand, it is natural to consider when a conditional expectation of

the O* ^{∗}*-algebraᏹonto the O

*-subalgebraᏺexists. Such a conditional expectation does not necessarily exist even for von Neumann algebras. In fact, Takesaki [2] has shown that there exists a conditional expectation of the von Neumann algebraᏹonto the von Neu- mann subalgebraᏺif and only ifΔ*

^{∗}

^{it}*0ᏺΔ*

_{ξ}

^{−}*0*

_{ξ}

^{it}*=*ᏺfor all

*t*

*∈*R, whereΔ

*ξ*0is the modular operator of the left Hilbert algebraᏹξ0. Here we consider a mapᏱ(

*· |*ᏺ) :

*A*

*→*

*P*ᏺ

*A*ᏺξ0

ofᏹinto the partial O* ^{∗}*-algebraᏸ

*(ᏺξ0,Ᏼᏺ). We will show thatᏱ(*

^{†}*· |*ᏺ) has proper-

*ties similar to those of conditional expectations, so it will be called a weak conditional-*

*expectation of*ᏹwith respect to (ᏺ,ξ0). Unfortunately, the rangeᏱ(ᏹᏺ) of the weak conditional-expectationᏱ(

*· |*ᏺ) is not necessarily contained inᏺ, and so we define an

*unbounded conditional-expectation*Ᏹ ofᏹontoᏺwith respect to

*ξ*0as follows: Ᏹis a map ofᏹontoᏺsatisfying

(i) the domain*D(Ᏹ) of*Ᏹis a*†*-invariant subspace ofᏹcontainingᏺsuch that
ᏺD(Ᏹ)*⊂**D(Ᏹ);*

(ii)Ᏹ(A)^{†}*=*Ᏹ(A* ^{†}*), for all

*A*

*∈*

*D(*Ᏹ) andᏱ(X)

*=*

*X, for allX*

*∈*ᏺ; (iii)Ᏹ(AX)

*=*Ᏹ(A)XandᏱ(XA)

*=*

*XᏱ(A), for allA*

*∈*

*D(Ᏹ), for allX*

*∈*ᏺ;

(iv)*ω**ξ*0(Ᏹ(A))*=**ω**ξ*0(A), for all*A**∈**D(Ᏹ),*

where*ω** _{ξ}*0is a positive linear functional onᏹdefined by

*ω*

*0(A)*

_{ξ}*=*(Aξ0

*|*

*ξ*0),

*A*

*∈*ᏹ. By restriction of the weak conditional-expectationᏱ(

*· |*ᏺ), we will show that there exists a maximal unbounded conditional expectationᏱᏺ of ᏹontoᏺwith respect to

*ξ*0. Furthermore, we will investigate unbounded conditional-expectations in case thatᏹ andᏺare generalized von Neumann algebras which are unbounded generalization of von Neumann algebras and that the von Neumann algebra (ᏺw) (the usual commutant of the weak commutantᏺw ofᏺ) satisfies the Takesaki condition. As an application of vector conditional expectations we will establish the existence of coarse graining for absolutely continuous positive linear functionals.

**2. Preliminaries**

In this section we introduce the basic definitions and properties of (partial) O* ^{∗}*-algebras.

We refer to [6–9] for O* ^{∗}*-algebras and to [10] for partial O

*-algebras.*

^{∗}LetᏴbe a Hilbert space with inner product (*· | ·*) andᏰa dense subspace ofᏴ. We
denote byᏸ(Ᏸ,Ᏼ) the set of all linear operators*X*inᏴsuch thatᏰ(X) (the domain of
*X)**=*Ᏸ, and

ᏸ* ^{†}*(Ᏸ,Ᏼ)

*=*

*X**∈*ᏸ(Ᏸ,Ᏼ); Ᏸ^{}*X*^{∗}^{}*⊃*Ᏸ^{},
ᏸ* ^{†}*(Ᏸ)

*=*

*X**∈*ᏸ* ^{†}*(Ᏸ,Ᏼ);

*XᏰ*

*⊂*Ᏸ,

*X*

*Ᏸ*

^{∗}*⊂*Ᏸ

^{}

*.*(2.1) Thenᏸ(Ᏸ,Ᏼ) is a vector space with the usual operations

*X*+

*Y*and

*λX*, andᏸ

*(Ᏸ,Ᏼ) is equipped with the following operations and involution:*

^{†}(i) the sum*X*+*Y*;

(ii) the scalar multiplication*λX;*

(iii) the involution*X**→**X*^{†}*≡**X** ^{∗}*Ᏸ, that is, (X+

*λY)*

^{†}*=*

*X*

*+*

^{†}*λY*

*,*

^{†}*X*

^{††}*=*

*X;*

(iv) the weak partial multiplication*X**Y**=**X*^{†∗}*Y*, defined whenever*X*is a left mul-
tiplier of*Y*, (X*∈**L*^{w}(Y) or*Y**∈**R*^{w}(X)), that is, if and only if*Y*Ᏸ*⊂*Ᏸ(X* ^{†∗}*) and

*X*

*Ᏸ*

^{†}*⊂*Ᏸ(Y

*).*

^{∗}Thenᏸ* ^{†}*(Ᏸ,Ᏼ) is a partial

*∗*

*-algebra, that is, the following hold:*

(i)*X**∈**L*^{w}(Y) if and only if*Y*^{†}*∈**L*^{w}(X* ^{†}*) and then (X

*Y*)

^{†}*=*

*Y*

^{†}*X*

*;*

^{†}(ii) if*X**∈**L*^{w}(Y) and*X**∈**L*^{w}(Z), then*X**∈**L*^{w}(λY+*μZ) for allλ,μ**∈*Cand*X*(λY+
*μZ)**=**λ(X**Y*) +*μ(X**Z).*

ᏸ* ^{†}*(Ᏸ) is a

*∗*-algebra with the usual multiplication

*XY*(which coincides with the weak partial multiplication

*X*

*Y*) and the involution

*X*

*→*

*X*

*. A partial*

^{†}*∗*-subalgebra ofᏸ

*(Ᏸ, Ᏼ*

^{†}*) is called a partial O*

^{∗}*-algebra on*Ᏸ, and a

*∗*-subalgebra ofᏸ

*(Ᏸ*

^{†}*) is called an O*

^{∗}*-*

*algebra on*Ᏸ. Here we assume that a (partial) O

*-algebra contains the identity operator*

^{∗}*I.*

In analogy with the notion of a closed symmetric (selfadjoint) operator, we define the
notion of a closed O* ^{∗}*-algebra (a selfadjoint O

*-algebra). Letᏹbe an O*

^{∗}*-algebra onᏰ. We define a natural graph topology onᏰ. This topology*

^{∗}*t*ᏹis a locally convex topology defined by a family

*{ ·*

*;*

_{X}*X*

*∈*ᏹ

*}*of seminorms

*ξ*

_{X}*≡*

*ξ*+

*Xξ*, (ξ

*∈*Ᏸ), and

*it is called the graph (or induced) topology on*Ᏸ. If the locally convex space Ᏸ[tᏹ] is complete, thenᏹ

*is said to be closed. We denote by*Ᏸ(ᏹ) the completion of the locally

^{}convex spaceᏰ[tᏹ] and put

*X**=**X*Ᏸ(ᏹ), *X**∈*ᏹ;

ᏹ*= {**X;* *X**∈*ᏹ*}**.* (2.2)

Thenᏹ^{}is a closed O* ^{∗}*-algebra onᏰ(ᏹ) in

^{}Ᏼwhich is the smallest closed extension of ᏹ, andᏰ

^{}(ᏹ)

*≡*

*X**∈*ᏹᏰ(X).^{}ᏹ*is called the closure of*ᏹ.

We next define the notion of selfadjointness ofᏹ. IfᏰ*=*Ᏸ* ^{∗}*(ᏹ)

*≡*

*X**∈*ᏹᏰ(X* ^{∗}*), then
ᏹ

*is said to be selfadjoint. If*Ᏸ

^{}(ᏹ)

*=*Ᏸ

*(ᏹ), thenᏹ*

^{∗}*is said to be essentially selfadjoint.*

It is clear that

Ᏸ*⊂*Ᏸ(ᏹ)*⊂*Ᏸ* ^{∗}*(ᏹ),

*X**⊂**X**⊂**X** ^{∗}*,

*∀*

*X*

*∈*ᏹ. (2.3)

We define commutants and bicommutants ofᏹ. The weak commutantᏹw ofᏹis de- fined by

ᏹw*=*

*C**∈*Ꮾ(Ᏼ); (CXξ*|**η)**=*

*Cξ**|**X*^{†}*η*^{},*∀**X**∈*ᏹ,*∀**ξ*,η*∈*Ᏸ^{}, (2.4)
whereᏮ(Ᏼ) is a*∗*-algebra of all bounded linear operators onᏴ. Thenᏹw is a weakly
closed*∗*-invariant subspace ofᏮ(Ᏼ) such that (^{}ᏹ)w*=*ᏹw. IfᏹwᏰ*⊂*Ᏸ, thenᏹw is a
von Neumann algebra; in particular, ifᏹis selfadjoint, thenᏹwᏰ*⊂*Ᏸ. The unbounded
commutants and unbounded bicommutants ofᏹare defined by

ᏹ_{δ}*=*

*S**∈*ᏸ(Ᏸ,Ᏼ); (SXξ*|**η)**=*

*Sξ**|**X*^{†}*η*^{},*∀**X**∈*ᏹ,*∀**ξ,η**∈*Ᏸ^{};
ᏹ_{σ}*=*ᏹ_{δ}*∩*ᏸ* ^{†}*(Ᏸ,Ᏼ);

ᏹc*=*ᏹ_{σ}*∩*ᏸ* ^{†}*(Ᏸ);

ᏹw^{ }*σ**=*

*X**∈*ᏸ* ^{†}*(Ᏸ,Ᏼ); (CXξ

*|*

*η)*

*=*

*Cξ**|**X*^{†}*η*^{},*∀**C**∈*ᏹw,*∀**ξ,η**∈*Ᏸ^{};
ᏹwc^{ }*=*ᏹw^{ }*σ**∩*ᏸ* ^{†}*(Ᏸ).

(2.5)

Then the following hold.

(i)ᏹ* _{δ}*is a subspace ofᏸ(Ᏸ,Ᏼ).

(ii)ᏹ* _{σ}* is a

*†*-invariant subspace ofᏸ

*(Ᏸ,Ᏼ) and (ᏹ*

^{†}*)*

_{σ}

_{b}*≡ {*

*S*

*∈*ᏹ

*;*

_{σ}*S*

*∈*Ꮾ(Ᏼ)

*} =*ᏹwᏰ.

(iii)ᏹcis a subalgebra ofᏸ* ^{†}*(Ᏸ).

(iv)ᏹ* ^{ }*w

*σ*is a

*†*-invariant

*τ*

_{s}*-closed subspace ofᏸ*

^{∗}*(Ᏸ,Ᏼ) containingᏹ, where the strong*

^{†}*topology*

^{∗}*τ*

*s*

*is defined by the family*

^{∗}*{*

*p*

^{∗}*(*

_{ξ}*·*);

*ξ*

*∈*Ᏸ

*}*of seminorms

*p*_{ξ}* ^{∗}*(X)

*≡*

*Xξ*+

^{ }

*X*

^{†}*ξ*

^{ },

*X*

*∈*ᏸ

*(Ᏸ,Ᏼ). (2.6) (iiv)ᏹ*

^{†}*wcis a*

^{ }*τ*

*s*

*-closed O*

^{∗}*-algebra onᏰsuch thatᏹ*

^{∗}*⊂*ᏹ

*wcand (ᏹ*

^{ }*wc)w*

^{ }*=*ᏹw. (iiiv) IfᏹwᏰ

*⊂*Ᏸ, thenᏹ

*w*

^{ }*σ*is a partial O

*-algebra onᏰsuch that*

^{∗}ᏹw^{ }*σ**=*

*X**∈*ᏸ* ^{†}*(Ᏸ,Ᏼ);

*X*is aﬃliated with

^{}ᏹw

*=*
ᏹw

Ᏸ^{τ}^{s}^{∗}^{}the*τ*_{s}* ^{∗}*-closure of

^{}ᏹw

Ᏸinᏸ* ^{†}*(Ᏸ,Ᏼ)

^{}, (2.7)

andᏹ* ^{ }*wcis an O

*-algebra onᏰsuch that ᏹ*

^{∗}*wc*

^{ }*=*

*X**∈*ᏸ* ^{†}*(Ᏸ);

*X*is aﬃliated with

^{}ᏹw

*=*
ᏹw

Ᏸ^{τ}^{s}^{∗}*∩*ᏸ* ^{†}*(Ᏸ). (2.8)

We introduce the notions of generalized von Neumann algebras and extended W* ^{∗}*-
algebras which are unbounded generalizations of von Neumann algebras. IfᏹwᏰ

*⊂*Ᏸ andᏹ

*=*ᏹ

*wc, thenᏹ*

^{ }*is said to be a generalized von Neumann algebra (or a GW*

^{∗}*-algebra)*on Ᏸ. A closed O

*-algebraᏹon Ᏸ*

^{∗}*is said to be an extended W*

^{∗}*-algebra (simply, an*

*EW*

^{∗}*-algebra) if (I*+

*X*

^{†}*X)*

^{−}^{1}exists inᏹ

*b*

*≡ {*

*A*

*∈*ᏹ;

*A*

*∈*Ꮾ(Ᏼ)

*}*for all

*X*

*∈*ᏹandᏹ

*b*

*≡*

*{*

*A;A*

*∈*ᏹ

*b*

*}*is a von Neumann algebra onᏴ.

We define the notion of strongly cyclic vectors for a closed O* ^{∗}*-algebraᏹonᏰinᏴ.
We denote bythe closure of a subsetofᏴwith respect to the Hilbert space norm
and denote byM

^{t}^{ᏹ}the closure of a subsetMofᏰwith respect to the graph topology

*t*ᏹ. LetMbe anᏹ-invariant subspace of Ᏸ. ThenᏹM

*≡ {*

*X*M;

*X*

*∈*ᏹ

*}*is an O

*- algebra onMand its closureᏹM(*

^{∗}*≡*(ᏹM)

^{∼}) is a closed O

*-algebra onM*

^{∗}

^{t}^{ᏹ}inM. If M

*is essentially selfadjoint, that is,*ᏹM is selfadjoint, then the projection

*P*

_{M}ofᏴonto Mbelongs to ᏹw,

*P*

_{M}Ᏸ

*(ᏹ)*

^{∗}*=*M

^{t}^{ᏹ}

*⊂*Ᏸ andᏹM

*=*ᏹ

*P*M

*≡ {*

*X*

_{P}_{M};

*X*

*∈*ᏹ

*}*, where

*X*

_{P}_{M}

*P*

_{M}

*ξ*

*=*

*P*

_{M}

*Xξ*for

*X*

*∈*ᏹand

*ξ*

*∈*Ᏸ. A vector

*ξ*0inᏰ

*is said to be strongly cyclic if*ᏹξ0

*t*ᏹ

*=*Ᏸ, and*ξ*0*is said to be separating if*ᏹw*ξ*0*=*Ᏼ.

We define the notions of (unbounded)*∗*-representations of*∗*-algebras. LetᏭbe an

*∗**-algebra with identity 1. A (**∗*-)homomorphism*π*ofᏭinto an O* ^{∗}*-algebraᏸ

*(Ᏸ) with*

^{†}*π(1)*

*=*

*I*is said to be a (

*∗*

*-)representation of*Ꮽ. In this case,ᏰandᏴare denoted, re- spectively, byᏰ(π) andᏴ

*π*.

Let*π*1and*π*2be (*∗*-)representations ofᏭon the same Hilbert space. If*π*1(x)*⊂**π*2(x)
for each*x**∈*Ꮽ, then*π*2*is said to be an extension ofπ*1and it is denoted by*π*1*⊂**π*2. Let*π*
be a (*∗*-)representation ofᏭ. We put

Ᏸ(π) *=*

*x**∈*Ꮽ

Ᏸ^{}*π(x)*^{}, *π(x)* *=**π(x)*Ᏸ(π), *x**∈**A.* (2.9)
If*π**=**π, then* *π* *is said to be closed;π*is a closed (*∗*-)representation ofᏭwhich is the
smallest closed extension of*πand it is called the closure ofπ. Letπ*be a*∗*-representation
ofᏭ. We put

Ᏸ* ^{∗}*(π)

*=*

*x**∈*Ꮽ

Ᏸ^{}*π(x)*^{∗}^{}, *π** ^{∗}*(x)

*=*

*π*

^{}

*x*

^{∗}^{}

*Ᏸ*

^{∗}*(π),*

^{∗}*x*

*∈*Ꮽ. (2.10) Then

*π*

*is a closed representation ofᏭsuch that*

^{∗}*π*

*⊂*

*π*

*⊂*

*π*

^{∗}*and is called the adjoint of*

*π. Ifπ*

*=*

*π*

*, then*

^{∗}*πis said to be selfadjoint. We remark thatπ*is closed (resp., selfadjoint) if and only if the O

*-algebra*

^{∗}*π(Ꮽ) is closed (resp., selfadjoint).*

**3. Vector conditional expectations**

Letᏹbe a closed O* ^{∗}*-algebra onᏰinᏴ,

*ξ*0

*∈*Ᏸa strongly cyclic and separating vector forᏹ, andᏺan O

*-subalgebra ofᏹ. Then*

^{∗}ᏺξ0*⊂*ᏺξ0*t*ᏹ

*⊂*ᏺξ0*t*ᏺ

*⊂*ᏺξ0

*∩*
ᏹξ0

*t*ᏹ

*=*Ᏸ

*⊂*Ᏼ. (3.1)

Ifᏺis closed, thenᏺξ0*⊂*ᏺξ0
*t*ᏹ

*⊂*ᏺξ0
*t*ᏺ

*⊂*Ᏸ*=*ᏹξ0
*t*ᏹ

. The following is easily shown.

*Lemma 3.1. Put*

Ᏸ^{}*π*ᏺ

*=*ᏺξ0,

*π*ᏺ(X)Yξ0*=**XYξ*0, *∀**X,Y**∈*ᏺ,
Ᏸ^{}*π*_{ᏺ}^{ᏹ}^{}*=*ᏺξ0*t*ᏹ

,

*π*_{ᏺ}^{ᏹ}(X)ξ*=**Xξ,* *∀**X**∈*ᏺ,*∀**ξ**∈*Ᏸ^{}*π*_{ᏺ}^{ᏹ}^{}*.*

(3.2)

*Thenπ*ᏺ*andπ*_{ᏺ}^{ᏹ}*are faithful**∗**-representations of*ᏺ*in*Ᏼᏺ*≡*ᏺξ0*such thatπ*ᏺ*⊂**π*_{ᏺ}^{ᏹ}*⊂**π*ᏺ*,*
*and*

Ᏸ^{}*π*ᏺ

*⊂*Ᏸ^{}*π*_{ᏺ}^{ᏹ}^{}*⊂*Ᏸ^{}*π*ᏺ

, Ᏸ^{∗}^{}*π*ᏺ

*=*Ᏸ^{∗}^{}*π*_{ᏺ}^{ᏹ}^{}*.* (3.3)
We denote by*P*ᏺ the projection ofᏴontoᏴᏺ*≡*ᏺξ0. Then we have the following
lemma.

Lemma 3.2. *P*ᏺᏰ* ^{∗}*(ᏹ)

*⊂*Ᏸ

*(πᏺ*

^{∗}*) andπ*

_{ᏺ}

*(X)Pᏺ*

^{∗}*ξ*

*=*

*P*ᏺ

*X*

^{†∗}*ξ, for allX*

*∈*ᏺ

*and for all*

*ξ*

*∈*Ᏸ

*(ᏹ).*

^{∗}*Proof. Take arbitraryX**∈*ᏺand*ξ**∈*Ᏸ* ^{∗}*(ᏹ). For any

*Y*

*∈*ᏺ, we have

*X*

^{†}*Yξ*0

*|*

*P*ᏺ

*ξ*

^{}

*=*

*X*^{†}*Yξ*0*|**ξ*^{}*=*

*Yξ*0*|**X*^{†∗}*ξ*^{}*=*

*Yξ*0*|**P*ᏺ*X*^{†∗}*ξ*^{}, (3.4)
and so*P*ᏺᏰ* ^{∗}*(ᏹ)

*⊂*Ᏸ

*(πᏺ) and*

^{∗}*π*

_{ᏺ}

*(X)Pᏺ*

^{∗}*ξ*

*=*

*P*ᏺ

*X*

^{†∗}*ξ.*

First we introduce the notion of a vector conditional expectation defined by Gudder and Hudson [1].

*Definition 3.3. A mapE*ofᏹintoᏰ* ^{∗}*(πᏺ

*) is said to be a vector conditional expectation of*ᏹgiven by (ᏺ,

*ξ*0) if the following hold.

(i)*E(XA)**=**π*_{ᏺ}* ^{∗}*(X)E(A), for all

*A*

*∈*ᏹ, for all

*X*

*∈*ᏺ.

(ii)*ω** _{ξ}*0(A)

*=*(E(A)

*|*

*ξ*0), for all

*A*

*∈*ᏹ.

A map*E*satisfying the conditions ofDefinition 3.3was called a conditional expecta-
tion ofᏹgiven by (ᏺ,*ξ*0) by Gudder and Hudson [1]. They gave the following theorem.

We prove the theorem for the sake of completeness.

*Theorem 3.4. A vector conditional expectationEof*ᏹ*given by (ᏺ,ξ*0*) exists uniquely, and*

*E(A)**=**P*ᏺ*Aξ*0, *∀**A**∈*ᏹ. (3.5)

*Denote byE(A**|*ᏺ) the unique vector conditional expectation ofᏹ*given by (ᏺ,ξ*0*), that is,*

*E(A**|*ᏺ)*=**P*ᏺ*Aξ*0, *∀**A**∈*ᏹ. (3.6)

*Proof. We put*

*E(A)**=**P*ᏺ*Aξ*0, *A**∈*ᏹ. (3.7)

ByLemma 3.2*E*is a map ofᏹintoᏰ* ^{∗}*(πᏺ). It is clear that

*E*is linear. For any

*A*

*∈*ᏹand

*X*

*∈*ᏺwe have, byLemma 3.2,

*E(XA)**=**P*ᏺ*XAξ*0*=**π*_{ᏺ}* ^{∗}*(X)Pᏺ

*Aξ*0

*=*

*π*

_{ᏺ}

*(X)E(A),*

^{∗}*ω*

*0(XA)*

_{ξ}*=*

*Aξ*0*|**X*^{†}*ξ*0

*=*

*P*ᏺ*Aξ*0*|**X*^{†}*ξ*0

*=*

*π*_{ᏺ}* ^{∗}*(X)E(A)

*|*

*ξ*0

; (3.8)

in particular,

*ω** _{ξ}*0(A)

*=*

*E(A)**|**ξ*0

*.* (3.9)

Hence*E*is a vector conditional expectation ofᏹgiven by (ᏺ,ξ0).

We show the uniqueness of vector conditional expectations. Let*E* be any vector con-
ditional expectation ofᏹgiven by (ᏺ,ξ0). For any*A**∈*ᏹand*X**∈*ᏺwe have

*E*(A)*|**Xξ*0

*=*

*π*_{ᏺ}^{∗}^{}*X*^{†}^{}*E*(A)*|**ξ*0

*=*

*E*^{}*X*^{†}*A*^{}*|**ξ*0

*=**ω**ξ*0

*X*^{†}*A*^{}

*=*

*Aξ*0*|**Xξ*0

*=*

*P*ᏺ*Aξ*0*|**Xξ*0

, (3.10)

which implies that

*E*(A)*=**P*ᏺ*Aξ*0*.* (3.11)

**4. Unbounded conditional expectations for O**^{∗}**-algebras**

We begin with the definition of unbounded conditional expectations of O* ^{∗}*-algebras. In
this section letᏹbe a closed O

*-algebra onᏰinᏴwith a strongly cyclic and separating vector*

^{∗}*ξ*0andᏺan O

*-subalgebra ofᏹ.*

^{∗}*Definition 4.1. A map*Ᏹofᏹontoᏺ*is said to be an unbounded conditional expectation*
ofᏹontoᏺwith respect to*ξ*0if

(i) the domain *D(Ᏹ) of*Ᏹis a*†*-invariant subspace ofᏹ containingᏺsuch that
ᏺD(Ᏹ)*⊂**D(Ᏹ);*

(ii)Ᏹis a projection; that is, it is hermitian (Ᏹ(A)^{†}*=*Ᏹ(A* ^{†}*), for all

*A*

*∈*

*D(*Ᏹ)) and Ᏹ(X)

*=*

*X, for allX*

*∈*ᏺ;

(iii)Ᏹisᏺ-linear, that is,

Ᏹ(AX)*=*Ᏹ(A)X, Ᏹ(XA)*=**XᏱ*(A), *∀**A**∈**D(*Ᏹ), *∀**X**∈*ᏺ; (4.1)
(iv)*ω** _{ξ}*0(Ᏹ(A))

*=*

*ω*

*0(A), for all*

_{ξ}*A*

*∈*

*D(*Ᏹ).

In particular, if*D(Ᏹ)**=*ᏹ, thenᏱ*is said to be a conditional expectation of*ᏹontoᏺ.

For unbounded conditional expectations we have the following lemma.

*Lemma 4.2. Let*Ᏹ*be an unbounded conditional expectation of*ᏹ*onto*ᏺ*with respect toξ*0*.*
*Then the following statements hold.*

(1)Ᏹ(A)ξ0*=**E(A**|*ᏺ*), for allA**∈**D(*Ᏹ*).*

(2)Ᏹ*is an*ᏺ-Schwarz map, that is,

Ᏹ^{}*A*^{†}^{}Ᏹ(A)*≤*Ᏹ^{}*A*^{†}*A*^{} *on*Ᏸ^{}*π*_{ᏺ}^{ᏹ}^{}*wheneverA**∈**D(Ᏹ) s.t.A*^{†}*A**∈**D(Ᏹ).* (4.2)
*Proof. (1) For allA**∈**D(Ᏹ) andX**∈*ᏺwe have

Ᏹ(A)ξ0*|**Xξ*0

*=*

*X** ^{†}*Ᏹ(A)ξ0

*|*

*ξ*0

*=*

Ᏹ^{}*X*^{†}*A*^{}*ξ*0*|**ξ*0

*=**ω**ξ*0

*X*^{†}*A*^{}*=*

*Aξ*0*|**Xξ*0

*=*

*P*ᏺ*Aξ*0*|**Xξ*0

, (4.3)

which implies

Ᏹ(A)ξ0*=**P*ᏺ*Aξ*0*=**E(A**|*ᏺ). (4.4)

(2) Take an arbitrary*A**∈**D(*Ᏹ) s.t.*A*^{†}*A**∈**D(*Ᏹ). Then we have
Ᏹ^{}*A*^{†}^{}Ᏹ(A)Xξ0*|**Xξ*0

*=* Ᏹ(A)Xξ0 ^{2}*=* Ᏹ(AX)ξ0 ^{2}

*=* *P*ᏺ*AXξ*0 ^{2}*≤* *AXξ*0 ^{2}
by (1)^{},
Ᏹ^{}*A*^{†}*A*^{}*Xξ*0*|**Xξ*0

*=*

Ᏹ^{}*X*^{†}*A*^{†}*AX*^{}*ξ*0*|**ξ*0

*=**ω**ξ*0

Ᏹ^{}*X*^{†}*A*^{†}*AX*^{}

*=**ω** _{ξ}*0

*X*^{†}*A*^{†}*AX*^{}*=* *AXξ*0 ^{2},

(4.5)

for each*X**∈*ᏺ, which byᏰ(π_{ᏺ}^{ᏹ})*=*ᏺξ0
*t*ᏹ

implies that

Ᏹ^{}*A*^{†}*A*^{}*≤*Ᏹ(A)* ^{†}*Ᏹ(A) onᏰ

^{}

*π*

_{ᏺ}

^{ᏹ}

^{}

*.*(4.6)

LetEbe the set of all unbounded conditional expectations ofᏹontoᏺwith respect
to*ξ*0. ThenEis an ordered set with the following order*⊂*.

Ᏹ1*⊂*Ᏹ2 iﬀ*D*^{}Ᏹ1

*⊂*Ᏸ^{}Ᏹ2

, Ᏹ1(A)*=*Ᏹ2(A), *∀**A**∈**D*^{}Ᏹ1

*.* (4.7)
InTheorem 4.6we will show that there exists a maximal unbounded conditional expec-
tation ofᏹontoᏺwith respect to*ξ*0.

*Definition 4.3. A map*Ᏹofᏹinto the partial O* ^{∗}*-algebraᏸ

*(Ᏸ(π*

^{†}^{ᏹ}

_{ᏺ}),Ᏼᏺ) is said to be a

*weak conditional expectation of*ᏹwith respect to (ᏺ,

*ξ*0) if

(i)Ᏹis hermitian, that is,Ᏹ(A)^{†}*=*Ᏹ(A* ^{†}*), for all

*A*

*∈*ᏹ; (ii)Ᏹ(ᏹ)Ᏸ(π

_{ᏺ}

^{ᏹ})

*⊂*Ᏸ

*(π*

^{∗}_{ᏺ}

^{ᏹ}) and

*π*_{ᏺ}^{ᏹ}(X)Ᏹ(A)*=*Ᏹ(XA), *∀**X**∈*ᏺ,*∀**A**∈*ᏹ; (4.8)
(iii)*ω**ξ*0(A)*=*(Ᏹ(A)ξ0*|**ξ*0), for all*A**∈*ᏹ.

For weak conditional expectations we have the following.

*Theorem 4.4. There exists a unique weak conditional-expectation*Ᏹ(*· |*ᏺ*) of*ᏹ*with re-*
*spect to (ᏺ,ξ*0*), and*

Ᏹ(A*|*ᏺ)*=**P*ᏺ*A*Ᏸ^{}*π*_{ᏺ}^{ᏹ}^{}, *∀**A**∈*ᏹ. (4.9)
*Proof. We first show the existence: we put*Ᏹ(A*|*ᏺ)*=**P*ᏺ*A*Ᏸ(π_{ᏺ}^{ᏹ}), *A**∈*ᏹ. It follows
fromLemma 3.2that for any*A**∈*ᏹ,Ᏹ(A*|*ᏺ) is a linear map ofᏰ(π_{ᏺ}^{ᏹ}) intoᏰ* ^{∗}*(π

_{ᏺ}

^{ᏹ}), and furthermore

Ᏹ(A*|*ᏺ)ξ*|**η*^{}*=*

*P*ᏺ*Aξ**|**η*^{}*=*(Aξ*|**η)**=*
*ξ**|**A*^{†}*η*^{}

*=*

*ξ**|**P*ᏺ*A*^{†}*η*^{}*=*

*ξ**|*Ᏹ^{}*A*^{†}*|*ᏺ^{}*η*^{} (4.10)
for each*ξ*,η*∈*Ᏸ(π_{ᏺ}^{ᏹ}), which implies thatᏱ(A*|*ᏺ)*∈*ᏸ* ^{†}*(Ᏸ(π

_{ᏺ}

^{ᏹ}),Ᏼᏺ) andᏱ(A

*|*ᏺ)

^{†}*=*Ᏹ(A

^{†}*|*ᏺ). ThusᏱ(

*· |*ᏺ) satisfies the condition (i) inDefinition 4.3. Furthermore, we show that it satisfies the conditions (ii) and (iii) inDefinition 4.3.

(ii) Take arbitrary*X**∈*ᏺand*A**∈*ᏹ. SinceᏱ(A*|*ᏺ)*∈*ᏸ* ^{†}*(Ᏸ(π

_{ᏺ}

^{ᏹ}),Ᏼᏺ) andᏱ(A

*|*ᏺ)Ᏸ(π

^{ᏹ}

_{ᏺ})

*⊂*Ᏸ

*(π*

^{∗}_{ᏺ}

^{ᏹ}) as shown above, it follows that

*π*

_{ᏺ}

^{ᏹ}(X)Ᏹ(A

*|*ᏺ) is well defined and

*π*_{ᏺ}^{ᏹ}(X)* ^{†}*Ᏹ(A

*|*ᏺ)

^{†}^{}

*ξ*

*=*

*π*

^{ᏹ}

_{ᏺ}(X)

^{∗}*P*ᏺ

*A*

^{†}*ξ*

*=*

*P*ᏺ

*X*

^{†}*A*

^{†}*ξ*

*=*Ᏹ^{}*X*^{†}*A*^{†}*|*ᏺ^{}*ξ* (byLemma 3.2) (4.11)
for each*A**∈*ᏹ,*X**∈*ᏺ, and*ξ**∈*Ᏸ(π_{ᏺ}^{ᏹ}).

(iii) This follows from the equality
*ω**ξ*0(A)*=*

*Aξ*0*|**ξ*0

*=*

*P*ᏺ*Aξ*0*|**ξ*0

*=*

Ᏹ(A*|*ᏺ)ξ0*|**ξ*0

(4.12)

for each*A**∈*ᏹ.

We next show the uniqueness: letᏱbe any weak conditional expectation ofᏹwith
respect to (ᏺ,*ξ*0). By (i) and (ii) inDefinition 4.3we have

Ᏹ(A)*π*_{ᏺ}^{ᏹ}(X)*=*Ᏹ(AX), *∀**A**∈*ᏹ,*∀**X**∈*ᏺ, (4.13)
which implies

Ᏹ(A)Xξ0*|**Yξ*0

*=*

*π*_{ᏺ}^{ᏹ}(Y)* ^{∗}*Ᏹ(AX)ξ0

*|*

*ξ*0

*=*

Ᏹ^{}*Y*^{†}*AX*^{}*ξ*0*|**ξ*0

*=**ω** _{ξ}*0

*Y*^{†}*AX*^{}

*=*

*AXξ*0*|**Yξ*0

*=*

*P*ᏺ*AXξ*0*|**Yξ*0

(4.14)

for each*A**∈*ᏹand*X,Y**∈*ᏺ. Hence, we have

Ᏹ(A)Xξ0*=**P*ᏺ*AXξ*0, *∀**A**∈*ᏹ,*∀**X**∈*ᏺ, (4.15)
which implies

Ᏹ(A)*=**P*ᏺ*A*Ᏸ^{}*π*_{ᏺ}^{ᏹ}^{}, *∀**A**∈*ᏹ. (4.16)
The weak conditional expectation ofᏹwith respect to (ᏺ,ξ0) has the following prop-
erties.

Proposition 4.5. Ᏹ(*· |*ᏺ) is a map ofᏹ*into the partial O*^{∗}*-algebra*ᏸ* ^{†}*(Ᏸ(π

_{ᏺ}

^{ᏹ}),Ᏼᏺ)

*satisfying*

(i)Ᏹ(A*|*ᏺ)Ᏸ(π_{ᏺ}^{ᏹ})*⊂*Ᏸ* ^{∗}*(π

_{ᏺ}

^{ᏹ}

*), for allA*

*∈*ᏹ;

(ii)Ᏹ(*· |*ᏺ) is linear;

(iii)Ᏹ(A*|*ᏺ)^{†}*=*Ᏹ(A^{†}*|*ᏺ*), for allA**∈*ᏹ Ᏹ(X*|*ᏺ)*=**X*Ᏸ(π_{ᏺ}^{ᏹ}*), for allX**∈*ᏺ*;*
(iv)Ᏹ(A^{†}*A**|*ᏺ)*≥**0, for allA**∈*ᏹ;

(v)Ᏹ(A*|*ᏺ)* ^{†}*Ᏹ(A

*|*ᏺ)

*≤*Ᏹ(A

^{†}*A*

*|*ᏺ) wheneverᏱ(A

*|*ᏺ)

^{†}*∈*

*L*

^{w}(Ᏹ(A

*|*ᏺ));

(vi)Ᏹ(A*|*ᏺ)*π*_{ᏺ}^{ᏹ}(X) and*π*^{ᏹ}_{ᏺ}(X)Ᏹ(A*|*ᏺ*) are well defined for eachA**∈*ᏹ*andX**∈*
ᏺ, and

Ᏹ(A*|*ᏺ)*π*_{ᏺ}^{ᏹ}(X)*=*Ᏹ(AX*|*ᏺ), *π*_{ᏺ}^{ᏹ}(X)Ᏹ(A*|*ᏺ)*=*Ᏹ(XA*|*ᏺ); (4.17)
(vii)*ω** _{ξ}*0(AX)

*=*(Ᏹ(AX

*|*ᏺ)ξ0

*|*

*ξ*0

*) for eachA*

*∈*ᏹ

*andX*

*∈*ᏺ.

*Proof. The statements (i), (ii), (iii), and (vi) follow from*Theorem 4.4.

(iv) This follows from the equality
Ᏹ^{}*A*^{†}*A**|*ᏺ^{}*ξ**|**ξ*^{}*=*

*P*ᏺ*A*^{†}*Aξ**|**ξ*^{}*=*

*A*^{†}*Aξ**|**ξ*^{}*= **Aξ*^{2} (4.18)
for each*A**∈*ᏹand*ξ**∈*Ᏸ(π_{ᏺ}^{ᏹ}).

(v) Take an arbitrary*A**∈*ᏹs.t.Ᏹ(A*|*ᏺ)^{†}*∈**L*^{w}(Ᏹ(A*|*ᏺ)). Then we have
Ᏹ(A*|*ᏺ)* ^{†}*Ᏹ(A

*|*ᏺ)

^{}

*ξ*

*|*

*ξ*

^{}

*=*

Ᏹ(A*|*ᏺ)* ^{∗}*Ᏹ(A

*|*ᏺ)ξ

*|*

*ξ*

^{}

*=*Ᏹ(A

*|*ᏺ)ξ

^{ }

^{2}

*=* *P*ᏺ*Aξ*^{ }^{2}*≤ **Aξ*^{2}*=*

Ᏹ^{}*A*^{†}*A**|*ᏺ^{}*ξ**|**ξ*^{} (by (4.18))

(4.19)

for each*ξ**∈*Ᏸ(π_{ᏺ}^{ᏹ}), which implies that

Ᏹ(A*|*ᏺ)* ^{†}*Ᏹ(A

*|*ᏺ)

*≤*Ᏹ

^{}

*A*

^{†}*A*

*|*ᏺ

^{}

*.*(4.20) (vii) This follows from

*ω** _{ξ}*0(AX)

*=*

*AXξ*0*|**ξ*0

*=*

*P*ᏺ*AXξ*0*|**ξ*0

*=*

Ᏹ(AX*|*ᏺ)ξ0*|**ξ*0

(4.21)

for each*A**∈*ᏹand*X**∈*ᏺ.

Here we put

*D*^{}Ᏹᏺ

*=*

*A**∈*ᏹ;Ᏹ(A*|*ᏺ)*∈**π*_{ᏺ}^{ᏹ}(ᏺ)^{}*.* (4.22)
Since*π*_{ᏺ}^{ᏹ}is faithful, for any*A**∈**D(Ᏹ*ᏺ) there exists a unique element*X** _{A}*ofᏺsuch that
Ᏹ(A

*|*ᏺ)

*=*

*π*

_{ᏺ}

^{ᏹ}(X

*). Hence, the mapᏱᏺfrom*

_{A}*D(Ᏹ*ᏺ) toᏺis defined by

Ᏹᏺ(A)*=**X**A*, *A**∈**D*^{}Ᏹᏺ

*.* (4.23)

Then we have the following.

Theorem 4.6. Ᏹᏺ*is a maximal among unbounded conditional expectations of*ᏹ*onto*ᏺ
*with respect toξ*0*.*

*Proof. We show that* *D(Ᏹ*ᏺ) is a *†*-invariant subspace of ᏹ containing ᏺ such that
ᏺD(Ᏹᏺ)*⊂**D(Ᏹ*ᏺ). In fact, it is clear that *D(Ᏹ*ᏺ) is a subspace ofᏹcontainingᏺ. By
Proposition 4.5(iii), *D(Ᏹ*ᏺ) is*†*-invariant, and it follows from Proposition 4.5(vi) that
Ᏹ(XA*|*ᏺ)*=**π*_{ᏺ}^{ᏹ}(X)Ᏹ(A*|*ᏺ)*∈**π*_{ᏺ}^{ᏹ}(ᏺ) for each*X**∈*ᏺand*A**∈**D(*Ᏹᏺ), which implies
thatᏺD(Ᏹᏺ)*⊂**D(Ᏹ*ᏺ). It is easily shown thatᏱᏺis a projection. Since

*π*^{ᏹ}_{ᏺ}^{}Ᏹᏺ(AX)^{}*=*Ᏹ(AX*|*ᏺ)*=*Ᏹ(A*|*ᏺ)*π*_{ᏺ}^{ᏹ}(X)*=**π*_{ᏺ}^{ᏹ}^{}Ᏹᏺ(A)^{}*π*_{ᏺ}^{ᏹ}(X)

*=**π*_{ᏺ}^{ᏹ}^{}Ᏹᏺ(A)X^{}, (byProposition 4.5(vi)) (4.24)
for each*A**∈**D(Ᏹ*ᏺ) and*X**∈*ᏺ, it follows thatᏱᏺ(AX)*=*Ᏹᏺ(A)X. Similarly,Ᏹᏺ(XA)*=*
*XᏱ*ᏺ(A). Hence,Ᏹᏺ isᏺ-linear. Furthermore, it follows fromProposition 4.5(vii) that
*ω** _{ξ}*0(Ᏹᏺ(A))

*=*

*ω*

*0(A) for each*

_{ξ}*A*

*∈*

*D(*Ᏹᏺ). ThusᏱᏺis an unbounded conditional expec- tation ofᏹontoᏺwith respect to

*ξ*0. Finally we show thatᏱᏺis maximal. LetᏱbe any unbounded conditional expectation ofᏹontoᏺwith respect to

*ξ*0. Take an arbitrary

*A*

*∈*

*D(*Ᏹ). Then it follows fromLemma 4.2(1) that

*P*ᏺ*AXξ*0*=**E(AX**|*ᏺ)*=*Ᏹ(A)Xξ0 (4.25)
for each*X**∈*ᏺ, which implies that

*P*ᏺ*Aξ**=*Ᏹ(A)ξ, *∀**ξ**∈*Ᏸ^{}*π*_{ᏺ}^{ᏹ}^{}*.* (4.26)
Hence, we have

*P*ᏺ*A*^{}Ᏸ^{}*π*_{ᏺ}^{ᏹ}^{}*=*Ᏹ(A)^{}Ᏸ^{}*π*_{ᏺ}^{ᏹ}^{}*∈**π*_{ᏺ}^{ᏹ}(ᏺ), (4.27)
which implies*A**∈**D(*Ᏹᏺ) andᏱᏺ(A)*=*Ᏹ(A). Thus,Ᏹ*⊂*Ᏹᏺ.

**5. Unbounded conditional expectations for special O**^{∗}**-algebras**

In this section we consider conditional expectations for special O* ^{∗}*-algebras (EW

*- algebras, generalized von Neumann algebras). For conditional expectations for von Neu- mann algebras Takesaki [2] has obtained the following.*

^{∗}*Lemma 5.1. Let*ᏹ*be a von Neumann algebra on a Hilbert space*Ᏼ*with a separating and*
*cyclic vectorξ*0*and*ᏺ*a von Neumann subalgebra of*ᏹ*. Then*Ᏹᏺ*is a conditional expectation*
*of*ᏹ*onto*ᏺ*with respect toξ*0*if and only if*Δ^{it}_{ξ}_{0}ᏺΔ^{−}_{ξ}_{0}^{it}*=*ᏺ*for allt**∈*R*, where*Δ*ξ*0*is the*
*modular operator of the left Hilbert algebra*ᏹξ0*.*

The following is our extension ofLemma 5.1to generalized von Neumann algebras.

*Lemma 5.2. Let*ᏹ*be a closed O*^{∗}*-algebra on*Ᏸ*in*Ᏼ*,ξ*0*∈*Ᏸ*a strongly cyclic and separating*
*vector for*ᏹ*and*ᏺ*a closed O*^{∗}*-subalgebra of*ᏹ. Suppose

(i)ᏺwᏰ*⊂*Ᏸ;

(ii)ᏺξ0*is essentially selfadjoint for*ᏺ*.*
*Put*

Ᏹ(A*|*ᏺ)*=**P*ᏺ*A**P*ᏺᏰ, *A**∈*ᏹ* ^{ }*wc

*.*(5.1)

*Then*Ᏹ(

*· |*ᏺ

*) is a linear map of the generalized von Neumann algebra*ᏹwc

^{ }*into the O*

^{∗}*-*

*algebra*ᏸ

*(PᏺᏰ) such that*

^{†}(a)Ᏹ(A*|*ᏺ)^{†}*=*Ᏹ(A^{†}*|*ᏺ), for all*A**∈*ᏹ* ^{ }*wc

*;*Ᏹ(X

*|*ᏺ)

*=*

*X*

*P*ᏺᏰ, for all

*X*

*∈*ᏺ

*wc*

^{ }*;*(b)Ᏹ(A

^{†}*A*

*|*ᏺ)

*≥*

*0, for allA*

*∈*ᏹ

*wc*

^{ }*;*

(c)Ᏹ(A*|*ᏺ)* ^{†}*Ᏹ(A

*|*ᏺ)

*≤*Ᏹ(A

^{†}*A*

*|*ᏺ), for all

*A*

*∈*ᏹ

*wc*

^{ }*;*

(d)Ᏹ(A*|*ᏺ)X*=*Ᏹ(AX*|*ᏺ*),XᏱ*(A*|*ᏺ)*=*Ᏹ(XA*|*ᏺ*), for allA**∈*ᏹ* ^{ }*wc

*, for allX*

*∈*ᏺ

*wc*

^{ }*;*

(e)*ω**ξ*0(AX)*=*(Ᏹ(AX*|*ᏺ)ξ0*|**ξ*0*), for allA**∈*ᏹ* ^{ }*wc

*, for allX*

*∈*ᏺ

*wc*

^{ }*.*

*Furthermore, suppose*

(iii)Δ^{it}_{ξ}_{0}(ᏺw)Δ^{−}_{ξ}_{0}^{it}*=*(ᏺw)*, for allt**∈*R*,*

*where*Δ*ξ*0 *is the modular operator of the left Hilbert algebra (ᏹ*w)*ξ*0*. Then,*Ᏹ(A*|*ᏺ)*∈*
(ᏺ*P*ᏺ)^{ }_{wc}*, for allA**∈*ᏹwc^{ }*.*

*Proof. By (i) we have*ᏹwᏰ*⊂*Ᏸ, and hence it follows from [6, Propositions 1.7.3, 1.7.5]

thatᏹ* ^{ }*wcis a generalized von Neumann algebra onᏰandᏺ

*wcis a generalized von Neu- mann subalgebra ofᏹ*

^{ }*wc. Since theᏺ-invariant subspaceᏺξ0ofᏰis essentially selfad- joint, it follows from [7, Theorem 4.7] that*

^{ }*P*ᏺ*∈*ᏺw, *P*ᏺᏰ*=*ᏺξ0
*t*ᏺ

*⊂*Ᏸ, (5.2)

ᏺξ0*=*
ᏺw

*ξ*0*.* (5.3)

By (5.2), Ᏹ(*· |*ᏺ) is a linear map of ᏹwc* ^{ }* intoᏸ

*(PᏺᏰ), and it is shown in a similar way to the proof ofProposition 4.5 thatᏱ(*

^{†}*· |*ᏺ) satisfies (a)–(e). Suppose (iii) holds.

We showᏱ(A*|*ᏺ)*∈*(ᏺ*P*ᏺ)^{ }_{wc}, for all*A**∈*ᏹwc* ^{ }*. By (5.3) we have

*P*ᏺ

*=*

*P*(ᏺw), and so by (iii) and by the Takesaki theorem [2] there exists a unique conditional expectationᏱ

*of*

^{ }the von Neumann algebra (ᏹw) onto the von Neumann algebra (ᏺw) with respect to
*ξ*0 such thatᏱ* ^{ }*(A)Pᏺ

*=*

*P*ᏺ

*AP*ᏺ for each

*A*

*∈*(ᏹw). Take an arbitrary

*A*

*∈*ᏹ

*wc. Then there exists a net*

^{ }*{*

*A*

_{α}*}*in (ᏹw) which converges strongly

*to*

^{∗}*A. From (5.2) it follows*immediately that

ᏺw

*P*ᏺ*=*
ᏺ*P*ᏺ

w, (5.4)

and by the basic theory of von Neumann algebras [11]

ᏺ*P*ᏺ

w

*=*
ᏺw

*P*ᏺ

*=*
ᏺw

*P*ᏺ*.* (5.5)

Hence we have

Ᏹ^{ }^{}*A*_{α}^{}_{P}_{ᏺ}*∈*
ᏺ*P*ᏺ

w

,
Ᏹ^{ }^{}*A*_{α}^{}_{P}_{ᏺ}*−−−−−→*

*τ**s*^{∗}*P*ᏺ*A**P*ᏺᏰ, (5.6)

which implies that*P*ᏺ*A**P*ᏺᏰ*∈*((ᏺ*P*ᏺ)w)^{τ}^{s}^{∗}*=*(ᏺ*P*ᏺ)* ^{ }*wc. Hence we have
Ᏹ(A

*|*ᏺ)

*=*

*P*ᏺ

*A*

*P*ᏺᏰ

*∈*

ᏺ*P*ᏺ

* *

wc, *A**∈*ᏹ* ^{ }*wc

*.*(5.7) In a similar way to the proof ofTheorem 4.4one can show thatᏱ(

*· |*ᏺ) is the unique weak conditional expectation of the generalized von Neumann algebraᏹwc

*with respect*

^{ }to ((ᏺ*P*ᏺ)^{ }_{wc},ξ0).

Now we put

*D*^{}Ᏹᏺ

*=*

*A**∈*ᏹ* ^{ }*wc;Ᏹ(A

*|*ᏺ)

*∈*ᏺ

*wc*

^{ }

*P*ᏺ

*.* (5.8)

Then, for any *A**∈**D(Ᏹ*ᏺ) there exists a unique element Ᏹᏺ(A) of ᏺ* ^{ }*wc such that
Ᏹᏺ(A)

*P*ᏺᏰ

*=*Ᏹ(A

*|*ᏺ), and in a similar way to the proof ofTheorem 4.6we can show the following.

Lemma 5.3. Ᏹᏺ*is an unbounded conditional expectation of the generalized von Neumann*
*algebra*ᏹ* ^{ }*wc

*onto the generalized von Neumann algebra*ᏺ

*wc*

^{ }*with respect toξ*0

*which is an*

*extension of*Ᏹᏺ

*.*

By Lemmas5.2and5.3we have the following.

*Theorem 5.4. Let*ᏹ*be a generalized von Neumann algebra on*Ᏸ*in*Ᏼ,*ξ*0*a strongly cyclic*
*and separating vector for*ᏹ*and*ᏺ*a generalized von Neumann subalgebra of*ᏹ. Suppose

(i)ᏺξ0*is essentially selfadjoint for*ᏺ*;*
(ii)Δ^{it}_{ξ}_{0}(ᏺw)Δ^{−}_{ξ}_{0}^{it}*=*(ᏺw)*, for allt**∈*R*,*

*where*Δ*ξ*0 *is the modular operator of the left Hilbert algebra (ᏹ*w)*ξ*0*. Then the following*
*statements hold.*

(1)Ᏹ(A*|*ᏺ)*=*Ᏹ(A*|*ᏺ)^{∼}*∈*(ᏺ*P*ᏺ)^{ }_{wc}*for eachA**∈*ᏹ*.*
(2)Ᏹᏺ*=*Ᏹᏺ*.*